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Astronomy
Dr. Jay Maron

Scales of the universe
Gravity and orbits
Orbital stability
Planets and moons
Tides

Core heating
Magnetic fields
Atmospheres
Water
Terrain

Exoplanets
Light
Stars
Galaxies
Hubble's law

Telescopes
Stellar nucleosynthesis



All sizes to scale, and the horizontal spacing between planets is to scale.
Moon sizes and distances

All sizes to scale, and the size of each moon is in scale with the distance to its host planet.

Pluto and Charon are faintly visible at this scale.

In the following figure all objects are magnified by a factor of 40 and the positions are unchanged. The planets are too large to display at this scale and so they are replaced by green dots. Pluto's moons Nix and Hydra are faintly visible at this scale.



Planet sizes are in scale with moon sizes.


Scales of the universe

Moon radius              /  Earth radius            =       .273
Jupiter radius           /  Earth radius            =     10.9
Sun radius               /  Earth radius            =    109
Moon orbit radius        /  Earth radius            =     60.3     =  384399 km / 6371 km
Earth orbit radius       /  Moon orbit radius       =    389
Neptune orbit radius     /  Earth orbit radius      =     30.1     Sets the size of the solar system
Light year               /  Earth orbit radius      =  63241
Alpha Centauri distance  /  Earth orbit radius      = 276100       Nearest star. Distance from sun = 4.366 light years
Galactic center distance /  Light year              =  27200       Sets the size of the Milky Way galaxy
Andromeda galaxy distance/  Galactic center distance=     94       Size of local group of galaxies.  Andromeda is 2.6e6 light years away
Virgo cluster distance   /  Andromeda distance      =     21       Nearest supercluster, 53.5 million light years away
Edge of universe         /  Virgo clulster distance =    256       Define the edge of the universe to be 13.7 billion light years

Mass of Earth      = 5.972e24 kg
Radius of Earth    = 6371 km
Earth-Sun distance = 1.496e11 meters

Planet data

         Semimajor    Mass      Radius   Escape  Orbit   Parent
         axis (AU)  (Earth=1)  (Earth=1) speed   speed   planet
                                         (km/s)  (km/s)
Sun                  333000     109.2   618.
Mercury      .387      .0553      .383    4.3    47.9
Venus        .723      .8150      .950   10.46   35.0
Earth       1.000     1.0000     1.000   11.2    29.8
Mars        1.524      .1074      .532    5.03   24.1
Jupiter     5.203   317.83      10.86    59.5    13.1
Saturn      9.537    95.16       9.00    35.5     9.64
Uranus     19.19     14.50       3.97    21.3     6.81
Neptune    30.07     17.20       3.86    23.5     5.43
Pluto      39.48       .00220     .184    1.23    4.74

Moon                   .0123      .273    2.38    1.02   Earth
Phobos                 2.5e-10    .0018    .0113  2.14   Mars
Deimos                 1.8e-10    .0010    .0056  1.35   Mars
Io                     .01495     .286    2.56   17.3    Jupiter
Europa                 .00804     .245    2.02   13.7    Jupiter
Ganymede               .0248      .413    2.74   10.9    Jupiter
Callisto               .0180      .379    2.44    8.2    Jupiter
Titan                  .0225      .404    2.64    5.6    Saturn
Triton                 .00358     .213    1.455   4.4    Neptune
Charon                 .000271    .093     .23     .21   Pluto

Asteroids              .0005                             Mass of all asteroids
Vesta       2.36       .0000447   .0413    .36   19.3    Asteroid
Ceres       2.766      .00016     .074     .51   17.9    Asteroid
Pallas      2.77       .0000359   .0427    .32   17.6    Asteroid

Kuiper belt            .03                               Mass of all Kuiper belt objects
Haumea     43.34       .00070     .0487    .84    4.48   Kuiper belt object
Makemake   45.79       .0007      .11      .74    4.42   Kuiper belt object
Eris       67.67       .00278     .183    1.34    3.44   Kuiper belt object
"Orbit speed" refers to speed around the sun for planets and to speed around the planet for moons.
Gravity at the surface of the Earth


The gravitational force on a test object on the surface of the Earth is

Mass of Earth              =  M                 =  5.972⋅1024 kg
Radius of Earth            =  R                 =       6371 km
Gravitational constant     =  G                 =  6.67⋅10-11 Newton meters2/kg2
Mass of a test object      =  m
Force on the test object   =  F  = -G M m / R2  =  g m
Gravitational acceleration =  g  = -G M   / R2  =  9.8 meters/second2
Gravitational energy       =  E  = -G M m / R

Circular orbit

Geosynchronous orbit

Mass of central object                       =  M
Gravity constant                             =  G
Distance of satellite from central object    =  R
Velocity for a satellite on a circular orbit =  Vc  =    (G M / R)½
Escape velocity for a satellite              =  Ve  = 2½ (G M / R)½  = 2½ Vc
For a satellite on a circular orbit,
Gravity force  =  Centripetal force
G M m / R2     =  m V2c / R
The escape velocity is obtained by setting
Gravity energy  =  Kinetic energy
G M m / R       =  ½ m V2e


         Escape    Circular orbit
         velocity    velocity
         (km/s)      (km/s)
Earth     11.2        7.9
Mars       5.0        3.6
Moon       2.4        1.7

Orbits

If a cannonball is fired at a speed of 7.8 km/s then it orbits the Earth in a circle. If the speed is lower then it crashes into the Earth.

     Speed (km/s)  Orbit type

Red       <7.8     Ellipse            Too slow to reach orbit.  Crashes into the Earth
Green      7.8     Ellipse (circle)   Critical speed for a circular orbit
Yellow             Ellipse            Orbits Earth as an ellipse
Cyan      11.2     Parabola           Min speed to escape Earth, the "escape speed"
Blue     >11.2     Hyperbola          More than the escape speed


Elliptic orbit:   Retraces its path each cycle
Parabolic orbit:  Departs the Earth and limits to a speed of zero
Hyperbolic orbit: Departs the Earth and limits to a positive speed

Orbital types

Elliptic orbit
Parabolic orbit
Hyperbolic orbit

Ellipticity

In general an orbit is either an ellipse, a parabola, or a hyperbola. A circle is a special case of an ellipse with an eccentricity of 0.

The degree of elongation of an ellipse is parameterized by its "eccentricity".

Eccentricity    Orbit type

  e = 0         Circle        (a circle is a special case of an ellipse)
0 < e < 1       Ellipse       (less than escape velocity)
  e = 1         Parabola      (exactly escape velocity)
  e > 1         Hyperbola     (more than escape velocity)
In describing an ellipse we replace the radius with the "semimajor axis". For a circle they are equal.
Semimajor axis   =  A                  (equal to the radius if the ellipse is a circle)
Semiminor axis   =  B  =  a (1-e2)1/2   (A=B for a circle)
Periapsis        =  X  =  A (1-e)      ("perigee" if the focus is the Earth)
Apoapsis         =  Y  =  A (1+e)      ("apogee" if the focus is the Earth)
Eccentricity     =  e  =  (1-B2/A2)½
Central mass     =  M                  (mass at the focus)
Gravity constant =  G
Orbit time       =  T  =  2πA (GM)   (depends on the semimajor axis and not the eccentricity)

Energy

For an object on a circular orbit,

Gravitational energy  =  -2 * Kinetic energy
The relationship between the kinetic and gravitational energy doesn't depend on R. If a satellite inspirals toward a central object, the gain in kinetic energy is always half the loss in gravitational energy. The total energy is negative.
Total energy     =  Gravitational energy  +  Kinetic energy
                 =  ½ * Gravitational energy
                 = -½ G M m / R

Angular momentum =  m V R
                 =  m (G M R)½
As R decreases, both energy and angular momentum decrease. In order for a satellite to inspiral it has to give energy and angular momentum to another object.
Lagrange points

A planet has 5 "Lagrange points". An object orbiting at any of these points orbits in synch with the planet. L4 and L5 are the "Trojan points" and objects here are stable. If an object at one of the points is jostled it will stay in the region of the Lagrange point and stay in synch with the planet. The L1, L2, and L3 points are unstable. If an object here is jostled it will exit the region and lose synch with the planet. A spaceship can park at these points but it requires an occasional small rocket firings to maintain its position. The Webb telescope will be at L2.


Zone of gravitational influence

If a planet orbits a star and a moon orbits a planet, and if the two orbit periods are equal, then

Star mass           =  M0
Planet mass         =  M
Planet orbit time   =  T
Moon orbit time     =  t  =  T
Planet orbit radius =  R
Moon orbit radius   =  r  =  R ( M/M0)1/3
Hill radius         =  H  =  R (3M/M0)1/3         Derivation
The "Hill radius" characterizes the range of a planet's gravitational influence. In the limit of M << M0, the Hill radius equal to the distance from the planet to the L1 or L2 Lagrange point.

Earth Lagrange points with distances to scale


Orbital stability

If a moon orbits too far from a planet then it gets stolen by the star. The boundary for this is around 1/3 of the planet's Hill radius. The Earth's moon is barely within this boundary.

Moon orbital radius / Earth Hill radius = .256

The moons of the gas giants are all well within their planet's Hill radius.

If two planets orbit within ~ 10 Hill radii of each other then they disrupt each other's orbits. This is a planet's "zone of gravitational dominance".

Lagrange points for the Earth and moon,
with size and distance to scale

Hill radii for the inner planets

Hill radii for the outer planets


Definition of a planet

The solar system exhibit at the American Museum of Natural History

An object is defined as be a planet if it is:

*) Large enough for gravity to squash it into a round shape (Ceres is near this threshold).
*) Capable of clearing its orbit of other objects.
*) Not a satellite of something else.

Pluto doesn't make the cut because its Hill radius is small and because it orbits within Neptune's zone of gravitational influence.


             Earth   Earth   Solar
             radii   masses  masses

Neutron star                   3.0      Maximum mass before a neutron star becomes a black hole
White dwarf                    1.4      Maximum mass before a white dwarf becomes a neutron star
Sun           109   333000     1.0
Red dwarf       9    25000      .075    Minimum mass to fuse hydrogen and be a star
Brown dwarf    10     4130      .0124   Minimum mass to fuse deuterium & max planet mass
Jupiter        10.9    318      .00095  Largest gas giant in the solar system
Uranus          4.0     14.5            Smallest gas giant in the solar system
Earth           1.0      1.0
Venus            .95      .82
Mars             .53      .11
Mercury          .38      .0553         Smallest object capable of clearing its orbital zone
Ganymede         .41      .025
Titan            .40      .022          Smallest object with an atmosphere
Callisto         .38      .018
Io               .29      .015
Moon             .27      .012
Europa           .24      .0080
Triton           .21      .0036
Eris             .18      .0028
Pluto            .18      .0022
Ceres            .074     .00016        Round asteroid.  Min mass to be round
Vesta            .041     .000045       Largest object that is not round
The minimum mass to be a gas is somewhere between the mass of Earth and Uranus.

The minimum mass to have an atmosphere is in the range of .02 Earth masses. If we base the definition of a planet on this mass, then Ganymede, Titan, and Callisto potentially qualify, and Pluto doesn't.

If the definition of a planet is based on gravitational roundness, and if orbital state is ignored, then a deluge of objects quality.

History of the discovery of solar system objects:

Year  Object  Semi major  Earth       Discoverer
              axis (AU)   masses

1781  Uranus    19.19    14.50        Herschel
1801  Ceres      2.77      .00016     Piazzi
1802  Pallas     2.77      .0000359   Olbers
1804  Juno       2.67      .0000045   Harding
1807  Vesta      2.36      .0000447   Olbers
1845  Astraea    2.57      .00000049  Hencke
1846  Neptune   30.07    17.20        Galle, using calculations by Verrier
1930  Pluto     39.48      .00220     Tombaugh
2002  Quaoar    43.37      .00023     Trujillo, Brown
2003  Sedna    506.2       .0002      Brown, Trujillo, Rabinowitzs
2004  Orcus     39.47      .000108    Brown, Trujillo, Rabinowitzs
2005  Haumea    43.22      .00070     Brown
2005  Makemake  45.79      .0007      Brown
2005  Eris      67.67      .00278     Brown
2007  OR10      66.99      .0005      Schwamb, Brown, Rabinowitz


1851  15 asteroids are known and they are grouped into their own category.
2000  The American Museum of Natural History builds a planet exhibit designed by Neil Tyson
2001  A New York Times article points out that Pluto is not present in the AMNH planet exhibit.
2006  Soter publishes a paper defining planethood in terms of "clearing the orbit".
2006  The International Astronomical Union redefines planethood, drawing from the
      Soter definition. Pluto is expelled from the planet club.
      Ceres, Pluto, Eris, Haumea, Makemake, Quaoar, Sedna, and Orcus are deemed
      to be "dwarf plaents".

Ecliptic plane
The ecliptic plane is defined by the Earth's orbit. All planets orbit close to the ecliptic plane except Pluto.
Planet eccentricity and inclination
Inner planets
Outer planets

Eccentricities of the planets. Dots indicate pericenter and apocenter. As the planets orbit the pericenter and apocenter stay fixed in space.

Mercury's large eccentricity and orbit speed allowed Einstein to use it to test general relativity.

Extrasolar planets tend to have larger eccentricities and inclinations than solar system planets.

All planets spin in the same direction as their orbit except for Venus, which is reverse, and Uranus, which is sideways.

"Orbit inclination" is with respect to the ecliptic plane.

"Spin tilt" is with respect to the planet's orbit.

       Eccentricity    Orbit      Spin   Spin tilt
                    inclination  (days)    (deg)
                       (deg)
Sun         -            -        25.05     7.25
Mercury     .2056       7.00      87.97      .034
Venus       .0068       3.39     243.02   177.36
Earth       .0167       0.00        .997   23.44
Mars        .0934       1.85       1.026   25.19
Ceres       .0758      10.59        .378    4
Jupiter     .0484       1.30        .414    3.13
Saturn      .0542       2.48        .440   26.7
Uranus      .0472        .77        .718   97.8
Neptune     .0086       1.77        .671   28.3
Pluto       .2488      17.14       6.39   119.6
Eris        .0437      43.89       1.08
Sedna       .857       11.93        .43

Seasons

The Earth's spin angular momentum is conserved and the north pole always points toward the north star. The Earth's tilt is more important than the distance from the sun for determining warmth.

Earth maximum orbital distance (apocenter)  =  1.017 AU
Earth minimum orbital distance (pericenter) =   .983 AU
Solar intensity at 1 AU                     =  1        (scaled units)
Solar intensity at apocenter                =   .967    (scaled units)
Solar intensity at pericenter               =  1.035    (scaled units)
Apocenter intensity / Pericenter intensity  =  1.070
Sun intensity in space                      =   1366 Watts/meter2
Earth surface, sun at zenith, clear day     =   1050 Watts/meter2
Earth surface, average over day and night   =    250 Watts/meter2
Earth tilt angle                            =     23 degrees
Projected intensity at  0 degrees           =  cos( 0)  =  1
Projected intensity at 23 degrees           =  cos(23)  =   .921
Projected intensity at 46 degrees           =  cos(46)  =   .695
The change from projecting at 23 degrees exceeds the change from orbital ellipticity.
Planets and moons

Mars system
Phobos (top) and Deimos (bottom)

Sizes and distances to scale

          Mass     Radius  Orbit  Orbit   Escape speed
        (Earth=1)   (km)   (Mm)   (km/s)    (km/s)

Mars     .1074     3390       -      -      5.04
Phobos   2.5e-10     11.3    9.38   2.14     .0114
Deimos   1.8e-10      6.2   23.5    1.35     .0056

Asteroid belt

Eccentricity as a function of radius

Inclination as a function of radius
Inclination as a function of eccentricity

The plot of inclination vs. eccentricity shows a set of asteroid families, which resulted from the breakup of larger asteroids.


Jupiter system

Io, Eurpa, Ganymede, and Calliso
Resonant orbits of Io, Europa and Ganymede

Io, Europa, and Ganymede are all tidally locked with Jupiter and they orbit in a 1:2:4 resonance. If the moons were alone, the tidal lock would be perfect and the moons would experience no tidal heating. Since the moons jostle each other's orbits, the tidal lock isn't perfect and tides heat the moon's interiors. Io is heated most and is intensely volcanic. Europa's tidal heating gives rise to a subsurface ocean.

             Mass   Diameter  Orbit   Orbit
           (e18 kg)  (km)     (Mm)    speed
                                      (km/s)
Amalthea       2.08   180      .18   26.57
Io         89319     3640      .422  17.33
Europa     48000     3122      .671  13.74
Ganymede  148190     5262     1.07   10.88
Callisto  107590     4821     1.88    8.20
Himalia        6.70   170    11.4     3.31

Saturn system

Artist's conception of a ring
Daphnis clear a gap and generates waves
Geysers on Enceladus

            Mass   Diameter  Orbit  Orbit speed
          (e18 kg)  (km)     (Mm)    (km/s)

Janus         1.90   179       .151
Mimas        37      396       .185
Enceladus   110      504       .238
Tethys      620     1062       .295
Dione      1100     1123       .377
Rhea       2300     1527       .527    8.48
Titan    135000     5150      1.22     5.58
Hyperion      5.6    270      1.48
Iapetus    1800     1470      3.56     3.26
Phoebe        8.3    213     12.9

Uranus system
             Mass   Diameter  Orbit   Orbit
           (e18 kg)  (km)     (Mm)    speed
                                      (km/s)
Portia         1.70    135      .066  9.37
Puck           2.9     162      .086  8.21
Miranda       66       472      .129  6.66
Ariel       1353      1158      .191  5.51
Umbriel     1172      1169      .266  4.67
Titania     3527      1577      .436  3.64
Oberon      3014      1522      .584  3.15
Sycorax        2.3     150    12.2

Neptune system

Triton orbits retrograde to Neptune's spin

             Mass   Diameter  Orbit    Orbit
           (e18 kg)  (km)     (Mm)     speed
                                       (km/s)
Despina        2.10    150       .053
Galatea        2.12    176       .062
Larissa        4.6     194       .074
Proteus       44       420       .118  7.62
Triton     21408      2705       .355  4.39
Nereid        27       340      5.51    .93

Pluto system

Pluto
Charon
Nix and Hydra


Pluto and moons, to scale. Click for larger picture

Pluto and Charon (New Horizons)
Pluto and Charon (New Horizons)
Pluto and Charon (Artist)


        Pluto    Orbit  10*HillRadius  Radius  Density  Eccentricity  Inclination
        masses  (Pluto  (Pluto radii)   (km)   (g/cm2)                  (deg)
                 radii)
Pluto    1                              1170    2.03
Charon    .0490   15.2      38.6         604    1.65         0           .001
Styx       tiny   36.4                                      ~0         ~0
Nix      <.00006  42.2    < 11.7                              .0030      .195
Kerbero    tiny   51.1                                      ~0         ~0
Hydra    <.00006  56.1    < 15.6                              .0051      .212

Pluto radius = 1153 km               Pluto mass = 1.305e22 kg

Kuiper Belt

Neptune and Pluto


The largest Kuiper belt objects are listed below, along with Neptune for comparison.

      Semi major   Earth
      axis (AU)    masses

Neptune   30.07   17.20
Eris      67.67     .00278
Pluto     39.48     .00220
Haumea    43.22     .00070
Makemake  45.79     .0007
2007OR10  66.99     .0005
Quaoar    43.37     .00023
Sedna    506.2      .0002
Orcus     39.47     .000108
Most of the objects in the Kuiper belt were thrown there by the gas giants. A planet's ability to throw objects into the outer solar system is given by the ratio of the escape velocity to the orbital velocity. The gas giants are sufficiently large and the rocky planets aren't, nor are Pluto and Eris.
         Semimajor    Mass      Escape  Orbit    Escape speed
         axis (AU)  (Earth=1)   speed   speed    / Orbit speed
                                (km/s)  (km/s)
Mercury      .387      .0553      4.3    47.9      .09
Venus        .723      .8150     10.46   35.0      .30
Earth       1.000     1.0000     11.2    29.8      .38
Mars        1.524      .1074      5.03   24.1      .21
Jupiter     5.203   317.83       59.5    13.1     4.54
Saturn      9.537    95.16       35.5     9.64    3.68
Uranus     19.19     14.50       21.3     6.81    3.13
Neptune    30.07     17.20       23.5     5.43    4.33
Pluto      39.48       .00220     1.23    4.74     .26
Eris       67.67       .00278     1.34    3.44     .39

Atmospheres

Earth
Titan
Veuns
Mars
                        Escape
      Density  Pressure speed  Gravity   N2     O2    N2     CO2    Ar    Temp
       kg/m3    (Bar)    km/s    m/s2   kg/m3  frac   frac   frac   frac   (K)

Venus    67     92.1     10.36   8.87  2.34           .035  .965           735
Titan     5.3    1.46     2.64   1.35  5.22           .984                  94
Earth     1.2    1       11.2    9.78   .94    .209   .781  .00039 .0093   287
Mars       .020   .0063   5.03   3.71   .00054 .0013  .027  .953   .016    210
No other object in the solar system has a meaningful atmosphere.

Titan is the smallest object with an atmosphere and Mercury is the largest object without an atmosphere.

Earth's atmosphere


Greenhouse gases in the Earth's atmosphere
Gas    Concentration   Contribution
      (ppm by volume)  to warming
H2O                    36-72%
CO2     394             9-26%
CH4       1.79          4-9%
O3       <=.07          3-7%

Gases
Molecules in a gas
Brownian motion
M  =  Mass of a gas molecule
P  =  Pressure
T  =  Temperature
Vol=  Volume
N  =  Number of gas molecules within the volume
D  =  Density in kg/m3
   =  N M / Vol
k  =  Boltzmann constant
   =  1.38*10-23 Joules/Kelvin
Mol=  Number of moles of gas molecules
   =  N /6.62*1023
V  =  Characteristic thermal speed of gas molecules
E  =  Mean kinetic energy of a gas molecule
   =  1/2 M V2
Ideal gas law:
P  =  k T N / Vol
   =  2/3 N E / Vol
   =  1/3 N M V2 / Vol
   =  1/3 D V2
   =  8.3 Moles T
For a system in thermodynamic equilibrium each degree of freedom has a mean energy of .5 k T.

A gas molecule moving in 3 dimensions has 3 degrees of freedom and so the mean kinetic energy is

E  =  3 * .5 k T  =  1.5 k T
This is also equal to the mean kinetic energy of a gas molecule.
E  =  .5 M V2
Hence
k T  =  2/3 E

Sound speed
Gamma  =  Adiabatic constant
       =  7/5 for diatomic molecules such as H2, O2, N2.  Air also has Gamma=7/5
       =  5/3 for monatomic molecules such as Helium and Xenon
The sound speed for an ideal gas is
SoundSpeed2  =  Gamma  P  /  Density
             =  1/3  Gamma  V2
For air,
Gamma       =  7/5
SoundSpeed  =  .63 ThermalSpeed
            =  343 meters/second at 20 Celsius
Gas properties simulation
Atmospheric escape

The "Balloons and Buoyancy" simulation at phet.colorado shows a gas with a mix of light and heavy molecules.

S = Escape speed
T = Temperature
k = Boltzmann constant
  = 1.38e-23 Joules/Kelvin
g = Planet gravity at the surface

M = Mass of heavy molecule                    m = Mass of light molecule
V = Thermal speed of heavy molecule           v = Thermal speed of light molecule
E = Mean energy of heavy molecule             e = Mean energy of light molecule
H = Characteristic height of heavy molecule   h = Characteristic height of light molecule
  = E / (M g)                                   = e / (m g)
Z = Energy of heavy molecule / escape energy  z = Energy of light molecule / escape energy
  = .5 M V2  / .5 M S2                          = .5 m v2 / .5 m S2
  = V2 / S2                                     = v2 / S2


For an ideal gas, all molecules have the same mean kinetic energy.

    E     =     e      =  1.5 k T

.5 M V2   =  .5 m v2   =  1.5 k T
The light molecules tend to move faster than the heavy ones. This is why your voice increases in pitch when you breathe helium. Breathing a heavy gas such as Xenon makes you sound like Darth Vader.

For an object to have an atmosphere, the thermal energy must be much less than the escape energy.

V2 << S2        <->        Z << 1


          Escape  Atmos    Temp    H2     N2      Z        Z
          speed   density  (K)    km/s   km/s    (H2)     (N2)
          km/s    (kg/m2)

Jupiter   59.5             112   1.18    .45   .00039   .000056
Saturn    35.5              84   1.02    .39   .00083   .00012
Neptune   23.5              55    .83    .31   .0012    .00018
Uranus    21.3              53    .81    .31   .0014    .00021
Earth     11.2     1.2     287   1.89    .71   .028     .0041
Venus     10.4    67       735   3.02   1.14   .084     .012
Mars       5.03     .020   210   1.61    .61   .103     .015
Titan      2.64    5.3      94   1.08    .41   .167     .024
Europa     2.02    0       102   1.12    .42   .31      .044
Moon       2.38    0       390   2.20    .83   .85      .12
Pluto      1.21    0        44    .74    .28   .37      .054
Ceres       .51    0       168   1.44    .55  8.0      1.14
Even if an object has enough gravity to capture an atmosphere it can still lose it to the solar wind. Also, the upper atmosphere tends to be hotter than at the surface, increasing the loss rate.

Titan is the smallest object with a dense atmosphere, suggesting that the threshold for capturing an atmosphere is on the order of Z = 1/25, or

Thermal Speed  <  1/5 Escape speed

Height of an atmosphere

M  =  Mass of a gas molecule
V  =  Thermal speed
E  =  Mean energy of a gas molecule
   =  1/2 M V2
H  =  Characteristic height of an atmosphere
g  =  Gravitational acceleration
Suppose a molecule at the surface of the Earth is moving upward with speed V and suppose it doesn't collide with other air molecules. It will reach a height of
M H g  =  1/2  M  V2
This height H is the characteristic height of an atmosphere.
Pressure of air at sea level      =  1   Bar
Pressure of air in Denver         = .85  Bar      One mile high
Pressure of air at Mount Everest  = 1/4  Bar      10 km high
The density of the atmosphere scales as
Density ~ (Density At Sea Level) * exp(-E/E0)
where E is the gravitational potential energy of a gas molecule and E0 is the characteristic thermal energy given by
E0 = M H g = 1/2 M V2
Expressed in terms of altitude h,
Density ~ Density At Sea Level * exp(-h/H)
For oxygen,
E0  =  3/2 * Boltzmann_Constant * Temperature
E0 is the same for all molecules regardless of mass, and H depends on the molecule's mass. H scales as
H  ~  Mass-1

Solar wind

The solar wind consists primarily of protons moving at ~ 600 km/s. The density of protons at Earth orbit is ~ 7 /cm3.


Water

Titan
        Dist from   Mass     H2O       H2O      Density
        sun (AU)  (Earth=1) (Earth=1)  frac     (g/cm3)

Mercury    .39     .055      -         -         5.60
Venus      .72     .82       -         -         5.20
Earth     1.00    1.00       .00025    .00025    5.52
Moon      1.00     .0123     -         -         3.35
Mars      1.52     .107      .00000027 .0000025  3.95
Phobos    1.52                                   1.88
Deimos    1.52                                   1.47
Vesta     2.36     .0000447  -         -         3.42
Ceres     2.77     .00016    .000033   .21       2.08
Pallas    2.77     .0000359  -         -         2.8
Io        5.20     .0150     -         -         3.528
Europa    5.20     .0080     .0033     .4        3.103
Ganymede  5.20     .0248     .012      .5        1.942
Callisto  5.20     .0180     .0084     .5        1.834
Pan       9.54      8.3e-10                       .42
Mimas     9.54     .0000063                      1.15
Titan     9.54     .0225     .012      .5        1.88
Triton   30.07     .00358    .00067   ~.2        2.061
Pluto    39.48     .00220                        2.03
Charon   39.48     .000271                       1.72
Jupiter   5.20  317.83                           1.326
Saturn    9.54   95.16                            .687
Uranus   19.19   14.50                           1.270
Neptune  39.48   17.20                           1.638

Ceres has an ocean's worth of H2O.
Ceres H2O / Earth H2O  =  .13

Earth mass = 5.972e24 kg

Earth's water fraction
Oceans                 .954
Ice caps and glaciers  .024
Lakes and rivers       .006
Underground            .016
Atmosphere             .00001

Gas giant composition
           Planet  Metal &     H2O        Gas      Density   H2O
           mass     rock                           (g/cm3)   frac

Ceres       .00016  .00013   .000033       0        2.08     .21
Europa      .0080   .005     .0033         0        3.103    .4
Mars        .107    .107     .00000027  .000000004  3.95     .0000025
Earth        1       1       .00025     .0000009    5.52     .00025
Jupiter    317.9    12-45              Most of it   1.33
Saturn      95.2     9-22              Most of it    .69
Uranus      14.5   0.5-3.7   9.3-13.5    .5-1.5     1.27     3/4
Neptune     17.2     1.2      10-15     1.0-2.0     1.64     3/4
Masses in Earth masses
Snow line
               Mean
           Temperature  Min   Max    Parent   Albedo
                (K)     (K)   (K)    planet

Mercury         340     100   700
Venus           735     735   735
Earth           288     184   330
Moon            220     100   390
Mars            210     130   308
Ceres           168       ?   235
Europa          102      50   125    Jupiter
Ganymede        110      70   152    Jupiter
Callisto        134      80   165    Jupiter
Titan            94                  Saturn
Titania          70      60    89    Uranus
Oberon           75                  Uranus
Nitrogen freeze  63
Oxygen freeze    54
Triton           38                  Neptune   .76
Nereid           50                  Neptune   .155
Pluto            44      33    55              .58
Hydrogen freeze  14
The boundary between rocky and icy objects is at Ceres' orbit.

The boundary for frozen nitrogen is at Neptune's orbit.


Liquids
            Freeze     Boil    Heat Capacity   Density
           (Kelvin)  (Kelvin)  (J/g/Kelvin)    (g/cm3)

Water        273       373         4.2         1.00
Ammonia      195       240         4.7          .73
Methane       91       112         1.6          .42
Ethanol      159       351         2.4          .79
Ethane        89       184                      .55
Propane       86       281                      .58
Hydrogen      14.0      20.3
Nitrogen      63.2      77.4
Oxygen        54.4      90.2
CO            68
CO2          194.7     216.6
Argon         83.8

Objects that we have landed spacecraft upon

Moon
Mars
Venus

Titan
Hydrocarbon lakes on Titan
Eros
Itokawa


Terrain

Mercury. Highest mountains are 10 km in height
Venus. Highest mountains are 12 km in height

Venus
Venus with the atmosphere removed

Plates

Mars

Olympus Mons on Mars
5 km ice mountain on Cers

Lava on Io


Density (grams/cm3)

Air at sea level   .00127
Ice                .92
Water             1.00
Rock            ~ 2
Iron              7.9
Nickel            8.9     Metallic asteroids are composted of mostly iron and nickel
Lead             11.3
Uranium          19.1
Gold             19.3
Osmium           22.6     Densest element
White dwarf        e9
Atomic nuclei     2e17
Neutron star      2e17
Planck density  5.1e96

Gravity for a uniform-density sphere
D  =  Density
R  =  Radius
M  =  Mass
   =  Density * Volume
   =  4/3 Pi D R3
A  =  Acceleration at the surface
   =  G M / R2
   =  (4/3) π G D R
Acceleration is proportional to R

        Density  Radius   Gravity
        g/cm3   (Earth=1)  m/s2
Earth    5.52    1.00      9.8
Venus    5.20     .95      8.87
Uranus   1.27    3.97      8.69
Mars     3.95     .53      3.71
Mercury  5.60     .38      3.7
Moon     3.35     .27      1.62
Titan    1.88     .40      1.35
Ceres    2.08     .074      .27

Mountains

If a mountain gets too high, pressure deforms the rock and the mountain sags. The height of a mountain is limited by the tensile strength of the rock. For a stout rock such as granite this is ~ 100 MegaPascals (Newton / meter2). The density of granite is ~ 3000 kg/m3.

The pressure at the base of a mountain is

Pressure = RockDensity * Gravity * Height
The height that gives a pressure of 100 MegaPascals is

Height ~ 103 N/m2 / 10 m/s2 / 3000 kg/m3 ~ 3 km

        Gravity  Radius  Tallest mountain
        (m/s2)    (km)       (km)

Earth     9.8     6371       8.9        Mount Everest
Venus     8.9     6052       8          Volcanic
Mercury   3.7     2440      10
Mars      3.7     3386      21.2        Mount Olympus.  Previously volcanic
Io        1.80    1822      18          Supervolcanos
Moon      1.62    1738       7          No geological activity
Titan     1.35    2576       2          Ice mountains
Ceres      .27     476       5          Ice mountains.  Round
Pluto      .66    1173       4          Ice mountains.
Vesta      .25     265    Tall          Not round
If a planet is substantially heavier than the earth and if it has enough water for oceans, gravity might make it impossible for dry land to exist.
Round or potato?

If we define the roundness of an object as the characteristic mountain height divided by the object's radius, then for the Earth,

Roundness ~ 10 km / 7*103 km ~ 10-3
The surface gravity for equal-density objects is proportional to radius, hence the roundness of an object of radius R scales as
Roundness  ~  10-3 * (R / RadiusOfEarth)2
For an object a tenth the size of the earth, the roundness is ~ 10-1. If we take this magnitude as the boundary between round and potato-shaped, then the smallest round objects should be a tenth the Earth's radius. The smallest round object in the solar system is Ceres, which has a radius of 487 km. The radius of the earth is ~ 6370 km.
Geology

Earth interior

The Moho discontinuity divides the crust from the mantle.


              Depth   Min density  Max density     Composition
              (km)      (g/cm3)      (g/cm3)

Ocean             4      1.0         1.0           Water
Crust            35      1.0         3.4           Granite, basalt, feldspar
Lithosphere     100      3.4         3.5           Peridotite, Dunite
Upper mantle    500      3.75        4.0           Mafic rock
Lower mantle   2890      4.5         5.5           Mafic rock
Outer core     5150     10.0        12.1           Iron, nickel
Inner core     6360     12.8        13.1           Iron, nickel

Rock
Limestone    Calcium carbonate CaCO3
Mafic        Rock containing iron or magnesium. Abundant below oceans. Denser than feldspar
Granite      Quartz, mica, feldspar, amphibole minerals
Basalt       Solidified lava. Mafic
Igneous      Granite, basalt
Feldspar     KAlSi3O8, NaAlSi3O8, CaAl2Si2O8.  Abundant above the Moho discontinuity and absent below.

Internal structure

The following diagrams are based on models and one shouldn't expect too much precision from them.

Uranus
Neptune. Similar structure as Uranus

Io
Europa, 2 models
Callisto

Moon
Titan
Ceres
Pluto


Radioactive heating of the Earth
              MicroWatts  Half life   Mantle     PicoWatts/kg
                 /kg      (billion    abundance  of mantle
                            years)    (ppb)

Uranium-238      94.6       4.47      30.8        2.91
Uranium-235     569.         .70        .22        .125
Thorium-232      26.4      14.0      124          3.27
Potassium-40     29.2       1.25      36.9        1.08
The Earth loses heat at a rate of .087 Watts/m2, for a global heat los of 4.42e13 Watts.

80% of the Earth's heat is from radioactivity and 20% is from accretion.

The radioactive heating rate 3 billion years ago is twice that of today.

The Earth's core temperature is ~ 7000 K.


Magnetic fields

Mercury
Venus

Jupiter magnetosphere
Jupiter plasma torus
Io's volcanos fill Jupiter's magnetosphere with plasma

Saturn magnetosphere
Saturn plasma torus

Uranus
Europa
Ganymede

          Dipole    Field at  Magneto-  Axis   Radius  Spin    Core     Core   Volcanic
          moment    equator   pause     angle  (Earth  (days) heating   Temp
         (Earth=1)  (Gauss)   (planet   (deg)   =1)           (1012     (K)
                               radii)                          Watts)
Sun    5000000                                 109      25.0   4e14   1500000
Jupiter  20000       4.28       80       9.6    10.9      .41  400000   36000
Saturn     600        .22       20      <1       9.0      .44  150000   11700
Uranus      50        .23       20      58.6     4.0      .72   18000    5000
Neptune     25        .14       25      47       3.9      .67   18000    5400
Earth        1        .305      10      10.8     1.0     1.00      48    6000    Yes
Europa        .0016   .0072      4.5              .24    3.55       1.6          No
Mercury       .0007   .003       1.5    14        .38   58.6                     No
Venus        <.0004  <.00003     -       -        .95  243.0                     Yes
Mars         <.0002  <.0003      -       -        .53    1.03                    No
Io           ?       ?           ?       ?        .29    1.77     100            Yes
The Earth's magnetic moment is 7.91e15 T m3.

Jupiter's mangnetic field is 0.00120 Gauss at Europa's orbit.

The sun rotates with a period of 25.0 days at the equator and 34.4 days at the poles. This extreme differential rotation powers a magnetic field dynamo.

For an object to have a magnetic field it needs size, heat, and spin.

Core heating drives convective turbulence in the mantle, and turbulence generates magnetic fields. If there is no spin, the fields will have random directions and the field at the surface will be small. Spin herds magnetic fields into a uniform orientation and produces a dipole shape, like the Earth's field. In this case the field at the surface will be larger.

Magnetic fields are generated by turbulence and lost by resistive diffusion, and the equilibrium field strength occurs when these are in balance. The larger the object, the longer it takes for diffusion to smooth away the field and the larger the equilibrium field.

Saturn's aurora


Plate tectonics

Dots represent volcanoes and lines represent plate boundaries
Earthquakes

The largest earthquakes occur at subduction zones and they cause the largest tsunamis. The pacific plate is being subducted by continental plates all around its circumference, making every city on the Pacific coast vulnerable to tsunamis.

The Atlantic Ocean is expanding and doesn't have subduction zones.

Volcanoes tend to form behind subductions zones, for example the mountains in Washington and Japan.

Earthquake tsunamis are more frequent than asteroid tsunamis.

The largest earthquakes have magnitude 10. Only an asteroid can cause a larger tsunami than this.


Extrasolar planetary systems

This is a plot of all extraterrestrial planetary systems with at least 3 planets. In the lower right there is an index relating star type with mass. The stars are depicted with the size, brightness, and color that your eye would see.

Planet size is scaled as the cube root of the mass. The solar system is 1/3 of the way down. Jupiter is the orange dot at the far right.

The "metallicity" of a star is defined as Metallicity = Mass of elements heavier than helium / Total mass

If the star has metallicity equal to the sun, orange dots are used for the planets. If the star is more metallic, yellow dots are used, and if it is less metallic, red dots are used.

Planet size tends to increase with star metallicity.

The purple dot indicates the location of the Goldilocks zone for each planetary system, where the temperature is right for liquid water.

Very few of the systems have an Earth-sized planet in the Goldilocks zone.

Most extraterrestrial planetary systems are more massive than the sun's planets. In the "Galactic Museum of Natural History", the solar system might be classified as a "Dwarf planetary system".


Habitable zone
Hoth
The "Goldilocks Zone"
Antarctica

Planet migration

Io, Europa, Ganymede, Callisto
Jupiter and its moons
Io, Europa, Ganymede

Goldreich & Tremaine (1980): "We present an illustrative application of our results to the interaction between Jupiter and the plantary disk. The angular momentum transfer is shown to be so rapid that substantial changes in both the structure of the disk and the orbit of Jupiter must have taken place on a time scale of a few thousand years."

Electromagnetic spectrum

Absorption of light by the atmosphere

The Earth's atmosphere transmits visible light and radio waves and it blocks all other kinds of radiation.


Color vision

Cone cells are sensitive to red, green, and blue light. Rod cells don't sense color and are sensitive to all light in the visible spectrum.


Photosynthesis

Photosynthesis uses red and blue light. The energy gainedfrom blue light is twice that from red light.


Blackbody radiation



This is a plot of the radiation intensity of blackbody radiation for various temperatures. Each curve is colored according to what your eye would perceive. Visible light ranges from 400 nm to 750 nm.

The strip of color at the right depicts the color as a function of temperature.


Laws of blackbody radiation:

T  =  Temperature in Kelvin
I  =  Radiation intensity in Watts/meter2
W  =  Wavelength of the peak of the radiation spectrum for a given temperature
F  =  Frequency of light
C  =  Speed of light  =  2.998e8 m/s


Stefan-Boltzmann law:   I   =  5.67e-8  T4

Wein's law:             W T =  2.897e-3  Kelvin meters

Wave equation:          F W =  C


                  Temperature      Intensity       Wavelength      Color
                   (Kelvin)     (Watts/meter2)    (micrometers)

The Earth            288              429             9.8          Infrared
Candle              1000            56700             2.9          Red
Incandescent bulb   2500          2000000             1.16         Orange
The sun             5778         60000000              .50         White
Sirius              9940        600000000              .29         Blue

The sun from space

Blue stars in the orion nebula

phet.colorado.edu: Blackbody radiation


Planck law

The Planck law gives the intensity of blackbody radiation as a function of temperature and frequency.

Planck constant     =  h  =  6.626e-34 Joule Seconds
Frequency           =  F               Hertz
Speed of light      =  C  =  2.998e8   meters/second
Boltzmann constant  =  k  =  1.381e-23 Joules/Kelvin
Temperature         =  T               Kelvin
Blackbody intensity =  I  =  2 h F3 C-2 (ehF/(kT)-1)-1  Watts/Hertz/meter2

UV power

The following table gives the power radiated in each band in percentages. A star with a temperature in the range of 4200 Kelvin is optimal for photosynthesis because it is both abundant in visible photons and sparse in UV photons.

 Temperature   UV   Visible   IR
  (Kelvin)     %       %      %

   2400        .00067 2.23   97.7
   2550        .0017  3.20   96.9     Trappist-1
   2600        .0023  3.56   96.4
   2800        .0067  5.26   94.8
   3000        .02    7.2    92.8
   3200        .03    9.5    90.5
   3400        .07   12.1    87.8
   3600        .12   14.8    85.1
   3800        .20   17.6    82.2
   4000        .32   20.5    79.2
   4200        .48   23.4    76.1
   4400        .69   26.3    73.0
   4600        .97   29.1    69.9
   4800       1.31   31.7    67.0
   5000       1.73   34.3    63.8
   5200       2.21   36.6    61.2
   5400       2.78   38.9    58.3
   5600       3.43   40.9    55.7
   5772       4.0    42.6    53.4     Sun
   5800       4.1    42.7    53.2
   6000       4.9    44.4    50.7
   6200       5.8    45.9    48.3
   6400       6.8    47.2    46.0
   6600       7.8    48.3    43.9

UV: 315 nanometers and beyond
IR: 680 nanometers and beyond

Stellar luminosity

The luminosity of a star scales with mass as

Luminosity  ~  Mass3.5
The lifetime scales as
Lifetime  ~  Mass  /  Luminosity
          ~  Mass-2.5
The sun burns for ~ 10 billion years.

The minimum mass for hydrogen fusion is 0.08 solar masses.


Nearby stars

These are the stars in a (x,y,z)=(40,40,16) light year volume centered around the sun, with the stars in the volume projected on the (x,y) plane. Dot size and brightness are in scale with the star's radius and surface brightness. The color of the dots reflects what your eye perceives. Overlapping dots indicate binary stars. There are many stars in this volume that are too dim to be seen compared to the sun.


In this figure the brigtnesses have been magnified by 20 to bring out dimmer stars. Any star brighter than the sun is reduced in brightness to be equal to the sun.


In this figure the brightness has been raised to the 1/16 power to bring out the dimmest stars. Brown dwarfs are barely visible as tiny red dots. There are 2 of them between the sun and Altair, one to the right of Alpha Cantauri and one below Sirius.


HR diagram

Hot blue stars in a nebula

HR diagram


Nearby stars
                Distance  Mass    Radius   Lum        Temp    Age    Metal    Rotation
                   (ly)                                       (my)            (days)
Sun                        1.00    1.00      1.00      5778    4540
Proxima Centauri    4.24    .123    .141      .0017    3042    4850           .21d
Alpha Centauri A    4.36   1.10    1.23      1.52      5790    4850  1.51
Alpha Centauri B    4.36    .907    .865      .500     5260    4850  1.60
Barnard's Star      5.96    .144    .196      .0035    3134   10000   .2      130
Wolf 359            7.78    .09     .16       .0010    2800    ~200            <3km/s
Lalande 21885       8.29    .46     .393      .025     3383   <5000  -.2d
Sirius              8.60   2.02    1.71     25.4       9940     250   .50d
Sirius B            8.60    .978    .0084     .026    25200                   white dwarf
Luyten 726-8 A      8.73    .10     .14       .00006   2670
Luyten 726-8 B      8.73    .10     .14       .00004
Wise 1541-2250    ~~9       .012    .1           dim    350
Procyon A          11      1.42    2.05      6.93      6530    3000  -.05d     23
Procyon B          11       .60     .0123     .00055   7740                   white dwarf
Altair             16.7    1.79    2.03     10.6       8500   <1000  -.2d        .37
Vega               25.04   2.135   2.26     37         9602     455   .52
Capella A          42      2.7    12        78         4940     520   .4      106
Capella B          42      2.6     9        78         5700     520   .4        9
Castor A           49.8    2.15    2.3      30        10300     200  9.5
Castor B           49.8    1.7     1.6      14         8840     200  2.8
Arcturus           36.7    1.10   25.7     170         4290          -.37d     <1.7km/s
Aldebaran          65      1.7    44       520         3910           .7      643
Achernar          139     ~7     ~10      3150        15000                   250 km/s
Canopus           310      8.5    65     13600         7350           .9
Polaris           434      7.54   46      2200         7200          1.12     ~17 km/s
Betelgeuse        643    ~18    1180   ~140000         3500      10
Zeta Orionis A    700     28     ~20    100000        30000                   140 km/s
Zeta Orionis B    700     23              1300            ?
Zeta Orionis C    700     20                 ?        24000
Rigel             860     24      71     66000        12130       8   .0d      40 km/s
Deneb            1400    ~20    ~110    ~54000         8525
A0620-00         3000     11                 0                     nearest known black hole
Eta Carinae     ~8000   ~125           5000000       ~38000       <3

Altair is a variable star, with radius varying from 1.63-2.03 and the temperature varying from 6900-8500 K.

Eta Carinae has a mass of 100-150 solar masses and a radius of 85-195 solar radii.

Alpha Centauri A and B:
Perigee    = 11.2 A.U.
Apogee     = 35.6 A.U.
Orbit time = 80 Years
The Sun and the Alpha Centauri system

Fate of stars

Star type     Mass   Luminosity   Color   Temp   Lifetime   Death       Remnant
                                          (K)    (109 yrs)

Brown Dwarf  <0.08                Red     1000  immortal
Red Dwarf     0.1          .0001  Red     2000   1000       Red giant   White dwarf
The Sun       1           1       White   5778     10       Red giant   White dwarf
Blue star     10      10000       Blue   10000      0.01    Supernova   neutron star
Blue giant    20     100000       Blue   20000      0.01    Supernova   black hole


Mass < 9 solar masses  ->  End as red giants and then turn into a white dwarf.
Mass > 9 solar masses  ->  End as supernova
9 < Mass < 20          ->  Remnant is a neutron star.
Mass > 20              ->  Remnant is a black hole.
130 < Mass < 250       ->  Pair-instability supernova (if the star has low metallicity)
Mass > 250             ->  Photodisintegration supernova, producing a black hole & relativistic jets.
Wikipedia: Main sequence
Supernovae in the Milky Way
Year  Distance  Apparent  Type
        (ly)    Magnitude
 185    8200     -4       1a(?)
 386   15000     +1.5     2
 393   34000      0
1006    7200     -7.5     1a    Brightest supernova, visible Earth-wide
1054    6500     -6       2     Remnant is the Crab Nebula Pulsar
1181    8500      0
1572    8000     -4       1a
1604   14000     -3       1a
1680    9000     +5       2b
1868   25000
1987  160000      2.9     2pec  Large Magellanic Cloud, a satellite galaxy of the Milky Way
The sun is 27200 light years from the center of the galaxy.



This is a plot of the radiation intensity of blackbody radiation for various temperatures. Each curve is colored according to what your eye would perceive. Visible light ranges from 400 nm to 750 nm.

The strip of color at the right depicts the color as a function of temperature.


Galaxies

Messier objects

The Messier objects as seen through a small telescope

Orion nebula as seen through the Hubble telescope

Milky Way
Andromeda galaxy

Milky Way
             Solar masses / 109

Total mass       1000       (Most of which is dark matter)
Stars              55
Gas                 7
Black holes         1
Dust                 .1
Central hole         .0042


Number of stars        =  3e11
Number of black holes  =  1/1000 times the number of stars
Number of planets      =  At least 1 per star
Earth-size planets in the goldilocks zone  =  Around .13 per star
Average mass of a star =  .18 solar masses
Disk diameter          =  100000 light years
Disk thickness         =  1000 light years
Star density near sun  =  .004 stars/lightyear3

Supermassive black holes
                 Galaxy  Central black  Distance
                 mass    hole mass      (million ly)

Small Magellanic   .007   none found      .21
Large Magellanic   .01    none found      .16
Triangulum         .05   <.000000003     2.64    Third-largest galaxy in local group
Andromeda         1.0     .0002          2.56    Second-largest galaxy in local group
Milky Way         1.2     .0000042        .0     Largest galaxy in local group
Virgo M87        17       .0035         53.5     Virgo cluster
Virgo M49                 .000565       55.9     Virgo cluster
NGC 3842          ?       .0097        331       Leo Cluster
NGC 4889          ?       .021         308       Coma Cluster
APM 08279         ?       .023       23600       Largest black hole observed
Masses in trillions of solar masses.


Nearby galaxies

This is a plot of galaxies within a 6x6x6 million light year volume centered on the Milky Way. The size of each dot is proportional to Luminosity1/3, and the sizes of the Milky Way and Andromeda are in scale with the distance between them. The two dots above the Milky Way are the Large and Small Magellanic Clouds.

Color indicates depth, where magenta corresponds to the top of the volume, red corresponds to points at the bottom, and green corresponds to points in the midplane.


This is the same as the previous plot but with dot size scaled as Luminosity1/6, to bring out the dimmer galaxies.


This is a plot of galaxies within a 20x20x20 million light year volume centered on the Milky Way, with dot size scaled as Luminosity1/3.


This is a plot of galaxies within a 160x160x160 million light year volume centered on the Milky Way, with dot size scaled as Luminosity1/3.

The Virgo Cluster is to the right and the Fornax Cluster is to the upper left. The Z=0 plane is chosen so that it contains the Virgo Cluster, the Fornax Cluster, and the Milky Way. This is why most of the points have the same color.


Cosmology

Hubble's law

                  Distance       Mass     Speed with  Speed      Escape    Gravity
                  (millions of   (solar   respect to  according  velocity  time
                  light years)   masses)  Milky Way   to Hubble  (km/s)    (billions
                                          (km/s)      law (km/s)           of years)
Sun                 1.6e-11      1.0           -          -        41.9      2.3e-10
Alpha Centauri         .0000044  1.1           -          -          .084     .031
Andromeda             2.54       1.0e12     -120         58       105       14.5
Virgo Cluster        54          1.2e15     1254       1240       790       41.0
Coma Cluster        321                     6950       7400
Edge of universe  14000                   300000     300000

Hubble constant  =  23 km/s/(million light years)
Hubble velocity  =  Hubble constant  *  Distance
Solar mass       =  1.99e30 kg
Light year       =  9.46e15 meters
Age of universe  =  13.8 billion years

Distance to the edge of the universe  =  Speed of light  *  Age of universe
Andromeda and the Milky Way are close enough for gravity to have reversed the Hubble expansion. The Virgo and Coma clusters are far enough away to be a part of the Hubble expansion. The edge of the universe is the horizon beyond which galaxies recede faster than the speed of light.

"Escape velocity" is the escape velocity of the Earth from the given object.

"Gravity time" is the amount of time it would take for the object's gravity to pull the Earth toward it, if the Earth were released from rest with respect to the object.

If an object's escape velocity is smaller than the Hubble velocity, then the gravity from that object cannot reverse the Hubble expansion.

The Andromeda galaxy will collide with the Milky Way in 4 billion years and in the distance future, the merged galaxy will collide with the Virgo Cluster.


Timeline of the universe

History of the Earth's atmospheric oxygen

The Earth's atmosphere became abundant in oxygen 600 million years ago, concurrent with the emergence of multicellular life.

Timeline of evolution
Early tetrapods


What could go wrong?

Meteor Crater, Arizona


Planet property    If too little                            If too much


Mass               Cannot capture atmosphere                Becomes gas giant
                   No volcanism
                   Cannot generate a magnetic field

Distance from      Too hot                                  Too cold for surface water
star               Inside the snow line

Atmospheric        Cosmic rays reach the surface            Blocks too much sunlight
thickness          Atmosphere loses heat at night           for photosynthesis

Water content      If you don't have oceans then you        No dry land
                   don't have enough photosynthesis
                   to generate an oxygen atmosphere

Planet spin        Does not generate a large-scale          Fine
                   magnetic field

Planet spin tilt   Fine                                     Extreme seasons


Star temperature   Not enough blue light for                Too much UV light
                   photosynthesis

Star metallicity   Small planets                            Too many gas giants

Star mass          Planet is so close to the star that it   Fine
                   is tidally locked to the star

Moon mass          Planet tilt becomes unstable, causing    Fine
                   extreme seasons

A moon of a gas giant can potentially be protected from the solar wind by the gas giant's magnetic field. It can also potentially have volcanism from tidal heating by the gas giant.
Mass extinctions

The Earth has been beset by asteroids, supervolcanoes, global ice ages, runaway global warming, supernovae, gamma ray bursts, and the industrial age.

Millions of
years ago

   66          Cretaceous–Paleogene extinction, caused by a 10 km asteroid.
               Dinosaurs become extinct.
  201          Triassic-Jurassic extinction.  Cause unknown.
  252          Permian-Triassic extinction.  Runaway global warming
  370          Late-Devonian extinction.  Cause unknown.
  445          Ordovician-Silurian extinction events.  Global glaciation.

Microwave background radiation from the big bang


Density of the universe
                 kg/m3
Planck density   5   *1096   =    PlanckMass / PlanckLength^3
Solar system     2   *10-8   =    Mass of sun / (30 AU)^3
Milky Way        3   *10-21  =    1.2e12 solar masses / (100000 lightyears)^3
Matter            .12*10-27  =    Mean density of protons & electrons in the universe
Dark matter       .66*10-27  =    Mean density of dark matter in the universe
Dark energy      1.67*10-27  =    Mean density of dark energy in the universe
As the universe expands the matter and dark matter density decrease and the dark energy density is constant.

In the early universe the dark matter density was vastly greater than the dark energy density. In the future dark energy will overwhelm dark matter and the universe will expand unchecked.


Telescopes

Galileo's telescope
Replica of Newton's telescope
Herschel 1.2 meter reflector
Herschel 1.2 meter reflector
Yerkes 1 meter refractor

Hale 1.5 meter
Hooker 2.5 meter
Palomar 5 meter
Palomar
Keck 10 meter

Keck 10 meter
Hubble 2.4 meter
Webb 5 meter
30 Meter Telescope

Telescope            Diameter  Resolution  Year
                     (meters)  (microrad)

Human eye                .005    291
Lippershey's telescope    ?        ?       1608   First telescope.  Refractor
Galileo's telescope #1   .015     34       1609   Refractor
Galileo's telescope #2   .038     10       1620   Refractor
Newton's telescope       .033     12       1668   First reflecting telescope
Herschel telescope      1.20       2.5     1789   Reflector
Yerkes refractor        1.02       2.5     1897   Refractor.  End of refractor age
Hale 60-inch            1.52       2.5     1908   Mount Wilson observatory
Hooker 100-inch         2.54       2.5     1917   Mount Wilson observatory
Hale 200-inch           5.08       2.5     1948   Palomar Observatory
Keck                   10          2.5     1993   Mauna Kea Observatory
Hubble                  2.4         .20    1990   Earth orbit
Webb Space Telescope    6.5         .10       ?   L2 Lagrange point
Thirty Meter Telescope 30           .20       ?   Mauna Kea Observatory

Refracting telescope

Focus
Focus
Refracting telescope

Refraction depends on wavelength, which introduces "chromatic aberration". This limits the size of refracting telescopes. Reflecting telescopes don't suffer this limitation.


Reflecting telescope

Parabolic mirror
Gregorian
Newtonian
Cassegrain
Ritchey-Chretien

1608  Lippershey constructs the first refracting telescope
1663  Gregory publishes a design for a "Gregorian" reflector
1668  Newton constructs the first reflecting telescope, a "Newtonian" reflector
1672  Cassegrain publishes a design for a "Cassegrain" reflector
1910  Ritchey-Chretien reflecting telescope, the basis for modern reflectors

Visual resolution

A person with 20/20 vision can distinguish parallel lines that are spaced by an angle of .0003 radians, about 3 times the diffraction limit. Text can be resolved down to an angle of .0015 radians.

   
        Resolution     Resolution    Diopters
        for parallel   for letters   (meters-1)
        lines          (radians)
        (radians)
20/20     .0003         .0015          0
20/40     .0006         .0030         -1
20/80     .0012         .0060         -2
20/150    .0022         .011          -3
20/300    .0045         .025          -4
20/400    .0060         .030          -5
20/500    .0075         .038          -6
"Diopters" is a measure of the lens strength required to correct vision to 20/20.

If you are looking at a screen that is .2 meters from your eyes, the minimum resolvable pixel size is

Pixel size  =  .0003 * .2  =  .00006 meters  =  .06 mm

Diffration

Aperture size = Wavelength
Aperture size = 5 * Wavelength

The resolution of a telescope is limited by diffraction.

Wavelength of light  =  L
Mirror diameter      =  D
Resolution angle     =  θ  =  1.22 * L / D
If we assume blue light with L=440 nm,
                     D        θ

Eye                 .005   .00011
10 cm telescope     .1     .0000054
Hubble telescope   2.5     .00000021

1 Degree    =  60 arcminutes  =  3600 arcseconds
1 arcsecond =  4.8e-6 radians

Twinkling stars

Moonlight distorted by the atmosphere
Mirage caused by refraction
Distortion of a wavefront by the atmosphere

A star without atmospheric distortion
A star with atmospheric distortion

The atmosphere blurs light from outer space, limiting telescopes to a resolution of 5e-6 radians or 1 arcsecond. This is the resolution of a 10 cm telescope. A telescope larger than 10 cm has the same resolution as a 10 cm telescope. The advantage of the larger telescope is more light.

Telescope     Resolution           Reason for
diameter      (radians)            resolution limit
(meters)

  < .1        5e-7 / Diameter      Diffraction
  > .1        5e-6                 Atmosphere
A space telescope doesn't experience atmospheric distoration and the limit is from diffraction only. The 2.5 meter Hubble space telescope has a better resolution than the 10 meter Keck Earth telescope.

Telescopes equipped with "adaptive optics" can correct atmospheric distortion and reach a resolution better than 5e-6 radians.


Sensitivity

Astronomers measure brightness in a goofy unit called "magnitudes". It was defined by Hipparcos in Ancient Greece and it's still with us.

The "Apparent luminosity" of a star is its brightness as viewed from the Earth.
The "Absolute luminosity" is the power generated by a star in Watts.

R  =  Distance to star
L  =  Luminosity in Watts
l  =  Flux in Watts/meter2
M  =  Absolute magnitude
m  =  Apparent magnitude              

Flux corresponds to the brightness of an object as viewed from the Earth.

l  =  L / (4 π R2)

L  =  3.29e28 * 10-M/2.5  Watts

l  =  2.16e-8 * 10-m/2.5  Watts/meter2

M  =  m + 87.71 - 2.17 log(R)
The fainter an object, the larger its apparent magnitude "m".

Nearby stars, with color, size, and brightness to scale. Overlapping stars are binary stars.

                  Distance  Luminosity  Flux        Absolute   Apparent
                   from       Watts     Watts/m2    magnitude  magnitude
                   Earth       (L)         (l)         (M)        (m)

Sun                1    AU   3.8e26      1360        4.8        -26.7
Full Moon        .00257 AU   4.8e15      2.6e-3     32.1        -12.7
Mars                .52 AU   2.3e16      3.1e-7     30.4         -2.9
Jupiter            4.2  AU   1.5e18      3.1e-7     25.8         -2.9
Saturn             8.5  AU   6.8e17      3.4e-8     26.7          -.5
Uranus            18.2  AU   1.5e16      1.6e-10    30.9          5.3    Discovered 1781
Ceres              1.77 AU   4.3e13      4.9e-11    37.2          6.6    Discovered 1801
Neptune           29.1  AU   3.8e15      1.6e-11    32.3          7.8    Discovered 1846
Pluto             28.7  AU   1.8e13      7.8e-14    38.2         13.6    Discovered 1930
Alpha Centauri A   4.36 ly   5.9e26      2.7e-8     4.4           0.0
WISE-0855          7.2  ly   3.4e21      5.9e-14   17.5          13.9    Rogue planet
Sirius             8.58 ly   9.8e27      1.2e-7     1.4          -1.5    Brightest star
Exo-Sun           10.0  ly   3.8e26      3.4e-9     4.8           2.0
Exo-Earth         10.0  ly   5.4e16      4.8e-19   29.5          26.6
Betelgeuse       640    ly   5.8e31      1.3e-7    -6.0            .4    Massive star
Andromeda    2560000    ly   9.9e36      1.3e-9   -21.6           4.2
Values for solar system objects are for when they are closest to the Earth.

Uranus is at the limit of human vision. It's conceivable that an ancient civilization could have detected Uranus. Ceres is just beyond human vision.

WISE-0855 is a brown dwarf with a mass somewhere between 3 and 10 Jupiter masses. It has a temperature of 240 Kelvin and was detected by the WISE infrared space telescope.

"Exo-Sun" and "Exo-Earth" are values for if the sun and the Earth are at a distance of 10 light years.

GRB 080319B is the most luminous recorded gamma ray burst.

The horizontal lines represent the limits of the Pan-STARRS, Keck, Hubble, and Webb telescopes. Exo-Earth is detectable by the Keck telescope.


Limit of telescopes
                  Flux        Apparent
                  limit       magnitude
                 Watts/m2      limit

Human eye          3.4e-11       7
Pan-STARRS         5e-18        24
Keck 10 meter      1e-19        28
Hubble             1e-20        31
Webb               5e-22        34

Atmospheric transmission

              Observable from the ground?
Gamma ray     No
X ray         No
Ultraviolet   No
Visible       Yes
Infrared      No
Millimeter    Yes, if the air is dry
Radio         Yes

Wavelength range of telescopes

Wavelength in meters

The SOFIA tel escope can see in the infrared because it flies on a Boeing 747 at an altutude of 12 km, which is above most of the atmosphere's water.


Radio telescopes

Square Kilometer Array
Arecibo
Green Bank

Square Kilometer Array
Square Kilometer Array
Square Kilometer Array, artist's conception

                                       Min & Max
                      Aperture  Year   wavelength
                         (m)            (meters)
Arecibo                  300    1963   .03    1.0  Puerto Rico
Green Bank Telescope     100    2000   .0026  3.0  Green Bank, West Virginia. Largest steerable dish
Astron                    10    2011   .01    1.0  High Earth orbit. Used for large-baseline interferometry
Square kilometer Array  1000    2019   .01    4.3  Australia and South Africa

Millimeter telescopes
                       Min & Max
      Aperture   Year  wavelength
      (meters)           (mm)
CSO      10.5    1986  .3    2.0  Mauna Kea                 Caltech
Maxwell  15      1987  .3    2.0  Mauna Kea
ALMA     12      2011  .3    9.6  Atacama Desert, Chile     54 12-meter dishes and 12 7-meter dishes
LMT      50      2011  .85   4.0  Sierra Negra, Mexico      Large millimeter Telescope
CCAT     25      2017             Cerro Chajnantor, Chile   Wavelength range similar to ALMA

CSO  = Caltech submillimeter observatory
CCAT = Cerro Chajnantor Atacama Telescope
ALMA = Atacama Large Millimeter Array

Space telescopes for the cosmic microwave background radiation
        Diameter  Resolution   Mass    Min    Max    Year    Location
          (m)      (urad)     (tons)   (mm)   (mm)

Planck      1.9    4100         .21     .3    11     2009    L2
WMAP        1.6   12000         .76     .32    1.30  2001    L2
COBE         .19               1.41                  1989    Geocentric

Infrared telescopes

The SOFIA telescope on board a Boeing 747

       Aperture  Year  Min    Max  Location
       (meters)        (mm)   (mm)

WMAP      1.6    2001  .003  .015  L2 Lagrange point     NASA   Observes cosmic microwave background
Spitzer    .85   2003  .003  .180  Sun orbit             NASA
WISE       .40   2009  .003  .025  Earth orbit           NASA
Herschel  3.5    2009  .055  .672  L2                    ESA
SOFIA     2.5    2010  .001  .655  Boeing 747 at an altitude of 12 km
SOFIA = Stratospheric Observatory for Infrared Astronomy
X ray telescopes
         Aperture  Resolution  Low   High   Focus   Mass   Year
           (m)      (urad)    (keV)  (keV)         (tons)

NuSTAR       .32       46       3      79  10.15    .17   2012  Geocentric  NASA
Hitomi soft                      .3    12   5.6    2.7    2016  Geocentric  JAXA
Hitomi hard                     5      80  12      2.7    2016  Geocentric  JAXA
INTEGRAL     .31     3500       3   10000          4.0    2002  Geocentric  ESA RKA NASA
Swift        .30                                    .61   2004  Geocentric  NASA GSFC
NuSTAR has a collecting area of 847 cm2 at 7 keV and a collecting area of 60 cm2 at 78 keV. The field of view is 12 arcminutes.

The Hitomi lost attitude control and went into an uncontrollable spin, destroying the telescope.


Gamma ray telescopes
X rays from the Vela pulsar
         Year    Low    High      Mass
                (MeV)   (MeV)    (tons)

Hitomi   2016      .06       .6    2.7    Geocentric     JAXA
Fermi    2008    30    300000      4.3    Lagrange #2    NASA

Shotgun telescopes
           Aperture Magnitude  Field of  Exposure   Full-sky      CCD      Year
           (meters)  limit      view     (seconds) survey time  (Gpixels)
                              (degrees)            (days)
Pan-STARRS   3.6      24.0       3.0        60         8          1.4      2010   Hawaii
LSST         8.4      24.5       3.5        15         2          3.2      2021   El Penon, Chile
The Pan-STARRS and LSST telescopes are designed to find solar system objects, which is why they use short exposures.
Photons
Frequency         =  F
Wavelength        =  W
Planck constant   =  h          =  4.1357⋅10-15 eV seconds
Speed of light    =  C  =  F W  =  2.9979⋅108   meters/second
Energy            =  E  =  h F
Aperture diameter =  D
Diffraction angle =  A  =  1.22 W / D


                Energy  Wavelength   Temperature
                 (eV)      (nm)       (Kelvin)

Gamma ray     124000          .01
X-Ray            124        10     290000
Bohr energy       13.6      91      32000
UV-Extreme min    12.4     100      29000
UV-C min           4.43    280      10350
UV-B min           3.94    315       9200
Human UV limit     3.10    400       7244
Violet             3.06    405       7155
Blue               2.79    445       6512
Cyan               2.58    480       6037
Green              2.33    532       5447
Yellow             2.10    589       4920
Orange             2.03    610       4750
Red                1.91    650       4458
Human IR limit     1.63    750       3864
Infrared            .12    104        290
Microwave           .0012  106          2.90       300 GHz
Radio                      109           .0029     300 MHz


1800   Herschel discovers infrared light by its effect on a thermometer
1801   Ritter discovers UV rays by their effect on AgCl
1835   Melloni builds a thermoelectric infrared detector
1878   UV rays are found to kill bacteria
1879   Stefan-Boltzmann law:  Power = Constant * Area * Temperature^4
1901   Planck hypothesizes that E=hF
1905   Einstein discovers the photoelectric effect
1960   UV rays found to be harmful to DNA

Antarctic telescopes

The Antarctic plateau

"Ridge A" in Antarctica is a 4 km high plateau where the environment is cold, dry, and has no wind, making it the best place in the world for a telescope. Resolution is up to 3 times higher than what can be achieved by telescopes at the equator. It is also ideal for submillimeter astronomy, which requires cold dry air.

Properties of Ridge A:
Altitude                     = 4053 meters.
Distance from the South Pole = 1000 km.
Distance from Dome A         = 144 km.     Dome A is the highest ice feature in
                                           Antarctica, with an altitude of 4091 meters.
Annual snowfall              = 2 cm.
Average temperature          = -70 Celsius.

Earth Lagrange point telescope farm

The Webb telescope will operate from the Earth L2 Lagrange point, an ideal place for telescopes. From there you can place a barrier that blocks out the sun, the Earth, and the moon all at the same time, allowing the telescope to reach a low temperature. This is essential for infrared astronomy, which is used to detect exoplanets and distant galaxies.

A manned base station at L2 can be used to contruct large space telescopes. Segmented mirrors are launched from Earth and assembled on site.

A manned base station needs 4 tons/meter2 of ice for radiation shielding, which can be obtained from the moon. It also needs to have both a zero-gravity module for telescope construction and an Earth-gravity module for habitation.

Assuming a launch cost/kg of 1000 $/kg and a cost of 1 billion dollars, 1000 tons of material can be launched. Each telescope will have a mass in this range, with most of the mass being mirrors.

In the following table, the first section lists the largest existing space telescopes and the second section lists the kinds of telescopes that could be constructed at L2.

          Diameter  Resolution  Flux    Focus  Mass     Min       Max
            (m)      (urad)     (W/m2)   (m)  (tons)    (um)      (um)

Spektr-R     10                            4.22   2.5   13000     930000     Geocentric
Planck        1.9   4100                           .21    300      11000     L2
Herschel      3.5                         28.5     .32     55        672     L2
Webb          6.5       .10      5e-22   131.4    6.5         .6       28.5  L2
Gaia          1.45      .000034                    .71                       L2
NuStar         .32    46                  10.2     .17    3 keV    79 keV    Geocentric
Fermi                                       -     4.3    30 MeV   300 GeV    L2

Radio       1000                       10000   1000                          L2
Millimeter   200                        2000   1000                          L2
Infrared     100                        2000   1000                          L2
Visible + UV 100                        2000   1000                          L2
X              5                        1000   1000                          L2
Gamma          5                           -   1000                          L2
Astrometry
Radio: Metal can be mined from the moon and used for a radio telescope. Radio telescope mirrors can be manufactured in space whereas all other mirrors have to be manufactured on the Earth. Multiple radio telescopes can be deployed at various points in the solar system for large baseline interferometry.

Millimeter: Useful for measuring the cosmic microwave background. Occultation can be used to reach high resolution.

Infrared: Use a segmented mirror coated with gold. This telescope is optimized for finding high-redshift objects and exoplanets.

Visible+UV: Use a segmented mirror coated with aluminum.

X-Ray: An extremely long focus length can be used, which is helpful because X-rays can only be deflected by small angles.

Gamma Ray: Deploy 4 high-mass telescopes separated by thousands of kilometers. The spatial resolution allows precision measurement of the source direction using time-of-arrival of gamma bursts.


Cosmic rays


Gravity wave telescopes

         Year   Range (Hertz)
LIGO     2002   300 - 7000          Washington
LISA    ~2025   .00003 - 0.001      Sun orbit

Crowd computing

Projects such as SETI, Kepler, LIGO, and Pan-STARRS require massive data and computation and are well suited to crowd computing.

                    TFlops

Folding@home       39700     Protein folding
GPUGRID.net         2570     Simulations of proteins
Milkyway@home       1175     Construct a 3D map of the Milky Way
Einstein@home       1138     Search for pulsars
Collatz conjecture   623
SETI@home            594     Search for alien radio signals
Asteroids@home       135
theSkyNet             46     Analyze data from the Square Kilometer Array

Neutrino telescopes

                                  Neutrino   Min   Volume    Source   Year
                                    type    energy
                                            (GeV)
Super-Kamiokande                    E M T            .00005  S A G  1996       Pure water
Cubic Kilometer Neutrino Telescope                  5                          Mediterranean
IceCube Neutrino Observatory                100                            1 km3 of Antarctic ice
India-Based Neutrino Observatory
ANNIE                                          .8            S A G           Neutrinos from Fermilab

Proton decay

Grand unification theories conjecture that protons can decay.

Proton  ->  Positron  +  Pion0                      The Pion0 has a charge of 0
Proton  ->  Kaon      +  Neutrino                   The Kaon has a charge of +1
Proton decay has thus far not been observed. The best limit on proton decay comes from the Super-Kamiokande experiment.
Lower limit on the proton halflife   =   1e34 years
Protons in 1 kg of hydrogen          = 6.0e26
Mass of protons for 1 decay/year     =  17000 tons

Future

Telescope size in meters


History of physics

Cosmology
-240     Eratosthenes measures the Earth's circumference to 20% error.
-240     Aristarchus proves that the sun is at least 10 times larger than the Earth
         using lunar eclipses.
 150     Ptolemy publishes The Almagest with the geocentric model
 550     Aryabhata publishes accurate measurements of size of the sun and moon
1543     Copernicus publishes a heliocentric model.
1600     Brahe measures accurate planet positions
1608     Lippershey invents the telescope
1609     Galileo builds a telescope and begins observing
1609     Kepler proves that planets orbit as ellipses using Brahe's data
1613     Galileo publishes observations of the phases of Venus, which support
         the heliocentric model
1632     Galileo publishes the "Dialogue Concerning the Two Chief World Systems",
         which contained a comparison of the systems of Ptolemy and Copernicus
1672     Richter and Cassini measure the parallax of Mars, producing a precise
         value for the size of the sun
1687     Newton publishes the Principia Mathematica, which contained the calculus,
         the laws of motion (F=MA), and a proof that planets orbit as ellipses.
1718     Halley finds that the stars move.  He found that Sirius, Arcturus and
         Aldebaran were 1/2 of a degree from the positions charted by the Ancient
         Greek astronomer Hipparchus
1783     Herschel finds that the solar system is moving with respect to the stars
1826     Olbers' paradox.  If the stars in the universe are uniformly distributed
         and if the universe is infinite, then the sky would appear infinitely bright
         with stars.
1863     Bessel measures the first stellar parallax, showing that the
         stars are more than 4 light years away.  This also implies that stars
         are as luminous as the sun.
         The parallax of stars is too small to see without a telescope.
1905     Einstein publishes special relativity
1915     Einstein publishes the general theory of relativity.
         Einstein shows that general relativity is consistent with the existence
         of a cosmological constant.  At the time the cosmological constant was a
         proposed explanation for why the universe hasn't collapsed gravitationally.
1920     Shapley finds that the sun is not at the center of the galaxy.
         Because starlight is absorbed by interstellar gas, we only see the nearby
         stars and it appears as though we live at the center of a disk of stars.
         Shapley measured the distances to globular clusters and found that they
         are centered on a point (the galactic center) that is far from the sun.
1922     Friedmann finds a solution to the equations of general relativity that
         are consistent with an expanding universe.
1923     Hubble measures the distances to Andromeda and Triangulum and finds that
         they are outside the Milky Way.  These were the first objects to be shown
         to be outside the Milky Way.
1929     Hubble's law published.  For distant galaxies, the recession velocity is
         proportional to distance.
1933     Zwicky's analysis of the Coma cluster of galaxies shows that they contain
         unseen matter that is not due to stars.
1965     Penzias and Wilson discover the cosmic microwave background radiation.
1970     Rubin and Ford measure galactic rotation and show that galaxies contain
         matter that is not due to stars.
1980     Guth and Starobinsky propose the theory of inflation to explain why the
         universe is flat
1998     Observations of supernovae show that the expansion of the universe is
         accelerating and the the cosmological constant is positive.
         Previous to this it was not known if the universe was destined for
         collapse (big crunch) or for infinite expansion (big chill).
2003     WMAP mission measures the Hubble constant to 5% precision, as well as
         other cosmological parameters.
         Previous to this, the Hubble constant had an error of ~ 20%.
         This settled once and for all the question of the overall structure of the
         universe.

Measurement of the distance to the sun

-260 Aristarchus established that the distance to the sun is at least 20 times the distance to the moon.

In 499, Aryabhata publishes a measurement of the distance to the sun.

Brahe's data consisted of measurements of angles between different objects. This data could be used to establish the shape of orbits but not their size. For example, if the size of the solar system were doubled along with the speeds of the planets, the angles would stay the same and you wouldn't be able to tell the difference.

In 1639, Horrocks used a transit of Venus to measure the distance to the sun, but this method is incapable of giving an accurate value, and it can only be done once per century.

In 1672, Richter and Cassini measured the parallax of Mars which gives a result for the distance to the sun that is more accurate than the Venus method. The Mars method has an advantage over the Venus method in that it can be done once every 26 months, when Mars is at closest approach.

In 1676, Romer used the moons of Jupiter to measure the time it takes for light to cross the Earth's orbit. This gives a value for R/C.

In 1729, Bradley measured the deflection of starlight due to the Earth's motion, which gives a measurement of V/C, or equivalently, a measurement of R/C.

In 1849, Fizeau produced the first measurement for the speed of light that was independent of the Earth-sun distance R.

Speed of light                           =  C
Earth-sun distance                       =  R
Earth orbital velocity                   =  V
Earth orbital time (1 year)              =  T  =  2 π R / V
Time for light to cross the Earth's orbit=  t  =  2 R / C

Energy

The unit of energy used for atoms, nuclei, and particle is the "electron Volt", which is the energy gained by an electron upon descending a potential of 1 Volt.

Electron Volt (eV)  =  1  eV  =  1.602e-19 Joules
Kilo electron Volt  =  1 keV  =  103 eV
Mega electron Volt  =  1 MeV  =  106 eV
Giga electron Vlt   =  1 GeV  =  109 eV

Fusion

Fusion of hydrogen into helium in the sun

Proton + Proton  ->  Deuterium + Positron + Neutrino
Hydrogen fusion requires a temperature of at least 4 million Kelvin, which requires an object with at least 0.08 solar masses. This is the minimum mass to be a star. The reactions in the fusion of hydrogen to helium are:
P    + P    -->  D    +  Positron + Neutrino +   .42 MeV
P    + D    -->  He3  +  Photon              +  5.49 MeV
He3  + He3  -->  He4  +  P   +  P            + 12.86 MeV

Helium fusion

As the core of a star star runs out of hydrogen it contracts and heats, and helium fusion begins when the temperature reaches 10 million Kelvin.


Heavy element fusion

A heavy star continues to fuse elements until it reaches Iron-56. Beyond this, fusion absorbs energy rather than releasing it, triggering a runaway core collapse that fuses elements up to Uranium. If the star explodes as a supernova then these elements are ejected into interstellar space.


Asteroid mining


Platinum group metals

Metallic asteroids are rich in platinum group metals, all of which are worth more than $5000 per kilogram. A typical metallic asteroid with a diameter of 100 meters has a mass of 1 billion kilograms and contains 19 tons of platinum with a value of 1 billion dollars. For a typical metallic asteroid,

         Mass in    Element   Value of element
         asteroid   cost/kg   in the asteroid
          (tons)    ($/kg)       (B$)

Platinum      19     55000    1.04
Nickel     67000        15    1.00
Rhodium        4.1   90000     .37
Iron      910000          .3   .27
Cobalt      6300        30     .19
Gold           1.8   60000     .108
Osmium         7.6   12000     .091
Germanium     37      2000     .074
Ruthenium     11      5500     .060
Palladium      3.8   14000     .053
Iridium        3.0   14000     .042
Gallium       80       280     .022
Zirconium      8      1600     .013
Rhenium         .85   5000     .004
The most profitable elements are platinum, nickel, rhodium, iron, cobalt, and gold.
Sudbury basin, Canada

Sudbury, Canada
Sudbury geology

The Sudbury basin mine in Canada is a meteor crater from as 12 km metallic meteor that struck 1.8 billion years ago. The Earth's crust is poor in these elements because they sink to the core and platinum mines tend to be at sites of metallic asteroid impacts.

The asteroid belt was formed from a planet that was shattered by collisions. The asteroid belt comsists of mostly rocky asteroids and some metallic asteroids.

Platinum fraction in early universe    =   .005 parts/million
Platinum fraction in the sun           =   .009 parts/million
Platinum fraction in the Earth's crust =   .004 parts/million
Platinum fraction in Sudbury mine ore  =   .5   parts/million
Platinum fraction in iron asteroid     = 19     parts/million
Annual platinum production             =500     tons/year
Platinum used in mufflers              =130     tons/year

Space still

Alcohol still
Laphroaig distillery

The valuable elements in a metallic asteroid tend to have high boiling points. These elements can be isolated by heating the asteroid to 3200 Kelvin to boil off the low-value iron, nickel, and cobalt. After boiling, the asteroid has 1/5000 of its original size, has a value of 10 thousand dollars per kilogram, and is easily transported to the Earth.

Heat can be obtained by focusing sunlight with mirrors. The elements ranked by boiling point are:

          Density Melt  Boil   $/kg   ppm in metallic
          g/cm3    K     K               asteroid

Rhenium    21.0   3459  5869   4600         .85
Tungsten   19.25  3693  5828     50        8.1
Tantalum   16.7   3290  5731    400         .06
Osmium     22.59  3306  5285  12000        7.6
Thorium    11.7   2115  5061     25         .04
Niobium     8.75  2750  5017     40         .2
Molybdenum 10.28  2896  4912     21        7.3
Hafnium    13.31  2506  4876    500        0
Iridium    22.4   2739  4701  14000        3.0
Zirconium   6.52  2128  4682     20        8
Ruthenium  12.45  2607  4423   5500       11
Uranium    19.1   1405  4404     75         .007
Platinum   21.45  2041  4098  55000       19
Rhodium    12.41  2237  3968  90000        4.1
Vanadium    6.0   2183  3680     12        6
Titanium    4.51  1941  3560     10      100
Palladium  12.02  1828  3236  13600        3.8
Cobalt      8.90  1768  3200     30     6300
Nickel      8.91  1728  3186     15    67000
Iron        7.87  1811  3134       .3 910000

Metallic core

Moon
Titan
Ceres
Pluto


Element classification

Siderophile:  Iron-living. Tends to sink to the core along with the iron.
Lithophile:   Rock-loving. Tends to become included in rock and escapes sinking
              to the core.
Chalcophile:  Ore-loving. Tends to combine with oxygen and sulfur and escapes sinking
              to the core.
Atmophile:    Is a gas at room temperature and tends to escape the crust into the
              atmosphere.
In the early solar system, a small planet formed in the region that is now the asteroid belt. The planet had a hot interior and there was enough time for the dense elements to sink to the core. Then the planet was shattered by collisions and became the present-day asteroid belt. Pieces of the planet that are from the core are now metallic asteroids.
Universe composition

Compositions are listed as parts per million by mass.

       Big Bang        Sun       Earth      Iron        Core
                                 crust    asteroid  Amplification

Hydrogen  750000      750000        1500        0        0
Helium    230000      230000                    0        0
Oxygen     10000        9000      460000        0        0
Carbon      5000        3000        1800     1100         .042
Iron        1100        1000       63000   910000        1
Silicon      700         900      270000       40         .000010
Tungsten        .0005       .004       1.1      8.1       .51
Platinum        .005        .009        .004   19      329
Gold            .0006       .001        .003    1.8     42
"Core amplification" is the degree to which the element is concentrated in the core, normalized so that the core amplification of iron is 1.
Core amplification

Platinum is more dense than iron and is hence more likely to sink to the Earth's core than iron. This is reflected in the "core amplification factor".

Platinum abundance in the crust =  cPt  =      .004 ppm
Iron abundance in the crust     =  cFe  = 63000     ppm
Platinum abundance in the core  =  CPt  =    19     ppm
Iron abundance in the core      =  CFe  =910000     ppm
Crust platinum/iron             =  cPt/cFe  =  .000000063
Core  platinum/iron             =  CPt/CFe  =  .000021
Core amplification factor       =  APt  =  (CPt/CFe) / (cPtcFe)  =  329
The elements with the highest core amplification factors are:
        Amplification   Density (g/cm3)

Ruthenium   762        12.4     Siderophile, Platinum group
Platinum    329        21.4     Siderophile, Platinum group
Rhodium     284        12.4     Siderophile, Platinum group
Osmium      263        22.6     Siderophile, Platinum group
Iridium     208        22.4     Siderophile, Platinum group
Nickel       52         8.9     Siderophile
Palladium    44        12.0     Siderophile, Platinum group
Gold         42        19.3     Siderophile
Rhenium      20        21.0     Siderophile
Cobalt       14.5       8.9     Siderophile
Selenium      4.2       4.81    Chalcophile
Germanium     1.8       5.32    Chalcophile
Bismuth       1.4       9.78    Chalcophile
Iron          1.0       7.9     Siderophile
Tungsten       .51     19.25    Lithophile
Molybdenum     .46     10.28    Siderophile
Mercury        .42     13.53    Chalcophile
Lead           .42     11.34    Chalcophile
Arsenic        .36      5.73    Chalcophile
Gallium        .29      5.91    Chalcophile
Copper         .13      8.96    Chalcophile
Tin            .063     7.26    Chalcophile
Silver         .030    10.49    Chalcophile
Zinc           .025     7.14    Chalcophile
Thorium        .00046  11.7     Lithophile
Uranium        .00027  19.1     Lithophile
Platinum group metals are the most likely to sink to the core. Almost all of the uranium and thorium resists sinking to the core, which is why nuclear energy is cheap.
Metallic asteroids

2011 UW158

"16 Psyche" is the largest metallic asteroid and is likely the core of a failed planet that had its mantle stripped away by collisions.

         Diameter   Perihelion   SemiMajor   Value    Value
            km          AU       Axis (AU)   (B$)     ($/kg)

16 Psyche   186         2.51       2.92     5⋅108      .02
Nereus         .33       .95       1.49        5      .033
Ryugu          .98       .96       1.19       95      .024
2011 UW158     .45      1.01       1.62        8      .021
Didymos        .8       1.01       1.64       84      .039
1989 ML        .6       1.10       1.27       14      .015
1992 TC       1.1       1.11       1.57       84      .015
The estimated value of each asteroid is from
Wikipedia. Values are given for all asteroids except Psyche. In the table, we assume a price/mass for Psyche of .02 $/kg which leads to a value of 500000 trillion dollars, far larger than the Earth's annual gross domestic product of 75 trillion dollars.
Asteroid delivery

If a 1010 kg asteroid is broken with a hydrogen bomb then

Mass of the asteroid            =  M  =  1010 kg
Energy of the hydrogen bomb     =  E  =  .5 M V2  =  4e16 Joules   (10 megatons of TNT)
Speed of the asteroid fragments =  V  =  3 km/s
A hydrogen bomb is capable of moving an asteroid.
Vaporizing a metallic asteroid

A space mirror can be used to vaporize a metallic asteroid, which leaves behind elements with high boiling points. These elements are the valuable ones. We assume that the asteroid is mostly iron.

Asteroid original temperature   =  220 Kelvin
Asteroid boiling temperature    = 3200 Kelvin      (Boiling point of cobalt)
Melting energy of 1 kg of iron  =  272 kJoules
Boiling energy of 1 kg of iron  = 6090 kJoules
Iron solid heat capacity        = .449 kJoules/kg/Kelvin
Iron liquid heat capacity       = .82  kJoules/kg/Kelvin
Iron solid temperature change   = 1591 Kelvin      (From  220 Kelvin to 1811 Kelvin)
Iron liquid temperature change  = 1323 Kelvin      (From 1811 Kelvin to 3134 Kelvin)
Iron solid heating energy       =  714 kJoules     (Energy to heat from  220 to 1811 Kelvin)
Iron liquid heating energy      = 1085 kJoules     (Energy to heat from 1811 to 3134 Kelvin)
Iron total heating energy       = 8161 kJoules     (Energy to heat, melt, and vaporize)
Energy to vaporize iron asteroid=  8.2⋅1015 Joules  (Assume a mass of 1 billion kg)
Power from 10 km3 space mirror  =  1.3⋅1011 Watts
Time to vaporize asteroid       =60000 seconds  =  17 hours
Mass of a 10 km3 space mirror   =  600 tons
Vaporization accounts for most of the energy requirement.

A 10 km3 space mirror can vaporize a 1 billion kg asteroid on 1 day.

The exiting vapor can be passed through tungsten pipes heated to a temperature of 3250 Kelvin, just above the temperature of the vapor. Any platinum group metals in the vapor will condense onto the pipes. Tungsten is used because it is the element with the highest melting point (3695 Kelvin).


Economic impact

Some of the metals in a metallic asteroid have the potential to overwhelm Earth production and reduce the price of the metal, opening new technological applications. These metals are osmium, ruthenium, iridium, rhodium, and platinum. For a 1 billion kg asteroid,

         Mass in   Annual Earth   Mass in asteroid
         asteroid     mining      / Annual Earth mining
          (tons)      (tons)

Osmium         7.6          1        7.6
Ruthenium     11           12         .92
Germanium     37          118         .31
Gallium       80          273         .29
Iridium        3.0         12         .25
Rhodium        4.1         30         .14
Platinum      19          245         .08
Cobalt      6300       110000         .057
Nickel     67000      2100000         .032
Rhenium         .85        50         .017
Palladium      3.8        250         .015
Gold           1.8       2800         .0006   Earth gold won't be eclipsed by asteroid gold
Iron      910000   1700000000         .0005
Zirconium      8       900000         .000009
Bringing a trillion kg asteroid to the Earth would satisfy the world's iron, nickel, and cobalt demand and then we wouldn't need to use energy for smelting.
Platinum production
           tons/year

World         161
South Africa  110
Russia         25
Zimbabwe       11
Canada          7.2
USA             3.7
Other           3.8
Data for 2014
Cosmic element abundance

Compositions are listed as parts per million by mass.

       Big Bang   Milky Way   Sun       Earth      Iron    Amplification
                                        crust    asteroid

Hydrogen  750000   739000   750000        1500        0        0
Helium    230000   240000   230000                    0        0
Oxygen     10000    10400     9000      460000        0        0
Carbon      5000     4600     3000        1800     1100         .042
Neon        1300     1340     1000                    0        0
Iron        1100     1090     1000       63000   910000        1
Nitrogen    1000      960     1000          20       33         .11
Silicon      700      650      900      270000       40         .000010
Magnesium    600      580      700       29000      320         .00076
Sulfur       500      440      400         420      360         .059
Calcium                                  50000      500         .00069
Potassium                                15000        0        0
Aluminum      50                60       82000       40         .000034
Sodium        20                40       23000        0        0
Phosphorus     7                 7        1000     2200         .15
Beryllium       .001              .0001      1.9      0        0
Lithium         .006              .0001     17        0        0
Boron           .001              .002       8.7      0        0
Fluorine        .4                .5       540        0        0
Chlorine       1                 8         170        0        0
Argon           .04             70                    0        0
Scandium        .03               .04       26        0        0
Titanium       3                 4        6600      100         .0010
Vanadium       1                  .4       190        6         .0022
Chromium      15                20         140       15         .0074
Manganese      8                10        1100      300         .019
Iron        1100     1090     1000       63000   910000        1
Cobalt         3                 4          30     6300       14.5
Nickel        60                80          90    67000       52
Copper          .06               .7        68      130         .13
Zinc            .3               2          79       28         .025
Gallium         .01               .04       19       80         .29
Germanium       .2                .2         1.4     37        1.8
Krypton         .04                                   0        0
Strontium                                  360        0        0
Ytterbium       .007              .01        2.8      0        0
Zirconium       .05               .04      130        8         .0043
Niobium         .002              .004      17         .2       .00081
Molybdenum      .005              .009       1.1      7.3       .46
Ruthenium       .004              .005        .001   11      762
Rhodium         .0006             .002        .001    4.1    284
Palladium       .002              .003        .006    3.8     44
Silver          .0006             .001        .08      .035     .030
Cadmium         .002              .006        .15      .02      .0092
Indium          .0003             .004        .16      .01      .0043
Tin             .004              .009       2.2      2         .063
Antimony        .0004             .001                 .34
Lutetium        .0001             .001                -
Hafnium         .0007             .001       3.3      0        0
Tantalum        .0001                        1.7       .06      .0024
Tungsten        .0005             .004       1.1      8.1       .51
Rhenium         .0002             .004        .003     .85    20
Osmium          .003              .002        .002    7.6    263
Iridium         .002              .002        .001    3.0    208
Platinum        .005              .009        .004   19      329
Gold            .0006             .001        .003    1.8     42
Mercury         .001              .02         .067    0         .42
Thallium        .0005             .001                0        0
Lead            .01               .01       10       60         .42
Bismuth         .0007             .01         .025     .5      1.4
Thorium         .0004             .0003      6         .04      .00046
Uranium         .0002             .001       1.8       .007     .00027
Neodymium                                   33
Lanthanum                                   34
Yttrium                                     29
Samarium                                     6
Cerium                                      60
Barium                                     340
Rubidium                                    60
Praseo                                       8.7
Gadolinium                                   5.2
Dysprosium                                   6.2
Erbium                                       3.0
Caesium                                      1.9
Europium                                     1.8
Arsenic                                      2.1     11         .36
Holmium                                      1.2
Terbium                                       .94
Thulium                                       .45
Bromine                                      3
Thallium                                      .53
Antimony                                      .2
Iodine                                        .49
Selenium                                      .05     3        4.2
Tellurium                                     .001

"Amplification" is the abundance of an element in a metallic asteroid divided by its abundance in the Earth's crust, with the value normalized so that it is "1" for iron.
Appendix

Calculating the Hill radius

In the limit of

Mass of Planet / Mass of Star   ->  0
the distance from the planet to L1 or L2 is the same and is called the "Hill radius" H.
G  =  Gravitational constant
M0 =  Mass of Star
M  =  Mass of Planet             R = Planet orbit radius          T = Planet orbit time
m  =  Mass of moon               r = Moon orbit radius            t = Moon orbit time
H  =  Planet Hill radius
To calculate the Hill radius, we set (without loss of generality)
G  =  M0  =  R  =  1
We assume that the star is vastly heavier than the planet and that the planet is vastly heavier than the moon:
m << M << 1      →        r << 1
In the limit M → 0, the distance from the planet to L1 or L2 is the same and is called the "Hill radius".

Suppose an object is orbiting at L2 with velocity U. Since this object orbits with the same angular velocity as the planet,

U  =  (1 + H)
The force balance on this object is
Force from star   +  Force from planet  =  Centripetal force

1/(1+H)2          +  M/H2               =  U2/(1+H)

1 - 2H            +  M/H2               =  1 + H

H = (M/3)1/3
Restoring units,
H = R (M/(3*M0))1/3

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