Atmospheres

```         Density     Pressure   Column     Atmos.    Escape           Atmos.
at surface  at surface  density    mass      speed   Gravity  Height  Temp  Spin
kg/m^3     10^5 N/m^2  tons/m^2  (Earth=1)  (km/s)  (m/s^2)   (km)   (K)  (days)

Venus     67.        94.        1038.      92.1      10.36    8.87      16    735  243.0
Titan      5.3        1.20       108.       1.46      2.64    1.35     130     94   15.9
Earth      1.2        1.01        10.2      1        11.2     9.78       8    287    1.00
Mars        .020       .0046        .17      .0063    5.03    3.71      11    210    1.03
Mars Hellas .037       .008         .30      .0063    5.03    3.71      11    240    1.03
```
"Air column mass" is the mass of air above a square meter of an object's surface.

"Mars Hellas" refers to the bottom of the Hellas Basin on Mars, which is 7 km below the mean surface.

```P  =  Pressure of atmosphere at the surface in Newtons/meter^2
g  =  Planet gravity at the surface in meters/s^2
C  =  Atmosphere column density in kg/meter^2
=  Mass of atmosphere above 1 meter^2 of the surface

P = C g
```
A column density of at least 4000 kg/meter^2 is required to block cosmic ray radiation. The Earth, Titan, and Venus qualify and Mars doesn't.

Atmospheric heat capacity

An ideal blackbody radiates power according to the Stefan-Boltzmann law:

```Radiation Power  =  Temperature  *  SurfaceArea^2  *  5.7*10^-8  Watts/K/m^2
```
Heat capacity of air = 1020 Joules/kg/Kelvin

For each of the above objects, how many days does it take for radiation to decrease the temperature of the air by 10 Kelvin?

Asteroid heating

Suppose an asteroid hits the Earth such that

```Velocity = 20 km/s
Diameter = 10 km
Density  =  2 g/cm^3
```
This is the size of the asteroid that caused the dinosaur extinction. If all of the kinetic energy of the asteroid goes into heating the atmosphere, by how many degrees does the temperature rise?

Atmospheric transmission
```A  =  Atmosphere thickness in kg/meter^2
a  =  Characteristic thickness in kg/meter^2
=  Thickness for which the transmitted fraction is 1/e
I  =  Intensity of light in space
T  =  Intensity of light that gets through the atmosphere and reaches the surface
```

The transmission of light through an atmosphere can be modeled as an exponential.

```T = I * exp(A/a)
```
Using data from the web, what is "a" for the Earth?

By what factor would you have to increase the Earth's atmospheric thickness to reduce the transmitted intensity by a factor of 2?

What would you predict is the transmitted intensity on Titan?

Nitrogen

Nitrogen is rare in the inner solar system. The abundance in the Earth's crust is 50 ppm by mass.

Let M be the mass of nitrogen in the Earth's crust down to a depth D. For what value of D is M equal to the mass of nitrogen in the atmosphere?

The abundance of Carbon in the Earth's crust is 300 ppm by mass. In Venus' crust, the temperature is hot enough to expel all carbon into the atmosphere. Suppose we assume the carbon content of Venus' crust was originally the same as the Earth's. Let M be the mass of carbon in Venus' original crust down to a depth D. For what value of D is M equal to the mass of carbon that is presently in the atmosphere?

Air drag

Suppose you want to estimate how far a soccer ball has to travel before losing half its velocity.

```M  =  Mass of the 2014 World Cup "Brazuka" ball  =  .437 kg
R  =  Ball radius                                =  .110 meters
D  =  Ball density                               =  78.4 kg/meters^3
A  =  Ball cross-sectional area                  =  .0380 meters^2
V  =  Ball initial velocity
d  =  Density of air                             =  1.2 kg/meter^3
F  =  Aerodynamic drag force                     =  .5 D A V^2
L  =  Characteristic distance the ball has
to travel to lose half its velocity
m  =  Mass of air that a ball passes through
after moving a distance L
=  A L d
```
Newton observed that the characteristic distance L is such that
```m = M

L = M / (A d)
= 9.6 meters
```
The depth of the penalty box is 16.45 meters (18 yards). Any shot taken outside the penalty box slows down substantially before reaching the goal.

Expressed in terms of densities,

```L = 4/3 R D / d
```
Newton was also the first to observe the "Magnus effect", where spin causes a ball to curve.

Balls
```           Diameter  Mass   Path  Court   Path/  Density
(mm)   (g)    (m)    (m)    Court  (g/cm^3)

Ping pong      40      2.7   1.8    2.74    .64   .081
Squash         40     24    15.6    9.75   1.60   .716
Golf           43     46    25.9  200       .13  1.10    Typical range for a driver
Badminton      54      5.1   1.8   13.4     .14   .062
Racquetball    57     40    12.8    9.8    1.3    .413
Billiards      59    163    48.7    2.7   18     1.52
Tennis         67     58    13.4   23.77    .56   .368
Baseball       74.5  146    27.3   19.4    1.4    .675   Distance from pitcher to batter
Whiffle        76     45     8.1                  .196
Football      178    420    13.8   20       .67   .142   Typical distance for a pass
Rugby         191    435    12.4   20       .62   .119   Typical distance for a pass
Bowling       217   7260   160     18.29   8.8   1.36
Soccer        220    432     9.3   16.5     .56   .078   Depth of penalty box
Basketball    239    624    11.4    7.24   1.57   .087   Distance to 3 point line
Cannonball    220  14000   945   1000       .94  7.9     Typical distance to enemy ship
```
"Path" is the Newton length and "Court" is the length of the court, unless otherwise specified. "Density" is the density of the ball.
Terraforming
```               Orbit      Mean      Parent
(A.U.)  Temperature  planet
(K)
Venus             .7      735
Earth            1.0      287
Mars             1.5      210
Ceres            2.8      168
Europa           5.2      102       Jupiter
Ganymede         5.2      110       Jupiter
Callisto         5.2      134       Jupiter
Titan            9.5       94       Saturn
Titania         19.2       70       Uranus
Oberon          19.2       75       Uranus
Nitrogen freeze            63
Triton          30.1       38       Neptune
Nereid          30.1       50       Neptune
Pluto           39.5       44
Hydrogen freeze            14

Triton surface ice composition:
Nitrogen  .55
H2O       .25
CO2       .15

```
Suppose we want to give Mars a nitrogen atmosphere by crashing a KBO into it.
```C = Column density of atmosphere in kg/meter^2
= Mass of atmosphere above a 1 meter^2 patch of the surface
g = Gravitational acceleration at the surface
P = Pressure in Newtons/meter^2
= C g
M = Total mass of a planet's atmosphere
= C * PlanetSurfaceArea

Earth pressure     =  10^5 Newtons/meter^2
Mars surface area  =  1.45*10^14 meters^2
```
If we assume the KBO is 1/10 nitrogen and that it has a density of 2 g/cm^3, what radius does it have to have to have enough nitrogen so that the atmospheric pressure it equal to the Earth's?

Solar wind ejection

Suppose a solar wind proton hits a target atom.

```Solar wind proton speed  =  800 km/s.
M  =  Mass of target atom / Mass of proton
```
What is the maximum possible recoil velocity of the target atom as a function of M?