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Hubble's Law

Cosmic history

Age in billions of years

0        Big bang.  The universe is a plasma.
.0003   The universe cools until the plasma becomes neutral matter.
The universe is a cold dark gas.
.15     Gravity collapses the gas and stars and galaxies begin forming.
1        Galaxies have formed, including the Milky Way.
9        Sun and Earth formed.
13        ~ 600 million years ago, oxygen became abundant in the Earth's atmosphere
and multicellular life emerged.  The tetrapod design emerged ~ 525 million
years ago.
13.7      Present
18        The Milky Way and Andromeda collide.
19        Sun becomes a red giant and consumes the Earth.
>20        The galaxies in the Local Group merge into one supergalaxy.
Dark energy overwhelmes gravity and causes a runaway expansion of
the universe and drives away all galaxies that haven't already
merged with the Milky Way.
1000        The last stars burn out.  The universe is dark hereafter.

Hubble's Law

If a galaxy is sufficiently distant (beyond 100 million light years), then its
speed with respect to the Milky Way is

Speed  =  Hubble_Constant * Distance           where the velocity is directed away from us.
Hubble_Constant = 23 km/s/megalightyear

Distance (million   Mass            Speed with respect   Speed from Hubble
light years)        (Solar masses)  to Milky Way (km/s)  law (km/s)
Edge of universe 13700                                 300000              300000
Coma Cluster       321                7*10^14            6950                7400
Virgo Cluster       54              1.2*10^15            1250                1240
Andromeda            2.54             3*10^12            -120                  58
Milky Way             .0248           3*10^12

Andromeda is near enough for gravity to have changed its velocity from the Hubble velocity.
The Virgo Cluster is too far.

Mass of sun     = 2.0e30 kg
Age of universe = 13.7 billion years
Speed of light  = 3.0e8 m/s

Escape velocity

M  =  Mass of object
R  =  Distance from object
G  =  Gravitational constant = 6.67*10^-11 m^3 / kg / seconds^2
m  =  Mass of object trying to escape
V  =  Escape velocity

Gravitational potential energy  =  Kinetic energy

G M m / R  =  1/2 m V^2

V = (2 G M / R)^.5

What is the escape velocity of the Milky Way with respect to Andromeda and the
Virgo cluster?

If Andromeda and the Milky are initially at rest at their present distance,
how long would it take for gravity to bring them together? How about the Virgo
cluster and the Milky Way?  To estimate this, assume a constant acceleration
equal to the acceleration when they are at their initial distance.

Given that the Hubble constant is equal to 23 km/s/megalightyear, estimate the
age of the universe. Assume that the universe has been expanding at a constant
rate since the big bang.

Suppose you are near a 1 solar mass blackhole and your feet are 2 meters closer to
the hole than your head. How close do you have to be to the hole for the difference