We will assume the universe is expanding at a constant rate. At t = t0Â another galaxy is a distance D0Â away from us as shown in the diagram. For the purposes of this exorcise it will be easier to think of the other galaxy being at 0 and we are moving away from it, with us being at location D0Â at t0.
1. If we assume the two galaxies were at the same location a t = 0, find an equation for the distance between the galaxy as a function of time, D(t).
2. Find the speed at which the galaxy are moving apart.
3. At t0Â light leaves the galaxy at 0 headed for us at a speed c. How long does it take the light to reach us? Do not forget to include the fact that we are moving away from the other galaxy, which means the light will have to travel farther than D0Â to reach us.
Solution
if we assume constant rate of expansion of the universe
then for a distance s between two galaxies
their rate of seperation is ds/dt = k ( where k is a constant)
given distance at t = to is s = Do
distance at t = 0 is s = 0
distance at t = t is s = D(t)
hence
ds/dt = k
s = kt + c ( where c is another constant)
hence
Do = k*to + c
0 = c
hence
k = Do/to
hence
s = Do*t/to
1. D(t) = Do*t/to
2. speed at which they are moving apart = v
v = ds/dt
v = d(D(t))/dt = Do/to
3. at to, light leaves galaxy at x = 0
it reaches us at time t = to + t
then
time taken for it to travel distace = t
distance = Do + v*t
hence
Do + v*t = c*t
hence
t = Do/(c - v) = Do/(c - Do/to)
.
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We will assume the universe is expanding at a constant rate- At t - t0.docx
1. We will assume the universe is expanding at a constant rate. At t = t0Â another galaxy is a
distance D0Â away from us as shown in the diagram. For the purposes of this exorcise it will be
easier to think of the other galaxy being at 0 and we are moving away from it, with us being at
location D0Â at t0.
1. If we assume the two galaxies were at the same location a t = 0, find an equation for the
distance between the galaxy as a function of time, D(t).
2. Find the speed at which the galaxy are moving apart.
3. At t0Â light leaves the galaxy at 0 headed for us at a speed c. How long does it take the light
to reach us? Do not forget to include the fact that we are moving away from the other galaxy,
which means the light will have to travel farther than D0Â to reach us.
Solution
if we assume constant rate of expansion of the universe
then for a distance s between two galaxies
their rate of seperation is ds/dt = k ( where k is a constant)
given distance at t = to is s = Do
distance at t = 0 is s = 0
distance at t = t is s = D(t)
hence
ds/dt = k
s = kt + c ( where c is another constant)
2. hence
Do = k*to + c
0 = c
hence
k = Do/to
hence
s = Do*t/to
1. D(t) = Do*t/to
2. speed at which they are moving apart = v
v = ds/dt
v = d(D(t))/dt = Do/to
3. at to, light leaves galaxy at x = 0
it reaches us at time t = to + t
then
time taken for it to travel distace = t
distance = Do + v*t
hence
Do + v*t = c*t
hence
t = Do/(c - v) = Do/(c - Do/to)
