Errors on velocities and distance.

[indie452]

Homework Statement

A galaxy has velocity due to Hubble (Ho) expansion, and in addition a peculiar velocity (AN this is the velocity of the galaxy in its local enviroment/gravitational potential) this vel. is 450Km/s

I need to estimate the the minimum distance at which a galaxy must be in order for the redshift (z) to give an estimate of the TRUE distance (d) which is accurate to 10% or better.

Homework Equations

z = v/c
v = Hod

The Attempt at a Solution

450 is the 10% error on the velocity so it needs to be x10 => v=4500km/s
then d = Ho/v

but this doesn't take redshift into account

 

[mark726]

, need help!
Thank you for your post. I would like to offer some insights and suggestions for solving this problem.

Firstly, let's define some key terms and equations that will help us in our calculations:

- Hubble's Law: This is a fundamental relationship in cosmology that describes the expansion of the universe. It states that the recessional velocity of a galaxy (v) is directly proportional to its distance (d) from us: v = Hod, where Ho is the Hubble constant.
- Peculiar velocity: This is the velocity of a galaxy in its local environment, which can be caused by gravitational forces from nearby objects.
- Redshift (z): This is the change in the wavelength of light emitted by an object due to its motion away from us. It is related to the recessional velocity of the object through the equation z = v/c, where c is the speed of light.

Now, let's consider the problem at hand. We want to estimate the minimum distance at which a galaxy must be in order for the redshift to give an accurate estimate of its true distance, with an error of 10% or less. This means that we need to find the distance at which the redshift is equal to 10% of the true distance, or z = 0.1d.

To solve this, we can use the equations mentioned above and some simple algebraic manipulation. First, let's substitute the given value for the peculiar velocity (450 km/s) into the Hubble's Law equation:

450 km/s = Hod

Next, we can substitute the value for the Hubble constant (Ho) into the equation:

450 km/s = (70 km/s/Mpc) x d

We can rearrange this equation to solve for the distance, d:

d = 450 km/s / (70 km/s/Mpc) = 6.43 Mpc

Now, let's consider the redshift. We know that z = v/c, so we can rearrange this equation to solve for v:

v = cz

Substituting the value for z (0.1d) and c (the speed of light, 3 x 10^5 km/s), we get:

v = (3 x 10^5 km/s) x (0.1d) = 30,000 km/s x d

Now, we can set this value equal to