Hubble's Law

In 1929, Edwin Hubble discover that the farther the galaxy is, the faster it is moving away from us, this is known as the Hubble law.

v = H_{0} d

In the formula, v is the recession velocity measured in km/s,

H_{0}

is Hubble Constant,

H_{0}

=68km/s/Mpc and d is the distance in Mpc. This provide us with a method to measure distance of extremely distant galaxies using redshift.

Define redshift $z$ , a dimensionless quantity that indicates how much the wavelength of light has increase relative to the original wavelength.

z = \frac{λ - λ_{0}}{λ_{0}}

Note that the value of redshift can be both positive and negative. By Doppler effect , one can determine the relative velocity with the observer.

v = c z

Here

c

is the speed of light in vaccum. click here

Let's first analyze the equation:

if the redshift is greater than 0, velocity is also greater than 0, which means we are moving away from the source of emission.
If the redshift is smaller than 0, velocity will also be smaller than 0, which means we are heading towards the source of emission.

We can calculate velocity through redshift, and through this, we can calculate the distance of galaxies.

Before you jump in and use this law, there are some things you should know in advance:

If redshift exceeds $0.1$ , we need to use the relativistic version of redshift formula: $v = \frac{{(1 + z)}^{2} - 1}{{(1 + z)}^{2} + 1} c$

When redshift is high, the linear model of

v = c z

will deviated from the relativistic results, so we should switch from the linear model to relativistic model.

When measuring objects in the local universe (distance is less than 10 Mpc), Hubble law is invalid because the proper motion is compariable with cosmological expansions, so the part of the redshift will be contributed by the proper motion, not the expansion of the universe.