Hubble’s Law

The distant galaxies are moving away from the earth. In other words, the universe is expanding. Considering this fact, Georges Lemaitre gave an important physical cosmological result known as Hubble’s Law, which explains the correlation between the distance to a galaxy and the redshift. According to the law, the recessional velocity or the redshift is directly proportional to the distance of the velocity. In layman’s language, the law of Hubble meaning states that the farther the galaxies are from the earth, the faster they recede away from it. The Hubble’s law is not only evidence of the expansion of the universe, but also an important explanation to the big bang model. Below we have given the formula of Hubble’s Law. 

(image will be uploaded soon)

The Numerical Expression for the Hubble’s Law

v = H0d

v = Recession velocity in km per second

d = Distance in Mpc

H0 = Expansion Rate Today [Hubble Parameter]

In this section, we’ll be deriving a numerical expression of the Hubble’s Law. As we discussed in the previous section, The Hubble effect shows the direct correlation between the earth's distance to a galaxy and its recessional velocity. Thus, numerically, Hubble’s law can be written as 

\[v \propto r\]

Where v represents the recessional velocity in km/s and r represents the distance of the galaxy in mpc.

For removing the proportionality, we require proportionality constant. This proportionality constant is known as the “Hubble constant” is denoted by H0. Thus, we get the Hubble’s law as

v = H0r

**The Hubble constant (H0) measured in Km/s/Mpc.

The Hubble Constant (H0) 

The Hubble constant is a unit representing the pace of universe expansion concerning a particular point in the universe. While solving problems for the Hubble's law, we'll be taking the Hubble constant in Km/s/Mpc.

Till now, there have been a lot of attempts for calculating the exact value of the Hubble constant as the Hubble constant keeps on changing. The current direct measurement of the Hubble constant is taken as 73.8 km/s or 2.4 km/sec/Mpc.

Redshift - Meaning and Definition

Redshift is a phenomenon resulting in an increase in the wavelength of electromagnetic radiation from an object. There are 3 main causes of redshift:

• Doppler Effect: The movement of objects closer or apart in space.

• Hubble Effect: The universe is expanding constantly. This expansion is resulting in the separation of objects without a change in their position. This is referred to as cosmological redshift.  

• Gravitational Redshift: Gravitational redshifts result in distortion of space-time and exertion of force on light and other particles due to strong gravitational fields.

\[z = \frac{\lambda - \lambda_{0}}{\lambda_{0}}\]

z = redshift

\[\lambda\] = measured wavelength shift

\[\lambda_{0}\] = true wavelength

The phenomena of redshift can be expressed numerically in an equation, known as the redshift equation, which is given by:

\[z = \frac{\delta \lambda}{\lambda}\]

Where z denotes the redshift, λ denotes the wavelength, and Δλ denotes the shift in the wavelength in the spectra.

Relativistic Redshift

\[z = \sqrt{\frac{c + v}{c - v}} - 1\]

z = redshift

v = recessional velocity

c = speed of light

Relativistic redshift is very useful for dealing with large values while solving problems related to the Hubble effect. Relativistic redshift perfectly substitutes the Doppler formula when the velocities are too large to be represented by the Doppler formula (generally for the velocities greater than 10% of the speed of light).

Let v represent the velocity of the object measured, c represents the speed of light and z represents the redshift, then the relationship can be expressed as

\[z = \sqrt{\frac{c + v}{c - v}} - 1\]

Limitations of Hubble’s Law

One can calculate the distance of the galaxy using Hubble’s equation after determining the shift of light into the redshift. Besides being an important result in physics, Hubble’s law still confers some limitations, which are as follows:

• The observed velocity gets influenced due to the intrinsic motion of the galaxies. 

• The reason behind the orbiting of galaxies is the gravitational movements. 

Did You Know?

(image will be uploaded soon)

The renowned Hubble space telescope is also named after Edwin Hubble. The telescope has been providing important information and images for research since 1990 when it was carried to the orbit. An asteroid and moon crater have also been named after Edwin Hubble for his valuable contribution to the fields of astrophysics.

Solved Examples on Hubble’s Effect

1. Galaxy NGC 123 has a velocity away from us of 1,320 km/s and the Hubble Constant's value is 70 km/s/Mpc. How far away is the galaxy according to Hubble's Law?

Answer: We are given a numerical based on the Hubble effect. 

According to the question, 

The recessional velocity (v) = 1320 km/s (the velocity at which the galaxy is moving away from the earth)

Hubble constant (H0) = 70 km/s/Mpc

We need to find the distance between the earth and the galaxy i.e. r.

Using Hubble’s law, we get

v = H0r

1320 = 70 r

r = 18.85 Mpc