Doppler redshift

What is Doppler redshift?

Doppler redshift is a phenomenon that happens when a light-emitting object, such as a star or galaxy, moves away from an observer. The emitting object stretches the wavelength of the light as it moves away. This causes the light to shift towards the red end of the spectrum. This shift in wavelength is known as a “redshift.”

Christian Doppler first described the effect in 1842, and he named it the Doppler effect. The Doppler effect of redshift is similar to the change in pitch of a siren on a moving vehicle as it approaches and moves away from an observer.

What is the redshift used for?

We can use the amount of redshift to calculate the velocity of the object away from the observer. The redshift equation establishes the link between the velocity of the object and the change in wavelength of the light.

In astronomy and astrophysics, Doppler redshift is widely used to study the dynamics of stars, galaxies, and galaxy clusters. It is also used to detect exoplanets and measure the expansion rate of the universe, to name a few things.

Formula

The formula for Doppler redshift is given by the Doppler shift equation for wavelength, which is:

z = \dfrac{\lambda_\text{observed} - \lambda_\text{emitted}}{\lambda_\text{emitted}}

where $z$ is the redshift, $λ_\text{observed}$ is the wavelength of the light observed by the observer, $λ_\text{emitted}$ is the wavelength of the light emitted by the object.

This equation can also be expressed in terms of the frequency of the light as follows:

z = \dfrac{\nu_\text{observed} - \nu_\text{emitted}}{\nu_\text{emitted}}

It can be further simplified by assuming that the object is moving at a velocity $v$ away from the observer. In this case, the equation becomes:

z = \dfrac{v}{c}

It shows that the redshift is directly proportional to the velocity of the object.

Relationship with Hubble’s law

The Doppler shift equation is related to several other important equations in astrophysics and astronomy. For example, Hubble’s law, which describes the expansion of the universe, is as follows:

v = H_0d

where $v$ is the velocity of the galaxy, $d$ is the distance to the galaxy, and $H_0$ is the Hubble constant. The Hubble’s law is based on the observed redshift of galaxies. It is used to study the expansion rate of the universe. The relationship between redshift and this law comes from the fact that the velocity of the object can be expressed via redshift as,

z = \dfrac{v}{c}

Relationship with Friedmann equations

Another equation related to Doppler redshift is the Friedmann equation. Friedmann equations describe the expansion rate of the universe in the context of the Big Bang model. The equation is as follows:

H_0^2 = \left(\dfrac{8\pi G}{3}\right)\rho - \dfrac{kc^2}{a^2} + \dfrac{\Lambda}{3}

here,

• $H_0$ is the Hubble constant,
• $G$ is the gravitational constant,
• $\rho$ is the density of matter and energy in the universe,
• $k$ is the curvature constant,
• $a$ is the scale factor,
• $\Lambda$ is the cosmological constant, and
• $c$ is the speed of light.

This equation is used to study the expansion rate of the universe, the density and composition of matter and energy, and the effects of dark matter and dark energy on the expansion. The Hubble constant can also be written as

H^2 = \left(\dfrac{\.{a}}{a}\right)^2

Where $a$ is the scale-factor and $\.{a}$ is the derivative of the scale factor. The scale factor here is related to the redshift $z$ as,

1 + z = \dfrac{1}{a}

A drawback of using this relationship is that this idea still requires rigorous testing. It is based on the assumption that the universe is the same everywhere and that light travels in a straight line. If this assumption is not true, the relationship between the redshift and the expansion of the universe may be different.

Summary

The wavelength of the light emitted by the object is stretched as it moves away from an observer. This causes the light to shift towards the red end of the spectrum, also known as redshift. Christian Doppler first described this phenomenon in 1842 and scientists and researcher use it widely in the fields of astronomy and astrophysics to study the dynamics of stars, galaxies and galaxy clusters, to detect exoplanets and to measure the expansion rate of the universe.