Physics note: Doppler effect and universe

Doppler effect

In the absorption spectrum, we see that blue shift (the spectrum lines appear at bluer side) happens when a star is approaching us. We see a red shift when it’s receding us.

When the observer and source of a wave is moving, the frequency of wave observed changes.

The new frequency is given by f’ = f(v+u_o)/(v+u_s), where f is the original frequency, v is the velocity of wave, u_o is the velocity of observer (Take the direction of source be positive) and u_s is the velocity of the source.

Now tweet the formula a bit: f’/f = (v+u_o)/(v+u_s), for EM wave v = c, c/λ=f, then

λ/λ’ = 1 + Δλ/λ = (c+u_o)/(c+u_s) = 1 + v_r/(c+u_s) ≈ 1 + v_r/(c) (since c>>u_s), therefore we get

Δλ/λ = v_r/c

Δλ/λ is called the fractional shift in wavelength. u_o-u_s is the relative velocity between Earth and the star, but being an observer on the Earth (which is not an inertial frame), we may imagine the star being the only moving object. v_r turns to be the radial velocity (the component parallel to the line linking star and the Earth) between the star and the Earth. Considering the sign, when the star is approaching us, radial velocity is negative then there’s a negative change in wavelength, causing a blue shift.

Note that this equation is valid only if v_r << c as shown in the proof. They are usually applicable for stars motion which has velocity about several kmh^-1 which is far less than c.

Application

1) Expansion of a star (red giants; supernova)

When θ is small, the radius can be given by R=Dθ, ΔR = D(Δθ). ΔR = vt = |Δλ/λ|tc, then

D =ΔR/Δθ = |Δλ|tc/λΔθ. t can be measured in a year [in the same period with Δθ], also note that Δθ in arcsecond should be converted into radian.

2) Binary star system

Consider a distant star of mass m orbiting a much bigger star of mass M, assume the bigger star is at rest. The radial velocity of smaller star is periodical with period equal to orbiting period. The maximum radial velocity is equal to the orbiting speed v (it’s at maximum when the velocity point towards the Earth, it can be obtained by Doppler effect.) By obtaining the period of the radial velocity T we can calculate the orbiting radius r = v/ω = Tv/2π. Considering the centripetal force, mv²/r = GMm/r², M = v²r/G = v³T/2πG. This helps us to estimate the mass of an unknown object in the universe since this can be applied beyond binary star system, they can be circular motion due to black hole or dark matter.

3) Existence of dark matter

Consider the galactic rotational curves (the stars in a galaxy orbits around the center of galaxy) In classical mechanics, v = (GM/r)^0.5 in star orbital motion as predicted as curve A. However the real observations show that the actual curve is B, which implies that the rotational speed v is independent of radius from the center of galaxy, and we may conclude that the mass spreads outside the galaxy, and called dark matter.

Dark matter (as well as dark energy), occupies 95% of matter in the universe, and they do not interact with any known physical phenomena except gravity.

The dark matter leads to the expansion of universe, and a red shift from distant galaxy is observed. The more distant galaxy, the faster recession velocity.

Hubble’s law states that the recession velocity of a galaxy is directly proportional to its distance. Mathematically, v = Hd, where recession velocity is measured in km s^-1, d is measured in Mpc and H is the Hubble constant = 73.5 km s^-1 Mpc^-1.

Note:

1) 73.5 is an approximated value. Up to now this constant are not precisely measured and have a percentage error of 3%.

2) This applies to distant and large enough galaxies only. Otherwise the recession velocity can’t dominant from gravitational pull between galaxies. For example, some small galaxies approach Milky Way Galaxy under gravitational pull.