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.

Physics: Stars and universe I

Stars and universe

Parallax determining the stellar distance

Taking photos towards a distant star in the time difference of 6 months (then the Earth is in the opposite side on the orbit) would make a observable parallax. Assuming the more distant stars as "fixed", and the stellar distance is d.

By trigonometry we get (1AU)/(tan p) = d.

The angle p is extremely small (less than 1'' = 1º/3600), tan p is approximately equals to p in radian.

One parsec (pc) is defined as the distance between Sun and the star whose parallax is 1 pc.

1pc = 1AU/1'' ≈ 2*10⁵AU or 3.26 ly.

Now we can simplify the equation into d = 1/p, where d is measured by pc, p is measured by arcsecond (1'').

Note that the closest star from sun is more than 1pc from sun, so the parallax is usually smaller than 1''.

The stellar size D = dθ where D is the stellar diameter and θ is the angular diameter.

Stellar Magnitude

Each magnitude represents a difference of 100^1/5 times in brightness, the lower the magnitude the brighter the star. For example, a star of magnitude 1 is 10 times brighter than a star of magnitude 6.

Apparent magnitude is the brightness of celestial body as observed from Earth. (In this case, Sun is the brightest with apparent magnitude -26.7)

Absolute magnitude is the brightness of celestial body if they are 10pc from Earth. Then this magnitude is independent of its distance, only depending on its brightness. (For example, the apparent magnitude of Sun is smaller than Sirius because Sun is much closer to Earth, but Sirius has a lower absolute magnitude than Sun.)

The brightness of a planet varies due of the position for reflection. The period of variation can be a hint of orbital motion of the planet.

Stellar Spectrum

Assuming stellar body as ideal black body, they emit EM waves as T > 0K.

1)       They emit EM waves with all range of frequency, but there's a peak for a specified frequency.

2)       The higher of its surface temperature, the higher peak frequency, while the magnitude of peak is higher too.

Stellar body has surface temperature around 2000~60000K, which corresponding to peak frequency of red and blue light, so we say hotter star is bluer while colder star is redder.

According to Harvard spectral classification the stars are classified as:

1)       O (blue): 30000 – 60000 K

2)       B (blue white): 10000 – 30000 K

3)       A (white): 7500 – 10000K

4)       F (yellowish white): 6000 – 7500 K

5)       G (yellow): 5000 – 6000 K

6)       K (orange): 3500 – 5000 K

7)       M (red): 2000 – 3500 K ("Oh! Be A Fine Girl Kiss Me" gives the order of OBAFGKM.)

Each class is divided from 0 to 9, e.g. B0 (hotter) to B9 (colder).

By observing the spectrum of star, dark lines may be found in the continuous spectrum. They are called the absorption spectrum, they represents the elements exists in the star.

Luminosity

The radiant power of a star J (it's Wm^-2) is given by the Stefan's law J = σT⁴, where σ is the Stefan-Boltzmann constant 5.67*10^-8 Wm^-2K^-4.

The equation works for black body which is ideal, but its approximation is also good for stars.

The luminosity of star L is given by J(surface area) = 4πR²J = 4πR²σT⁴ which has unit W.

For example, the luminosity of Sun = 4π(6.69*10⁸)²(5.67*10^-8)(5780)⁴ = 3.85 * 10²⁶ W.

While comparing the luminosity with Sun, we can simplify the equation to the following form:

The Hertzsprung - Russell diagram shows the spectral classes, luminosity and surface temperature of the stars. Note that the left side of the diagram shows the higher surface temperature.

1)       Main sequence: They are the initial stage of a star, releasing heat through fission and fusion.

2)       Red giants and supergiants: When a star (with mass quite bigger than Sun) used up its fuels for nuclear reaction, it expands as a giants with larger size, higher luminosity but lower surface temperature. They are usually located at the right-top corner in the diagram.

3)       White dwarfs are the final fate of a smaller star. They are hot but dark and small.

Considering the life of a star: when its mass is smaller than the Sun, it becomes white dwarfs directly.

For bigger star they many change into red giants or supergiants, then becomes a neutron star or supernova. For stars with 3.2 times of mass of Sun, they finally become black holes. They are called black holes because their gravitational field produced is sufficiently high that even light (matter of highest speed) can't escape from it.

Consider the escape velocity c < (2GM/r)^0.5, we know that the Schwarzschild radius (which the light can't escape beyond this radius) is equal to 2GM/c². For a planet of mass equals to that of Earth, its Schwarzschild radius is only 9 mm.