**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*10**.

^{5}AU or 3.26 lyNow 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__, 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.

^{1/5}times in brightnessApparent 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 = σT**, where σ is the Stefan-Boltzmann constant 5.67*10^{4}^{-8}Wm^{-2}K^{-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) = 4πR**which has unit W.^{2}J = 4πR^{2}σT^{4}For example, the luminosity of Sun = 4π(6.69*10

^{8})^{2}(5.67*10^{-8})(5780)^{4}= 3.85 * 10^{26}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/c^{2}. For a planet of mass equals to that of Earth, its Schwarzschild radius is only 9 mm.
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