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D.1 Stellar Quantities

Light year (Ly) & Astronimical unit (AU)

A Light year is the distance travelled by light in one year, which is 9.46×10^15^m

An Astronimical unit is the average radius of the Earth’s orbit around the Sun, which is 1.5×10^11^m = 3.26ly

Stellar parallax

The parallax angle is the angle, at the position of the star, that is subtended by a distance equal to the radius of the Earth’s orbit around the Sun (1AU).

image-20230221111231609

A parsec is the distance to a star whose parallax is 1 arc second, which is 3.09×10^16^m

An arc second is 1/3600 of a degree

  • The average distance between stars in a galaxy is about 1pc
  • Obveratory on Earth could see star within 100pc (Because of the distortion caused by atmosphere)
  • High resolution satellite could see star within 100,000pc
  • Distance toward the edge of milky way galaxy is 30,000pc
    • Galaxy cluster (30 galaxies): 1,000,000pc

The parallax method is also a means of measuring astronomical distances:

image-20230221111948107

$$ d \textcolor{grey}{\text{ (parsecs)}}=\frac{1}{p\textcolor{grey}{\text{ (arc seconds)}}} $$

But if the star is too far, p would be too small to be measured accureately!

Apparent brightness & Luminosity

Apparent brightness: The received power per unit area $$ b=\frac{L}{4\pi d^2} $$ $b$: Apparent brightness ($Wm^{-2}$)

$L$: Luminosity ($W$)

$d$: Distance ($m$)

Luminosity: Total power radiated by the star

$$ L=\sigma AT^4 $$

$\sigma$: Stefan-Boltzmann constant

$A$: Area of the star ($m^2$)

$T$: Temperature ($K$)

A star is assumed to radiate like a black body.

$L_{\bigodot}$ symbolized the luminosity of the sun, the same for other variables.

The nature of stars

The stability of a star depends on the equilibrium between radiation pressure and gravitational pressure

  • Radiation pressure make the star expand
    • Nuclear fusion provides the energy the star needs to keep it hot so that the star’s radiation pressure is high enough to oppose gravitational contraction.
  • Gravitation make the star collapse

Proton–proton cycle (fusion reactions)

  • $^1_1\text H+^1_1\text H\rightarrow^2_1\text H+^0_1e^++v_e$
  • $^2_1\text H+^1_1\text H\rightarrow^3_2\text He+\gamma$
  • $^3_2\text He+ ^3_2\text He\rightarrow^4_2\text He+2^1_1\text H$

Main sequence star

A normal star that is undergoing nuclear fusion of hydrogen into helium.

  • 90% of stars are main sequence star
Composition Proportion
Hydrogen 70%
Helium 28%
Other heavier elements 2%
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