Question

1. a) Express the distance of the star Proxima Centauri from the star Sirius B and the planet
Earth in light years.

1. d) The average temperature of the interior of a sun-like star is of the order 10^8 K. Estimate
the mass of the star in terms of the solar mass if it has a radius of order 10^10 cm.

3. b) A supernova explosion takes place at the distance of 4 pc. If the absolute magnitude of
the supernova is −25, calculate its apparent magnitude. Will the supernova be brighter
than the Sun? Justify your answer.

Answer 1

1.d)

$F_{gr} = G\frac{mM}{(R / 2)^2}$ Gravitational force

$F_{rad}\Delta t = \Delta p$ - Newton's Law for radiational force

$\Delta p = mc = E / c$ - Photons impulse

$F_{rad}\Delta t = E / c$

$L = E / \Delta t$ - luminosity of the star

$F_{rad} = \frac{E}{c\Delta t} = \frac{L}{c}$ Radiational force

$F_{gr} = F_{rad}$ - equilibrium of the forces

$G\frac{mM}{(R / 2)^2} = \frac{L}{c}$ equilibrium of the forces

$L = 4\pi r^2\sigma T^4$ - luminosity of the star

$G\frac{mM}{(R / 2)^2} = \frac{4\pi r^2\sigma T^4}{c}$

$G\frac{\pi r^2R\rho M}{(R / 2)^2} = \frac{4\pi r^2\sigma T^4}{c}$

$G\frac{\rho M}{R} = \frac{\sigma T^4}{c}$

$G\frac{4\rho\pi R^3\rho}{3R} = \frac{\sigma T^4}{c}$

$G\frac{4\rho^2\pi R^2}{3} = \frac{\sigma T^4}{c}$

$\rho = \sqrt{\frac{3\sigma T^4}{4\pi R^2Gc}}$ - density of the star

$$\rho = \sqrt {\frac {3 * 5.67 * 10 ^ {- 8} * 10 ^ {32}}{4 \pi 10 ^ {16} * 6.67 * 10 ^ {- 11} * 3 * 10 ^ {8}}} = 82247 \frac {kg}{m ^ {3}}$$ $$M = \frac {4 \pi R ^ {3}}{3} \rho \text{-mass of the star}$$ $$M = \frac {4 \pi 10 ^ {24}}{3} * 82247 = 2 * 10 ^ {31}$$

ANSWER: $M = 10 M_{\text{sun}}$ - mass of the star

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