PX158 - PS3

Q1

In the Earth’s early history, the Sun’s luminosity is thought to have been only $70 %$ of the present day value (owing to a higher ratio of hydrogen to helium in its core). Calculate the temperature of the early Sun.
stefan-boltzmann law :

L_{⊙} \propto T_{e f f}^{4}

T_{e f f} = \sqrt[4]{0.7} T_{e f f, n o w} \approx 5305 K

What is the peak wavelength of this early Sun? What colour would it appear relative to the present Sun?
wien's law :

λ_{m a x} T = 2.9 \times 10^{- 3} ⟹ λ_{m a x, ⊙} = 546.7 n m

The bright star Antares has a measured colour of $(B - V) = 1.83$ , while our Sun has a colour of $(B - V) = 0.65$ . State, with reasons, whether Antares is hotter or cooler than our Sun. What does this tell us about the luminosity of Antares?
Antares is redder than the sun $⟹$ cooler than the sun

Using the definition of effective temperature $T_{e f f}$ from the lectures, calculate the luminosities of the following stars in terms of solar luminosities:
(a) A white dwarf with $R = 0.01 R_{⊙}$ and $T_{e f f} = 25000 K$
(b) A red giant with $R = 100 R_{⊙}$ and $T_{e f f} = 3000 K$ .
(The effective temperature of the Sun $T_{e f f, ⊙} = 5800 K$ .)
stefan-boltzmann law :

L = 4 π σ R^{2} L^{4} ⟹ \frac{L}{L_{⊙}} = {(\frac{R}{R_{⊙}})}^{2}

∴ L_{W D} \approx 0.035 L_{⊙}

L_{g i a n t s t a r} \approx 716 L_{⊙}

A neutron star has radius, $R = 10 k m$ and luminosity, $L = 0.5 L_{⊙}$ . Calculate (i) its effective temperature, and (ii) the wavelength of maximum flux density in its spectrum, assuming that the star radiates as a black-body.
stefan-boltzmann law :

T_{e f f} = {(\frac{L}{4 π σ R^{2}})}^{\frac{1}{4}} \approx 1.28 \times 10^{6} K

λ_{m a x} \approx 2.3 n m

What would be the best region of the electromagnetic spectrum to observe the neutron star?
X-rays

Jupiter’s moon Io orbits $421600 k m$ from the centre of Jupiter with a period of $1.77 d$ . Calculate the mass of Jupiter. (You may neglect Io’s mass compared to that of Jupiter.)

\frac{4 π^{2}}{P^{2}} = \frac{G (m_{1} + m_{2})}{R^{3}}

m_{j u p} \approx 1.9 \times 10^{27} k g

Spectra of a star of mass $0.8 M_{⊙}$ show, through the Doppler effect, that it has an orbital speed of $17 m s^{- 1}$ owing to a planet orbiting it on a period of $7 y r$ . Calculate the mass of the planet.) Use the small mass of the planet compared to the star to make approximations to help your calculations.)
In practice the Doppler effect only gives the component of stellar velocity along our line of sight to the target. In most cases it is not possible to determine the orientation of orbits with respect to the line of sight. Show that this will cause us to underestimate the masses of planets found in this manner.
$m_{s} >> m_{p} ⟹ (m_{s} + m_{p}) \approx m_{s}$ :

m_{p} \approx 1.87 \times 10^{27} k g

Two stars in a binary of orbital period $2.7 h r$ have orbital speeds of $65 k m s^{- 1}$ and $350 k m s^{- 1}$ . Calculate the masses of the two stars.

M_{1} = 0.7 M_{⊙}; M_{2} = 0.13 M_{⊙}