Download Coursework 7 File

Our Universe
Coursework 7
Due in on Wednesday of week 9 at 16:00
Useful information
Boltzmann’s constant k = 1.38 × 10−23 J K−1
Planck’s constant h = 6.63 × 10−34 m2 kg s−1
Mass of hydrogen molecule mH2 = 3.34 × 10−27 kg
Gravitational constant G = 6.67 × 10−11 N kg−2 m2
Solar mass Msun = 2 × 1030 kg
Solar luminosity Lsun = 3.8 × 1026 W
Solar radius Rsun = 7 × 108 m
Jupiter radius RJup = 7 × 107 m
1 Astronomical unit 1.5 × 1011 m
1 parsec = 3.26 light years
Speed of light c = 3 × 108 m s−1
Stefan-Boltzmann constant σ = 5.7 × 10−8 W m−2 K−4
Permittivity of free space 0 = 8.85 × 10−12 F m−1
Proton charge e = 1.6 × 10−19 C.
Exercise class question - not to be handed in
1. The atoms in a gas of temperature T have kinetic energies Eke = 32 kT on average. Assuming
that this is the typical energy associated with collisions between hydrogen nuclei at the centre
of the Sun, leading to fusion reactions, calculate the distance of closest approach between two
hydrogen nuclei, using the conservation of energy principle. The temperature at the centre
of the Sun is approximately T = 1.55 × 107 K. You should note that the electrostatic force
attempts to repel the particles as they approach one another, and the associated electrostatic
potential energy is given by Epe = −q1 q2 /(4π0 d), where d is the distance between the charges
q1 and q2 .
2. The radius of a proton is 8.5 × 10−16 m. Assuming that hydrogen nuclei are required to
physically collide with each other before undergoing fusion, comment on the answer that you
obtained in part (1).
3. Calculate the temperature that is required at the centre of the Sun for close approaches
between hydrogen nuclei to result in physical collisions.
4. What missing physics may explain the fact that fusion reactions provide the energy source
for the Sun?
Please note that all questions below should be handed in for assessment
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Homework question 1
1. Consider a recently formed giant planet orbiting a solar-type star with semimajor axis 50 AU.
The giant planet has a radius, Rp , that is twice Jupiter’s radius and a surface temperature
T = 1500 K. Estimate the ratios of the luminosities emitted by the central star and the
planet: (i) between wavelengths 500-510 nm; (ii) between wavelengths 2000-2010 nm. Based
on your answers, what approach would you recommend for attempting to directly image the
giant planet?
* Homework question 2
1. Write down an expression that relates the apparent magnitudes of two stars, m1 and m2 , to
their measured fluxes, F1 and F2 .
2. If m1 = −7.05 and m2 = 0.45 calculate the ratio of the fluxes.
3. Parallax measurements show that the distance to both stars is d = 1.34 parsecs. Furthermore,
the flux measured from star 2 is F2 = 1.77 × 10−8 W m−2 . Calculate the luminosities of the
two stars.
4. Spectroscopic measurements show that star 2 is a G2 star with temperature 5800 K, and star
1 is an M2 star with temperature 3300 K. Calculate the radii of the two stars, and express in
units of the solar radius. What types of stars are these ?
5. Define what is meant by the absolute magnitude of a star.
6. Show that the absolute magnitude, M , and apparent magnitude, m for a star are related by
the expression
m − M = 5 log10 d − 5,
where d is the distance to the star in parsecs.
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