S2: Astrophysics: From Planets to The Cosmos Trinity term 2012

S2: Astrophysics: From Planets to The Cosmos
Trinity term 2012
1. The orbital period of Mars is 1.881 yrs. When Mars is opposite the Sun in the sky, its
position with respect to the background of fixed stars is measured at sunset and again at
sunrise. The effective baseline between these two positions is 11,700km and the change in
its position on the sky is 30.8 arcseconds. Calculate the length of the Astronomical Unit
(AU) in metres.
2. The orbital period of Venus is 0.6152 yrs. When Venus is between the Earth and Sun (at
inferior conjunction) radar signals are launched from the Earth and the reflected signal is
detected after a delay of 276.2s. Calculate the length of the Astronomical Unit in metres.
3. What is the shift in position of the Sun due to the influence of Jupiter? Express your
answer in solar radii, R ⊙ = 6.96 × 10 5 km. [The Sun’s mass is 1047 × Jupiter’s mass;
distance from Sun to Jupiter is 5.2AU; 1AU = 1.5× 10 6 km.]
The nearest star is 4.2 light years away. When viewed from a planet orbiting the nearest
star what is the angular shift of the Sun due to Jupiter?
4. What is the velocity of the Sun due to the influence of Jupiter? (The Sun’s mass, M ⊙ =
2.0 × 10 30 kg, is 1047 × Jupiter’s mass and the semi-major axis of Jupiter’s orbit is 5.2
AU).
In 1995 the first exo-planet detected by the radial velocity technique was that around the
star 51 Peg. The radial velocity amplitude was measured to be 60 m.s −1 and the period is
4.2days. Assuming the orbit is circular and edge-on and that host star has the same mass
as the Sun estimate:
(i) the mass of the planet (give your answer in Jupiter masses) and
(ii) the radius of the orbit in Astronomical Units.
5. Use the Virial Theorem to estimate how long the Sun can radiate at its current luminosity
by contracting under gravity.
6. Computer models of the evolution of the Sun show that conversion of hydrogen to helium
will stop when about 10 percent of its mass has been converted. Estimate the lifetime of
the Sun in its hydrogen-burning phase.
Hint: Use the difference in mass between four protons and one Helium nucleus to verify
that 0.7% is converted into energy. For each Helium nucleus produced this ”mass deficit”
is converted into energy (m H = 1.0078 amu.; m He = 4.0026 amu.). Then, assuming the
present luminosity, use the total mass converted & efficiency to determine the lifetime.
7. The strongest lines in the visible wavelength spectrum of the Sun are those of calcium,
sodium, iron and magnesium. Why are these elements so prominent in the solar spectrum
when we believe it to be composed mostly of hydrogen & helium?
8. γ Virginis is a visual binary system with period 171 yr. It consists of two identical main
sequence stars of spectral type F0 separated by 3.75 arcsec and each with apparent bolometric
magnitude of 3.43. By assuming the main sequence mass-luminosity relation, L ∝
M 4 , obtain values for the parallax of the system and the mass of each star in M ⊙ . [the
absolute bolometric magnitude of the Sun is +4.72]
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Figure 1: The primary minimum in the light curve of the eclipsing binary system AR Lacertae,
plotted as a function of phase (i.e. fraction of period). This system, whose period is 1.98 days
is also a spectroscopic binary: the radial velocity curves of the two components are sinusoidal
and have amplitudes of 116km s −1
9. AR Lac is an eclipsing spectroscopic binary with period 1.98 days. The radial velocity
curves of the two stars are sinusoidal with equal amplitudes of 116km.s −1 . The diagram
(Fig. 1) shows the primary minimum in the light curve as a function of orbital phase.
Determine the masses and radii of the two components. Assume the orbit is edge on.
10. By equating the gravitational acceleration at the surface of a neutron star to the centrifugal
acceleration at the equator, calculate the size of a 3M ⊙ neutron star that can spin with
a period of 1ms without disintegrating. What is the density implied? Compare this with
nuclear densities.
11. A mass m falls towards a black hole and ends up in an orbit at three Schwarzschild radii
from the balck hole (this the closest stable orbit to the black hole). Use the Newtonian
expression for the decrease in potential energy, and the assumption that half this decrease
will be radiated away (from the Virial theorem), to estimate the fraction of the rest mass
that will be radiated away in this process. How does that compare with the fraction of
the rest mass released in the p-p chain that you calculated in Ques 6?
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12. The period-luminosity relation for Cepheid variable stars is:
M V = −2.81LogP − 1.43 (1)
where P is the period in days and M V is the absolute V-band magnitude. Cepheid variable
stars are found in the M31 with a period of 31 days and an apparent V-band magnitude,
m V = 18.6. Calculate the distance to M31 in kiloparsecs, assuming no absorption along
the line-of-sight (either in M31 or the Milky Way).
The radial velocity of M31 with respect to the Milky Way is -300km/s i.e. towards the
Milky Way. How do you interpret this?
13. The star S2 has an orbit around the Galactic Centre with a period of 15.9 years and a
semi-major axis of 125 milli-arcsecs. Taking the distance to the Galactic Centre to be 8kpc
calculate the mass of a compact object around which S2 orbits.
Sgr A* is the radio source at the centre of the Galaxy. When it it flares at X-ray wavelengths
the flux increases by × 4 in 40 minutes. Estimate the size of the emitting region
and compare it’s linear size to the semi-major axis of S2.
14. The RAVE experiment measures the radial velocities of stars in the neighbourhood of the
Sun and finds that the distribution of velocities implies an escape velocity from the Milky
Way (at 8kpc radius) in the range 430-560 km.s −1 . What mass range interior to 8kpc do
you estimate from this measurement?
15. The Tully-Fisher relation calibrated from Hubble Space Telescope measurements of Cepheid
variable stars in local galaxies is:
M V = −7.8(log∆v − 2.5) − 20.5 (2)
where M V is the absolute magnitude of the galaxy in the V-band and ∆v is the full
width of the observed integrated line profile of neutral hydrogen emission. For the spiral
galaxy NGC 3198 the width of the HI profile, ∆v = 300 km.s −1 , and the apparent V-band
magnitude, m V = 10.4. Calculate the distance to NGC3198 in Mpc.
The recession velocity of NGC3198 is 680 km.s −1 , what value of the Hubble constant is
implied? If in fact H 0 = 72 km.s −1 .Mpc −1 what conclusion do you draw?
16. A rich cluster of galaxies of radius 2Mpc contains 1000 elliptical galaxies with average
luminosity 10 10 L ⊙ . An analysis of the colours and spectra of typical elliptical galaxies
indicates that on average it takes 3M ⊙ to generate one L ⊙ . The rms velocity of galaxies
in the cluster is 1300 km.s −1 . Make an estimate of the mass of the cluster. Compare the
total mass in galaxies with mass of the cluster. What possibilities are suggested by this
result?
17. The present mass density of the Universe is 2.5 × 10 −27 kg.m −3 (about 2 protons m −3 ) and
the present day energy density in CMB photons is 5 × 10 5 eV.m −3 . Compare the energy
density in matter with the energy density in the CMB radiation today, and calculate the
redshift at which the two energy densities were equal.
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