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

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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