PX158 - PS5

Q1

The spectrum of a galaxy shows an atomic line, identifiable as the main optical line of hydrogen, Hα, at a wavelength λ = 810 nm. What is the redshift of the galaxy? How far away is it? (Rest wavelength of $H α$ = 656 nm; Hubble’s constant $H_{0}$ = 72 km s−1 Mpc−1 .)

\begin{aligned} 1 + z & = \frac{λ}{λ_{0}} \\ z & \approx 0.235 \\ v = c z & = H_{0} d \\ d & \approx 978 M p c \end{aligned}

The Cosmic Microwave Background (CMB) is a near perfect blackbody spectrum at 2.73 K. What is the peak wavelength of the CMB?

\begin{aligned} λ_{m a x} T & = σ = 2.898 \times 10^{- 3} m K \\ λ_{m a x} & \approx 1.06 m m \end{aligned}

A distant quasar is found to have a luminosity of 4 × 1012 L . Calculate the accretion rate of the central black-hole in solar masses per year. (Assume a 10% efficiency for converting mass into energy)

\begin{aligned} L_{q} & = 4 \times 10^{12} L_{⊙} \\ = 0.1 m c^{2} \\ m & = \frac{4 \times 10^{12} L_{⊙}}{0.1 c^{2}} \\ \approx 1.69 \times 10^{23} k g \\ a c c r e t i o n r a t e & \approx 1.69 \times 10^{23} k g s^{- 1} \\ \approx 1 M_{\oplus} m i n^{- 1} \\ \approx 2.7 M_{⊙} y r^{- 1} \end{aligned}

A star passing close to the Milky Way’s central black-hole (known as Sgr A $^{*}$ ) was found to have a velocity of 7000 km s−1 at a distance of just 120 AU from the inferred location of the black hole. Assuming it was on a circular orbit, calculate the mass of the central black hole in solar masses.

\begin{aligned} \frac{G M m}{R^{2}} & = \frac{m v^{2}}{R} \\ M & = \frac{v^{2} R}{G} \\ \approx 1.32 \times 10^{37} k g \\ \approx 6.7 \times 10^{6} M_{⊙} \end{aligned}

As a result of gravitational perturbations, galaxies have random motions known as “peculiar velocities” with respect to the Hubble expansion. Assuming that these add an uncertainty of order 300 km s−1 to the measured recession velocity of any galaxy, estimate the distance at which this causes a 20 % uncertainty in the distance to a galaxy when it is estimated using Hubble’s law. As a result of peculiar velocities, does Hubble’s law work best for nearby or distant galaxies?

\begin{aligned} 20 % & \to 300 k m s^{- 1} \\ v & = 1500 \frac{k m}{s^{- 1}} \\ d & \approx 20.8 M p c \end{aligned}

The figure shows the rotation curve of the galaxy UGC12591. Calculate the mass (in solar masses) interior to radii of 10, 25 and 40 kpc for this galaxy.

\begin{aligned} v (10 k p c) & = 500 k m s^{- 1} \\ v (25 k p c) & = 475 k m s^{- 1} \\ v (40 k p c) & = 450 k m s^{- 1} \\ \frac{G M m}{R^{2}} & = \frac{m v^{2}}{R} \\ M & = \frac{v^{2} R}{G} \\ M (10 k p c) & = 5.8 \times 10^{11} M_{⊙} \\ M (25 k p c) & = 1.3 \times 10^{12} M_{⊙} \\ M (40 k p c) & = 1.9 \times 10^{12} M_{⊙} \end{aligned}