Problems

#04

The expression to be used is

$$M/M_\odot = a^3/P^2$$

where $a$ is in AU and $P$ is in years. The mass $M$ is the total mass of the two orbiting bodies.

We have a total separation of 500 kpc. I assume this is twice the major axis, so the semi-major axis is half this value, giving $a=250$ kpc. There are about 200,000 AU in 1 pc, so the orbital size is $a = 250 \times 1000 \times 200,000 = 5\times 10^{10}$ AU. The period is $P=3\times 10^{10}$ years. The mass is then $M = (5\times 10^{10})^3/(3\times 10^{10})^2 = 140$ billion solar masses.

#07

The average speed of a gas particle is given roughly by $v=\sqrt{3kT/m}$, for $m$ the particle mass, $T$ the temperature, and $k=1.38\times 10^{-23}$ Joules/Kelvin the Boltzmann constant. For H atoms, and $T=20\times 10^6$ K, I find $v\approx 700$ km/s.

The circular orbital speed is given by $v_c = \sqrt{GM/r}$. The mass is $M=10^14 M_\odot = 2\times 10^{44}$ kg. The orbital radius is $r=10^6$ pc, or $r=3\times 10^{22}$ m. The orbital speed is then $v_c \approx 670$ km/s, or quit comparable to the speed of H particles.

#13

Distance is given by $d = cz/H_0$, for $z$ the redshift value and $c$ the speed of light. We have two redshifts, with two distances $d_1=0.15c/H_0 = 640$ Mpc and $d_2=0.155c/H_0 = 660$ Mpc. The difference in distance between these two absorbing galaxies is about 20 Mpc.