Astrophysics PS4

Author

Will St. John + Theo Darci-Maher + Elliott Lewis

Problem 1

The Milky Way disk is approximately 50 kpc in diameter. Imagine that you observe a spaceship, which is 100 meters in length, traveling at 0.994 c.

  1. While the ship is in flight, how long does it appear to you?
  2. What is the diameter of the galaxy as measured by someone on the spaceship?
  3. How long does the journey across the galaxy take, as measured by you?
  4. How long does the journey across the galaxy take, as measured by someone on the spaceship?

Answer:

import astropy.units as u
import astropy.constants as c

D_mw = 50 * u.kpc
D_s = 100 * u.m
v = 0.994 * c.c

gamma = 1 / (1 - v**2 / c.c ** 2) ** 0.5

# moving rulers are compressed
D_s_prime = D_s / gamma

D_mw_prime = D_mw / gamma

# Moving clocks run slow
t_rest = D_mw / v

t_moving = t_rest / gamma

print(f"length of moving ship from MW reference frame: {D_s_prime:.2f}")
print(f"length of MW from ship perspective: {D_mw_prime:.2f}")
print(f"time for ship to cross MW from MW perspective: {t_rest.to(u.yr):.2f}")
print(f"time for ship to cross MW from ship perspective: {t_moving.to(u.yr):.2f}")
length of moving ship from MW reference frame: 10.94 m
length of MW from ship perspective: 5.47 kpc
time for ship to cross MW from MW perspective: 164062.56 yr
time for ship to cross MW from ship perspective: 17945.17 yr

Problem 2

Starting with the equation for superluminal motion, derive the following equations:

  1. v/c < 1 for angles satisfying \(\frac{\frac{v_{app}^2}{c^2} - 1}{\frac{v_{app}^2}{c^2} + 1} < \cos\phi < 1\)
  2. That the smallest possible value for v/c for the source is \(\frac{\frac{v_{app}^2}{c^2} - 1}{\frac{v_{app}^2}{c^2} + 1} < \cos\phi < 1\)
  3. The smallest possible v/c will occur at an angle \(\phi_{min}\) such that \(\cot\phi_{min} = \frac{v_{app}}{c}\)
  4. The Lorentz factor corresponding to the minimum v/c is \(\gamma_{min} = \frac{1}{\sqrt{1 - v_{min}^2}} = \sqrt{1 + v_{app}^2/c^2} = \frac{1}{\sin\phi_{min}}\)
  5. The inner 6” of the jet from M87 is observed to have an apparent velocity of 4.5c https://ui.adsabs.harvard.edu/abs/2013ApJ…774L..21M/abstract. Using your newly derived equations, estimate the minimum velocity, minimum approaching angle, and Lorentz factor of the jet.

Answer:

Problem 3

Use the ADS and/or ArXiV search engine tools to identify a recent (within the past few years) peer-reviewed scholarly manuscript that deals with a relativistic concept and/or observations of superluminal motion. Provide a brief summary of the main results of the paper, including a discussion of how the material connects to the concepts that we discussed in Chapter 4 of the textbook.

Answer: