Solutions for homework 4 : Galaxies

 

 

 

  1. [5 points] Describe two properties possessed by Cepheid variable stars that make them very important for measuring the distance to nearby galaxies.

 

Property 1 : Cepheid variable stars are very good “standard candles”.  In more detail, there is a close relationship between their period of variability and their intrinsic luminosity.  Once we measure their period, we can use this relationship along with the apparent brightness of the star to determine the distance.

 

Property 2 : Cepheids are very bright stars.  Thus it is possible to see them even in other galaxies.

 

Grading scheme

q      1 pt for recognizing that Cepheids are good standard candles

q      2 pt for mentioning the period-luminosity relation

q      2 pts for mentioning that Cepheids are very bright

 

 

  1. [5 points] State Hubble’s law.  Suppose we observe a galaxy to be receding from us at a speed of 13,000km/s.   If Hubble’s constant has a value of 65 km/s/Mpc, how far away is this galaxy?

 

Hubble’s law says:

 

 

We rearrange this to get a formula for the distance.

 

 

So, putting in the numbers we have

 

 

Grading scheme

q      2 pt for getting the basics of Hubble’s law right

q      1 pt for rearranging the formula correctly

q      2 pts for the right answer

 

 

  1. This question concerns the mass of galaxies:
    1. [5 points] Suppose a star orbits around the center of its galaxy at a distance of 10,000pc and with a speed of 250km/s.   How much mass does the region of the galaxy within this radius possess?  Show your working.  [Hint:  you will need to know that 1pc=3´1016m, and that Newton’s constant of Gravitation is G=6.67´10-11 N m2 kg-2. To check that your answer makes sense, you might want to compare it with the mass of the Sun.]

 

The appropriate formula is:

 

 

Now, we need to convert velocity and distance to standard units

 

 

So, we then get…

 

 

Dividing by the mass of the Sun (2´1030 kg), this mass is 1.4´1011 times more than the mass of the Sun.   This is a reasonable number for a galaxy.

 

Grading scheme

q      1 point for identifying correct formula

q      2 points for converting units

q      2 points for correct answer (partial credit of 1 pt if they get the wrong answer but realize that its wrong!)

 

 

 

 

 

 

 

 

 

 

    1. [5 points] What measurements lead us to believe that galaxies contain large quantities of dark matter?

 

By studying the velocity of gas and stars at different radii in a galaxy, we have used the above formula to determine that there are large quantities of mass even in the “dark” parts of galaxies.  In other words, there are large amounts of mass that we can’t “see” but can detect via its gravitational effects. 

 

Grading scheme

q      2 pts for general effort

q      upto 3 points for a clear description.

 

 

 

  1. [5 points] A non-scientist friend of yours is skeptical about the existence of supermassive black holes at the centers of galaxies.  Using language that they can understand, write a brief and concise summary of the current evidence for supermassive black holes (you might want to make your case by focusing on two or three specific cases).

 

Grading scheme

q      1 pt for general effort

q      2 pt for using non-technical language that anyone could understand

q      2 pt discussing valid evidence for supermassive black holes