What formulas do I need to solve the above problems?

[Red_Night]

I was going to put this up in the homework forum, but I feel the folks down here are more capable of helping me out.

I have to do an assignment, but I cannot go in for office hours to get help from my professor. Regretfully, I lost the sheet and forgot to grab another until just a few days ago and now I'm on my own. All I really need are the formulas.

Here goes:

1. A 10-km-radius neutron star is spinning one-thousand times per second.

a) Calculate the speed of a point on its equator, and compare it with the speed of light.

b) Calculate the orbital speed of a particle in a circular orbit just above the neutron star's surface, and compare with your answer to part (a).

2. Calculate the total mass of the Galaxy lying within 20 kpc of the Galactic center if the rotational speed at that radius is 240 km/s.

3) Centaurus A lies at a distance of 4 Mpc from Earth. This galaxy has radiu jets that span across the sky -- from the end of one lobe to the end of the other lobe-- with an angular diameter equivalent of 28.5 full moon widths. If the jets are equal in length how long is one of them in pc. (hint: one full moon is .5 degrees in angular diameter)

4. According to Hubble's law, with Ho=70 km/sec/Mpc, how long will it take for the distance from the Milky Way Galaxy to the Coma Cluster to double? (Hint: You don't need to look up the distance to the coma cluster)

1. I can do part a, but I am completely flabbergasted at part b.

2. Need formula

3. Need formula

4. I believe I can do this one, but don't I need the velocity of the coma cluster to solve it?

 

[Astronuc]

1b, and 2. How would one determine orbital velocity at a given altitude from a large mass? In other words, what forces are balanced?

 

[Matterwave]

1b) Use the circular orbit acceleration formula: a=v^2/r. Now just think what supplies the acceleration?

2) Use the same formula as 1b. Keep in mind what supplies the acceleration. I believe we can assume spherical symmetry since the dark matter distribution (majority of the mass) is more or less spherical.

3) This is just simple trig. Draw an isosceles triangle with and angle of 28.5*.5 degrees as the angle. The opposite leg is the length you want to find. The bisector of that leg to your angle is 4 Mpc. Do some trig (split the triangle into 2 right triangles), and you should get the answer.

4) You don't need the velocity or the distance. Just find how long it takes the scale factor a to double. For uniform expansion, a governs the expansion of the whole universe. So when a doubles, the proper distances to every point doubles. Remember:

$$H=\frac{1}{a}\frac{da}{dt}$$

 

[Red_Night]

Thank you guys very much for your replies. This has really helped.

@ Astronuc: The force of gravity pulling the particle down and the force of movement to the side?

@ Matterwave: I'm really confused as to how one gets mass out of the equation you listed. It seems that there's a bit more to questions 1b and 2 than I thought.

I'm thinking I need to figure out how massive certain volumes-

I just realized v represented volume. Now I get it.

 

[Red_Night]

One more thing: what does it mean to compare the answers?

Edit:

Okay, for 1b, do you take the cube root of what you get from the formula?

I'm confused about where to get the density for number 2. Am I on the right track?

Edit 2: Yeah, I'm still completely confused how one calculates mass from that info.

 

[Chronos]

Rotational speed has no real effect on orbital velocity.