# Pulsar Radius from its rotational period

1. Feb 11, 2009

### TFM

1. The problem statement, all variables and given/known data

A pulsar emits bursts of radio waves with a period of 10 ms. Find an upper limit to the radius of the pulsar.

2. Relevant equations

Not Sure

3. The attempt at a solution

Can anyone help with this, I cannot see how the period will help tell you the upper limit to the radius. I know that pulsars are basically neutron stars, and they have high densities (10^15 kg/m^3), but I ams lightly unsure how to get the radius of the pulsar from its period.

TFM

2. Feb 11, 2009

### mgb_phys

What happens when the rotation rate is fast enough that centrifugal force on a point on the surface as is stronger than gravity?

3. Feb 11, 2009

### TFM

Well, Gravity is pulling down, the centrifugal force is pushing outwards, so items on the surface would be "pushed" off of the surface.

4. Feb 11, 2009

### mgb_phys

Correct - so at a certain speed the surface would break off, doesn't this set a maximum radius for a given rotation rate?

5. Feb 11, 2009

### TFM

Indeed it would,

So:

$$mg = m\omega^2r$$

$$g = \omega^2r$$

And since:

$$Omega = \frac{2\pi}{Period}$$

Thus:

$$g = \frac{4\pi^2}{Period^2}r$$

Does this look okay?

6. Feb 11, 2009

### mgb_phys

You will need to write 'g' for the star in terms of it's mass (or density) and radius.

7. Feb 12, 2009

### TFM

True, but we aren't given a mass for the star? Would we use the density as being 10^15?

8. Feb 12, 2009

### TFM

Okay, so if we use:

$$g = -\frac{MG}{r^2}$$

and

$$M = density*volume$$

$$M = density*(\frac{4}{3}\pi r^3)$$

$$g = -\frac{(density*(\frac{4}{3}\pi r^3))G}{r^2}$$

Thus:

$$-\frac{(density*(\frac{4}{3}\pi r^3))G}{r^2} = \frac{4\pi^2}{Period^2}r$$

Since we need the magnitude only for g:

$$\frac{(density*(\frac{4}{3}\pi r^3))G}{r^2} = \frac{4\pi^2}{Period^2}r$$

$$(density*(\frac{4}{3}\pi ))G = \frac{4\pi^2}{Period^2}$$

Does this look better?

TFM