# Pulsar Radius from its rotational period

• TFM
In summary, the conversation discusses finding an upper limit for the radius of a pulsar based on its period of radio wave emissions. The attempt at a solution involves using equations for gravity and centrifugal force, as well as the density of the pulsar, to determine the maximum radius. However, there is uncertainty about the exact equation to use without knowing the mass of the star.
TFM

## Homework Statement

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

Not Sure

## 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.

Any helpful suggestions would be most helpful,

TFM

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

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

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

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?

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

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

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

## 1. What is a pulsar?

A pulsar is a type of neutron star that emits beams of electromagnetic radiation from its magnetic poles, which can be observed as regular pulses by astronomers.

## 2. How is the radius of a pulsar calculated from its rotational period?

The radius of a pulsar can be calculated using the formula R = (30 x c x P) / (2π x ν), where R is the radius, c is the speed of light, P is the rotational period, and ν is the frequency of the pulsar's electromagnetic pulses.

## 3. Why is the rotational period of a pulsar important in determining its radius?

The rotational period of a pulsar is important because it is directly related to the pulsar's size. As the pulsar rotates, its radius is affected by its angular momentum, which in turn affects the pulsar's rotational period.

## 4. How accurate is the calculation of a pulsar's radius from its rotational period?

The calculation of a pulsar's radius from its rotational period is generally considered to be very accurate, with a margin of error of less than 1%. However, it is important to note that this calculation is based on several assumptions and may vary for different types of pulsars.

## 5. Can the radius of a pulsar change over time?

Yes, the radius of a pulsar can change over time due to a variety of factors, including the pulsar's rotation rate, its magnetic field, and interactions with its environment. However, these changes are relatively small and may not be noticeable over short periods of time.

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