## A First Problem in Astrophysics

Posted in Education, The Universe and Stuff with tags , , , , on February 25, 2011 by telescoper

When I first arrived at Cambridge University (nearly 30 years ago) to begin my course in Natural Sciences, eventually leading to a specialism in Physics, one of the books we were all asked to buy was the Cavendish Problems in Physics. One of the first problems I had to solve for tutorial work was from that collection, and I have been setting it (in a slightly amended form) for my own students ever since I started lecturing. I thought I’d put it up here because I think there might be a few budding theoretical astrophysicists who’ll find it interesting and because it provides a simple refutation of a crazy theory that has been doing the rounds on Twitter all morning.

I like this problem because it involves a little bit of lateral thinking, because not all the information given seems immediately relevant to the question being asked, but you can get a long way by just writing down the pieces of information given and thinking about how you might use simple physical ideas to connect them to the answer.

If you haven’t seen this problem before, why not have a go?

Using only the information given in this Question, estimate the ratio of the mean densities of the Earth and Sun:

i) the angular diameter of the Sun as seen from Earth is half a degree

ii) the length of 1° of latitude on the Earth’s surface is 100km

iii) the length of a year is 3×107 seconds

iv) the acceleration due to gravity at the Earth’s surface is 10 m s-2.

HINT: You do not need to look up anything else, not even G!

The answer you should get is that the mean density of the Earth is something like 3.5 times that of the Sun, although the information given in the question isn’t all that accurate.

In fact the mean density of the Earth is about 5500 kg per cubic metre, and that of the Sun is about 1400 kg per cubic metre; the average density of the Sun is just 40% higher than water, which is perhaps surprising to the uninitiated….

The density of solid iron on the other hand is about 7900  kg per cubic  metre, and even higher than that if it is compressed…