A First Problem in Astrophysics

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…

UPDATE: I’ve added my Solution.

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14 Responses to “A First Problem in Astrophysics”

  1. Bryn Jones Says:

    My answer was 3.3 for the ratio of the mean density of the Earth to that of the Sun using the information in the question (although two-figure precision is not justified given the figures in the question).

    I did wonder when I used to give physics tutorials whether there was some book giving problems suitable for tutorial discussions. It seems that there may well be such a book after all.

    I came across the bizarre theory about the Sun being made of iron earlier today in an article in The Guardian online aboout a threat from the editor of some magazine called Energy & Environment to sue some scientists for criticising the academic quality of the magazine, which appears to publish articles that dispute that anthropogenic climate change is occurring. I note that the odd theory about the Sun being made of iron was submitted to the ArXiv preprint archive (though under the category “General Physics” rather than astrophysics). This does confirm a reservation I have held for many years that the ArXiv archive could suffer from dilution by material that would not be considered suitable for publication in mainstream research journals.

    • telescoper Says:

      Bryn

      The book of Cavendish Problems is an excellent source of tutorial problems. Some of them are straightforward, some are fiendish, but they’re all educational.

      I have previously mentioned them in an old blog post occasioned by the death of Brian Pippard (who edited the Cavendish Problems).

      https://telescoper.wordpress.com/2008/09/24/death-and-conkers/

      Peter

    • “This does confirm a reservation I have held for many years that the ArXiv archive could suffer from dilution by material that would not be considered suitable for publication in mainstream research journals.”

      Well, “General Physics” is just a code-word for “crackpot”. The idea might be that if crackpots publish there, they will be so happy that they have made it to the ArXiv that they will shut up, which is less work than dealing with their complaints if their papers are rejected outright. (Those silly enough to submit a paper to this category probably don’t notice that it is essentially a honey-pot for crackpots.)

      Steer clear of General Physics and you should be OK quality-wise.

    • Well, I didn’t even know that “General Physics” contained crackpot ideas: I don’t think I had ever looked at it. I’ll steer clear!

  2. telescoper Says:

    You’ll see this question incorporates a nice example of an examiner’s trick. The item ii) gives an inaccurate value for the radius of the Earth, which translates into a slightly erroneous final result. That’s a deliberate ploy to make it easy to check that students have actually done the problem as stated rather than simply looking up the answer!

    • Bryn Jones Says:

      Yes, I did wonder about that, but assumed that the figure 100 km for the length of one degree of latitude was chosen to be a round number to make calculations simpler.

      The length of the metre was of course chosen in revolutionary France so that the distance from the equator to the poles would be 10 million metres, but a surveying error meant that the real distance is a little larger than that. As a consequence 1 degree of latitude corresponds to about 111 km.

    • Check this out for even more fun setting such problems: http://xkcd.com/1047/

  3. “a crazy theory that has been doing the rounds on Twitter all morning”

    More proof that Twitter is, to a large degree, a bullshit multiplier. Like the idea that astrology is somehow wrong because the precession of the equinoxes isn’t taken into account, this is not even something new. I appreciate the internet (I even met my wife there), but I like to be spared from things “trending” in Twitter.

  4. Garret Cotter Says:

    I’m lucky enough to have a copy of the problem sets from Caltech freshman/sophomore physics in 61/62 and 62/63. If you know their significance, and you can find me, then maybe you can call…

    • Bryn Jones Says:

      Well, nobody has said it, so I’ll state it: Richard Feynman.

      (Just in case there was some reader who hadn’t understood the significance.)

  5. Ian Douglas Says:

    The simplicity of the refutation of the batty iron sun theory was one of the first things that hit me when I started reading about it (and tweeted about its battiness). I’d really love to know why anyone would believe such a thing. Any ideas?

  6. I’d rather drop out the approximations much later and have an exact answer based on these inputs, to check for errors.

    Using F=Gm1m2/R^2 a couple of times you can drop out in a few lines:

    [Density ratio]= (5* g/4 * 10E9 * pi^2)/(720^3 * LOD)

    where LOD is the length of a degree. For simplicity I have taken the half-degree solar diameter approximation as exact but that’s an easy adjustment (the result is a function of its third power, obviously). Putting in the terms as given you get 3.305. As stated, true LOD is about 11% higher, and g is about 2% smaller than given so the result on better inputs would be about 13% lower, in the 2.88 area.

  7. Josh Hill Says:

    8 Year old Josh Hill asks ” What would happen to dark matter if it comes into contact with a black hole?” Sorry way over Dads head.

    • telescoper Says:

      Dark matter interacts via gravity in the same way that ordinary matter does, so it fall in and make the black hole more massive. You wouldn’t see it, of course, but then you wouldn’t anyway because it’s dark even when it isn’t inside a black hole!

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