Black Holes, Mercury, and Einstein: The Role of Dimensional Analysis

In last week’s posts we looked at basic astronomy and Einstein’s famous E=mc2 through the lens of the secret weapon of theoretical physicists, “dimensional analysis”, which imposes a simple consistency check on any known or proposed physics equation.  For instance, E=mc2 (with E being some kind of energy, m some kind of mass, and c the cosmic speed limit [also the speed of light]) passes this consistency condition.

But what about E=mc or E=mc4 or E=m2c3 ? These equations are obviously impossible! Energy has dimensions of mass * length2 / time2. If an equation sets energy equal to something, that something has to have the same dimensions as energy. That rules out m2c3, which has dimensions of mass2 * length3 / time3. In fact it rules out anything other than E = # mc2 (where # represents an ordinary number, which is not necessarily 1). All other relations fail to be consistent.

That’s why physicists were thinking about equations like E = # mc2 even before Einstein was born. 

The same kind of reasoning can teach us (as it did Einstein) about his theory of gravity, “general relativity”, and one of its children, black holes.  But again, Einstein’s era wasn’t first to ask the question.   It goes back to the late 18th century. And why not? It’s just a matter of dimensional analysis.

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This Weekend, Measure the Distance to Venus Yourself

This Sunday, Venus reaches a special position from which it is easy to estimate roughly how large the average Venus-Sun distance (RVS) is relative to the average Earth-Sun distance (RES).  (I say “on average” because the Venus-Sun distance isn’t quite constant.  Venus’s orbit, like that of all the planets, isn’t quite circular.  But this is a small effect that we can ignore for the purpose of rough estimates.)

If you are a true diehard astronomy-geek, by all means get up at 5 or 5:30 in the morning on Sunday (or really any of the next few days) to check this directly.  I can assure you (since I have been up at that time recently, due to chronic insomnia more than astronomy-geekhood) that Venus looks absolutely gorgeous against the deep blue of the pre-dawn sky.  But if you have no intention on getting up that early, or clouds intervene, there’s a shortcut — on your phone.

Greatest Elongation, Near-Circular Orbits, Half-Lighting and Right Angles

On Sunday, Venus moves to a position where, from Earth’s perspective, the angle between Venus and the Sun on the sky reaches its maximum.  This position is called “greatest elongation“, and it is reached twice per cycle, once in the evening sky and once in the morning.  If Venus’s orbit were perfectly circular, this would also be the moment when Venus appears half-lit; as we’ve been seeing in two recent posts (1,2), that’s an effect of simple geometry:

  • if Venus’s orbit were circular, then at greatest elongation, the triangle formed by Earth, Venus and the Sun would be a right angle where Venus is located, and Venus would be half-lit.

This holds for Mercury too, as it would for any near Sun-orbiting planet.

Since Venus’s orbit isn’t quite circular, this isn’t precisely true; half lit and the right angle come together, but greatest elongation is off just a few days. This is a minor detail unless you’re an astronomy-geek, and won’t keep us from getting a good estimate of the Venus-Sun distance.

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