Mars

How to Estimate the Distance to the Outer Planets Yourself

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Now, the last step in mapping out the other planets, before heading for more intriguing territory.

In a previous post I showed you how you can measure the distance between Venus and the Sun, RVS, relative to the distance between Earth and the Sun, RES. Under the assumption that Venus’s orbit around the Sun is circular (or nearly so), you can use the fact that when the angle between Venus and the Sun reaches its maximum (the moment of greatest elongation, and also approximately the moment when Venus appears half lit by the Sun), there’s a simple right-angle triangle in play. High school trigonometry then gives you the answer: RVS/RES ≈ 0.72 ≈ 1/√2. The same trick works for Mercury, which, like Venus, is a near Sun-orbiting planet, closer to the Sun than Earth.

But there’s no maximum angle for Mars, Jupiter, or the other far planets. These planets are further out than Earth and can even appear overhead at midnight, when they are 180 degrees away from the Sun. Fortunately there’s another right triangle we can use, again under the assumption of a (almost-)circular orbit, and that can give us a decent estimate.

The Triangle for the Far Sun-Orbiting Planets

Let’s focus on Mars first, although the same technique will work on the outer planets. Mars has a cycle in which it disappears behind the Sun, from Earth’s perspective, on average every 780 days. (That start of the cycle is called “solar conjunction,” or just “conjunction” when the context is clear.) About half a cycle later, after on average 390 days, it is at “opposition”: closest to Earth, largest in a telescope, appearing overhead at midnight, and at its brightest. But if we wait only a quarter cycle, on average 195 days after conjunction, then the Mars-Sun line is at a 90 degree angle to the Earth-Sun line. That means that Mars, Earth and the Sun form a right-angle triangle with the right angle at the location of the Sun.

So on the day of first quarter we should measure the angle on the sky between Mars and the Sun. That’s the angle A on the figure below. Then the Mars-Sun distance RMS and the Earth-Sun distance RES are the two sides of a right-angle triangle. That means they are related by the tangent function:

  • RMS/RES = tan A.
For a planet whose distance RPS from the Sun is greater than Earth’s, an estimate of its distance compared to the Earth-Sun distance RES can be made using the fact that (were its orbit perfectly circular) it would make a right-angle triangle after the first or third quarter of its cycle, with the right angle located at the Sun. Sizes of Earth, Sun and planet not shown to scale.

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Who Orbits Who, and Where? Check it Yourself

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So far the arguments given in recent posts give us a clear idea of how the Earth-Moon system works: Earth’s a spinning sphere of diameter about 8000 miles (13000 km), and the size of the Moon and its distance are known too (diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter). We also know that the Sun is much further than the Moon and larger than the Earth, though we don’t know more details yet.

What else can we learn just with simple observations? Since the stars’ daily motion is an illusion from the Earth’s spin, and since the stars do not visibly move relative to one another, our attention is drawn next to the motion of the objects that move dramatically relative to the stars: the Sun and the planets.  Exactly once each year, the Sun appears to go around the Earth, such that the stars that are overhead at midnight, and thus opposite the Sun, change slightly each day.  The question of whether the Earth goes round the Sun or vice versa is one we’ll return to.   

Let’s focus today on the planets (other than Earth) — the wanderers, as the classical Greeks called them.  Do some of them go round the Earth?  Others around the Sun?  Which ones have small orbits, and which ones have big orbits? In answering these questions, we’ll start to build up a clearer picture of the “Solar System” (in which we include the Sun, the planets and their moons, as well as asteroids and comets, but not the stars of the night sky.)

The Basic Patterns

If we make the assumption (whose validity we will check later) that the planets are moving in near-circles around whatever they orbit, then it’s not hard to figure out who orbits who. For each possible type of orbit, a planet will exhibit a different pattern of sizes and phases across its “cycle“ when seen through binoculars or a small telescope. Even with the naked eye, a planet’s locations in the sky and changes in brightness during its cycle give us strong clues. Simply by looking at these patterns, we can figure out who orbits who.

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Physics and Curiosity on Mars

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The promised follow-up article on the workshop last week in Waterloo, Canada will have to wait til Monday; I had too many scientific activities and chores to take care of today, and I want to make sure the article, which is a bit complicated, is nevertheless clear.   But in the meantime, let’s celebrate Martian Curiosity!

First, a big congratulations to the NASA folks!  Very impressive, and fantastically cool.  I was a huge fan of the Spirit and Opportunity rovers, especially of their 3D photography.  Looking at those photos on a big screen, through red/green 3D glasses. brought me to sweeping Martian vistas and deep Martian craters — as vivid and as close as I’ll ever see them.  It was amazing stuff, and I look forward to more from the new rover.

Next: some perspective.

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