Estimating the Distance to Jupiter in My Backyard

Last night, using the methods I described as part of my check-it-yourself astronomy series, I estimated the distance to the planet Jupiter using nothing more than my eyes, a protractor, and a simple calculator. It took about 30 seconds of measuring something before and after sunset, and about 15 more seconds using my cell phone’s calculator. You can do it too, if you have clear skies over the weekend.

There are only two parts of the process:

1. know which week to ask the question, and
2. during that week, measure the angle A in the sky between the Sun and Jupiter.

Then the distance from Jupiter to the Sun R_JS is the distance from Earth to the Sun R_ES times the tangent of the angle A

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The reason is simple geometry, corresponding (for a general planet P) to the figure below.

When to Measure?

The day on which you should make the measurement is the one when Earth, Jupiter and the Sun make a right triangle as in Figure 1. You can estimate this date quite easily. Once every 13 months or so, Jupiter is at “opposition”, meaning that there is a straight line connecting the Sun, Earth and Jupiter, as at the far left of Figure 1. At opposition, Jupiter is directly overhead at midnight. Let’s call the time between oppositions the length of Jupiter’s cycle. Then, about 1/4 or 3/4 of the way through the cycle, the three heavenly bodies form a right-angle triangle, drawn in Figure 1, and the angle between Jupiter and the Sun as seen from Earth is enough to determine the ratios of the triangle’s sides.

Looking at Figure 1, you can see that if R_JS were just a little larger than R_ES, the angle A would be close to 45 degrees. If R_JS were enormously larger than R_ES, then the angle would be very close to 90 degrees. But wandering into your backyard just after sunset, and looking up, reveals the angle to be much larger than 45 degrees, yet clearly less than 90.

How to Measure?

This is a bit harder, but how hard it is depends on how accurate an answer you are seeking.

The first problem is that you can’t see Jupiter at the same time you see the Sun — the former is too dim and the latter too bright — so you can’t measure the angle between them directly. You need to watch where the Sun sets. Call that the “sunset point”. Then you need to measure the angle between Jupiter and that point of sunset, let’s say, 30 minutes after sunset. Next, you need on an additional angle that accounts for how far the Sun has moved below the horizon during that 30 minutes. (Perhaps the best way to do this is to look at the Sun 30 minutes before sunset, estimate its angle away from the sunset point, and then, assuming the Sun moves by that same angle in the next 30 minutes, add that angle on to your 30-minutes-post-sunset measurement of Jupiter.)

You’re not going to get a perfectly precise answer, and that leads to the second problem. R_JS is quite sensitive to the angle A — the further out the planet is from the Sun, the worse this problem becomes. So if you don’t measure the angle accurately, your estimate of R_JS may be rather far off. (Note: Even if you do measure the angle accurately, this method cannot get you a precise answer, because the whole method is based on idealizing Earth and Jupiter as traveling at constant speeds on circular orbits, which is it an approximation of the truth.) This can be seen in the graph below, which shows that as A approaches 90 degrees, the inferred planetary-sun distance grows very rapidly.

My Estimates

I started with a rough qualitative estimate, with no intention to be precise. Just a quick glance at the sky revealed that the angle between Jupiter and the point of sunset was less than 80 degrees and more than 60. Accounting for the Sun’s motion after sunset, that put the angle A between 85 and 65 degrees. That only tells us, as shown in the dashed lines of Figure 4, that the Jupiter-Sun distance lies between 2.2 and 11.6 times the Earth-Sun distance. Still, it already reveals that the ratio of the distances is neither 1.1 nor 100; Jupiter orbit is significantly, though not spectacularly, larger than Earth’s.

Then I measured things a bit more carefully, taking about thirty seconds to do it, and found the estimate shown in the solid lines of Figure 4: the angle is somewhere between 77 and 82 degrees. That’s accurate enough to tell us that R_JS lies between 4.3 and 7.1 R_ES . Although that’s still not precise, it is much better than my first estimate.

In fact Jupiter’s orbit isn’t circular, nor is Earth’s — R_JS moves between 4.95 and 5.46 times the average of R_ES , as shown by the red bar in Figure 4 — so it’s not possible to achieve better than 10%-20% precision using this method. But it’s pretty good for something that requires nothing more than pre-college math and the naked eye! This approach works that much better for Mars (because the angle for Mars is smaller, and the result less sensitive to how well you measure the angle). Conversely, it works less well for Saturn, Uranus or Neptune.

If you like this kind of challenge, try it, and let me know if you succeeded in getting a more precise measurement of the angle and what you found for R_JS. Or if you’re a science teacher, give it as a challenge to your students.

A shortcut — less fun, but much more precise — is to look up the positions of Jupiter and the Sun in the sky, using the coordinates on the sky known as “right ascension” and “declination”. This information is easily found on many websites, such as this one and this one. I won’t go into the details here, but it isn’t hard then to extract the angle between Jupiter and the Sun from this information, and obtain a more precise estimate of R_JS.

Of course, so far we’ve only found the ratio of two distances, R_JS /R_ES. If we want to know R_JS in miles or kilometers, then we need first to measure R_ES in miles or kilometers. Fortunately, this isn’t that hard to do… using meteor showers. But that’s another story.

Final Thoughts

The distance R_EJ from Earth to Jupiter changes dramatically as the two planets orbit the Sun, from a maximum of R_JS +R_ES when the planets are on opposite sides of the Sun to R_JS –R_ES when Jupiter is at opposition. But last night, using the right-angle triangle of Figure 1, we know that the distance was given by the secant (=1/cosine) of the angle A:

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For angles close to 90 degrees, secant and tangent are almost equal, and so it turns out that my estimate reveals that Jupiter is currently between 4.4 and 7.2 times further away than is the Sun.

With a telescope, there are other methods for obtaining R_JS, which you can infer from Figure 1. Jupiter appears larger near opposition, when it is as close as it gets to Earth, and smaller when it lies almost at the other side of the Sun; comparing its angular size at these times would tell you R_JS relative to R_ES . The fraction of the planet which is sunlit from our perspective at various times in the orbit also reveals R_JS .

I hope that some of you go out over the weekend and give this a try! It just takes a few minutes. The same approach works for Mars and Saturn, the other naked-eye outer planets, and a similar approach works for Mercury and Venus. But in each case, we have to wait for the right week.