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Orbital Dynamics

Dave North


Our first point about orbits? There will be two full Moons this month. Doubtless this will bring up the term "Blue Moon" being the second full Moon in a month, but as best I have been able to determine, that's hogwash.

But it's also becoming common usage, which is how "hogwash" becomes "English." And considering how much of said wash is going around, it's a miracle there's a dirty pig in the US.

I kind of enjoy double-full-moon months. It's fun to hear twice the whining from the (often imaginary) smudgie crowd!

Last month's little notes about the inconstant lunar orbit did provoke a seemingly straightforward question: how long would it take before it was necessary to recenter the Moon in an eyepiece if one were using sidereal rates?

We'll skip right past the "what eyepiece" part and assume we have a fairly typical 1-degree field of view.

Let's see. The Moon completes a sidereal orbit (that's the one we're interested in - not the 29.5-day synodic month) in roughly 27.3 days. That means it will (on average) traverse 13.2 degrees per day, or .55 degrees per hour. Or, if you prefer, 33 minutes per hour.

So, in just under half an hour we're at the edge right?


On average, the Moon takes up a hair over half a degree of the sky. If you've got a 1 degree field of view, when you center up the Moon you've only got .25 degree to spare on each side (about 15 minutes of arc).

If you want to keep the whole enchilada (okay, the whole tostada at full, and maybe the whole taco at quarter) you've only got the time that it takes to travel 15 minutes of arc, which just happens to be about 27.3 minutes.

That sounds awfully familiar. Where have we seen that number before?

It's the sidereal period in days! How did that happen?

It's a coincidence. Really. This is decidedly not a Face On Mars thang. For example, you might consider what the result would be if we used a 3/4-degree field of view instead.

Cheap magician's trick. Nothing to see here ...

Okay, so we don't exactly have to be correcting our view very often. But what happens when we're not at an 'average' point in the orbit, but rather when the Moon is close and zooming by?

Ah, an interesting problem for someone who took orbital dynamics in college, then forgot it completely!

We recall from our fuzzy Keplerian memories that the Moon will sweep out an equal area in any given equal amount of time (barring other influences).

Without getting too hairy, we can approximate this by noting then that the radius times arc should be equal in all cases, and that over a short period of time the arc very closely approaches a line.

Let's see. The Moon's orbit varies by about 14 percent from apogee to perigee, an ellipse with each axis offset by 7 percent from center. (The rough numbers are 356,400 Km and 406,700 Km respectively).

If the average angular travel is given as 33 minutes per hour (coincidentally, that's the rough angular size of the Moon at perigee - another Woo!) we can choose an arbitrary radius (or use the real radius) and vary it by 7 percent each way, run the simple calculation and roughly arrive at a perigee angular velocity of 35.5 minutes of arc per hour and about 30.8 at apogee.

Aha! So we just plug these numbers in and see how much the time varies, right?

Wrong again! Nothing about the Moon is simple!

We also have to account for the enlarged Moon at perigee (33.5 minutes of arc approximately) and the shrunken orb at apogee (about 29.4 minutes).

This shrinks the 'available white space' from 15 to 13 arcminutes at perigee and eats them up in a mere 22 minutes!

That, by the way, will be our definitive answer to the 1-degree eyepiece field question.

Should you observe at apogee, you can correct at the lazy rate of about once every half hour.

Now that wasn't such a tough question, was it?

It does illustrate something about lunar rates on telescopes, however. Though they cannot actually keep the Moon centered (except near the "middle" of the orbit) they can certainly slow down the need to correct the eyepiece. Quite a bit, actually.

You might say, gee Dave, then why do you sniff at them as relatively silly?

Same old same old. You'd better have Really Good Alignment (be it equatorial or star alignment of a goto) before that slight difference will work in your favor.

Small errors add up fast.


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