## Formula for Jupiter's central meridians and Great Red Spot transits

Recently, someone who saw my Web page about Great Red Spot transit times asked for the formulae or algorithms I used to determine my results. I was not able to give a really simple answer to that question.

In determining GRS transit times, you start by finding out the longitude of the GRS on Jupiter. There are two complications to this: first, the GRS is a cloud feature, and like any other cloud feature, it moves slowly around the planet. Sky & Telescope often publishes updated values for the GRS longitude, as do some other Web sites.

The second complication is that Jupiter has three different systems of longitude. This problem arises because Jupiter doesn't rotate as a solid object; clouds near the equator rotate a little faster than those closer to the poles. So System I is used for measuring features close to the equator, and System II for features in the temperate regions. And when radio bursts were detected on Jupiter, it was found that they rotate at yet a third rate, so a System III had to be created. This is the rate of rotation of the core of Jupiter, and is of essentially no interest to visual observers. The GRS is far enough from the equator to be measured in System II.

Actually computing these central meridians tends to be painful. At one time, I used the method described by Jean Meeus in Astronomical Algorithms . I suspect that Meeus' method is in widespread use, since Astronomical Algorithms is perhaps the most commonly used reference work for astronomical computing. He provides a precise method that relies on getting exact positions of Jupiter and the Earth, plus a great deal of spherical trigonometry. I later switched to an equally precise, but still very complex, method based on vector and matrix manipulations.

There is a BASIC program to determine the Jovian central meridian in Systems I and II from the Sky & Telescope site. With small modifications, it could also include System III, though I dunno if anyone would care very much. The same site also gives a BASIC program for the central meridian of Mars. Both are based on slightly more complicated (and probably more accurate) algorithms than those given below, but aren't too tough to deal with.

As if that weren't enough, Dan Bruton has posted a different BASIC program to determine Jovian central meridia, based on one published in Astronomy magazine. It only gives System II longitudes (which is all you need for the GRS.)

When I originally got the inquiry on this subject, it occurred to me that it might be possible to compute the central meridians more simply. After a bit of work, I found a method that is accurate to about a degree, good enough to provide GRS transit times to within about a minute. This method is really simple, and accurate enough for most purposes.

The crux of the method is the realization that the central meridians are essentially simple linear functions of time, plus a correction.

```CM( System I) =   156.84 + 877.8169147 * jd + correction
CM( System II) =  181.62 + 870.1869147 * jd + correction
CM( System III) = 138.41 + 870.4535567 * jd + correction *
```

where 'jd' is the Julian Day, and the 'correction' is computed as follows:

```jup_mean = (jd - 2455636.938) * 360. / 4332.89709
eqn_center = 5.55 * sin( jup_mean)
angle = (jd - 2451870.628) * 360. / 398.884 - eqn_center
correction = 11 * sin( angle)
+ 5 * cos( angle)
- 1.25 * cos( jup_mean) - eqn_center
```

(All of the above angles are in degrees.) That's essentially all there is to the method, aside from the usual discarding of multiples of 360 degrees.

Once you have the current System II central meridian for a given time, computing the next (or previous) GRS transit time is a snap. Suppose you know that the current System II central meridian is 123.5 degrees, and you've looked on the Sky & Telescope Web site (or in a current issue of the magazine) and found that the GRS longitude is about 74 degrees. So we've gone (123.5 - 74), or 49.5, degrees past the last transit. Look at the above formula for System II, and you'll see that Jupiter rotates at about 870.187 degrees a day. So the last transit must have been (49.5 / 870.187), or about .057, days ago.

Similarly, the next transit is (74 + 360 - 123.5) degrees ahead of us, or 310.5 degrees. Dividing again by 870.187, the next transit lies .357 days in the future. (In these GRS computations, we're taking advantage of the fact that the "correction" varies slowly with time. It won't change much over these day-to-day intervals.)

So how do these formulae work? You don't need to know this if you just want to use the formulae, but you may be curious as to where they came from. If so, read on:

If you were observing from the sun, and if Jupiter moved in a circular orbit, the "correction" would be zero, all the time. Jupiter would appear to rotate at absolutely constant rates in each system. Of course, most of us observe from Earth, and Jupiter is in a somewhat eccentric orbit. So we see Jupiter from a slightly varying angle, and it appears to slow down and speed up very slightly in a periodic way. That's where the "correction" comes in. It consists of four sine and cosine terms. (For greater accuracy, still more terms could be added. But these are the four largest terms.)

The first term has an amplitude of about eleven degrees and period about 400 days, the average time between oppositions of Jupiter (or "synodic period"). We have to include it because, as seen from Jupiter, the Earth can be as much as eleven degrees from the sun. This term contributes nothing when the Earth, Sun, and Jupiter are in a straight line (conjunction or opposition), but up to 11 degrees at times in between.

The second term, with an amplitude of about five degrees, reflects the time it takes for light to cross the orbit of the earth. When Jupiter is at opposition, we're about sixteen light-minutes closer to it than we are when Jupiter is in conjunction with the sun. Over those sixteen minutes, Jupiter can rotate about ten degrees.

The remaining two terms involve the eccentricity of Jupiter's orbit. The third term, of amplitude 1.25 degrees, is also a "light-time lag" term. In this case, it means that Jupiter is about .5 AU closer to us at perihelion than it is at aphelion. .5 AU is about four light-minutes, and during that time, Jupiter could rotate about 2.5 degrees.

The final term, again of amplitude 11 degrees, is caused by Jupiter moving faster in its orbit at perihelion than it does at aphelion. To include this, the "correction" term must include the "equation of the center", the difference between Jupiter's true anomaly and mean anomaly. This can be approximated quite closely as a sine term.

Several of the constants in the above formulae came from Astronomical Algorithms. I got the rotation rates from the Bureau des Longitudes, and derived the remaining data.

Small but important change: Jan Šlégr pointed out that the value for System III wasn't quite right. It looks as if I used a slightly inaccurate value for the rotation rate in that system (it wasn't given to enough decimal places). To fix this, the formula

`CM( System III) = 272.29 + 870.4529147 * jd + correction `

should be replaced (as it is above) with

`CM( System III) = 138.41 + 870.4535567 * jd + correction `