[Physics] Planetary alignment of the gas giants

Question

There is this nice and new applet of the orbits of the planets around the sun:

Solar System Orrery

You can click and drag any planet and watch their relative orbits evolve in time. I was curious about planetary alignments$^\color{red}{1}$ and have these questions:

1. I have observed that the first three gas giants (Jupiter, Saturn, and Uranus) align at years 1623, 1941, 2259, etc, with an approximate interval of 317 years. How do we explain $317$ from the orbital period of these three planets?
2. In what years will the fourth gas giant (Neptune) join in the alignment, and in what intervals?
3. Out of curiosity, in what years will Earth be aligned to all four gas giants, and in what intervals?

$\color{red}{1}$. The orrery is divided into 12 sections. We will define an alignment as "in a line in the same section".

P.S. This PSE post says there were alignments on 1665 and 2673. With this orrery as a visual aid, these years don't look impressive. Also, this Cornell Univ site says there will be at 2854 but it doesn't look much either. (A picture really is worth a thousand words.)

Answer 1

How long time does it take before three planets achieve the same relative position?

The answer is never, except for the case when their orbital periods can be expressed with low integers, like the 4:2:1 resonance of Io, Europa and Ganymede

However, what you are asking about is when they are going to be in almost the same position again, a quazi-period.

To find those periods, we are pretty much only left by brute forcing as our method. A nice little detail about the case with three planets is that the inner planet is always aligned with one of the other ones at the closest three-planet alignments. That allows us to calculate accurate solutions. In the cases of four or even five planets I simply give up.

To check all the possibilities, we can use a program. Here is an example of a function in JavaScript returning a list of quazi-periods and alignment error:

sameLine = function (period1,period2,period3,limit){
    results = [];
    newMargin = 1;
    synodic1 = 1/(1/period1 - 1/period2);
    anomaly1 = (synodic1/period1) % 1;
    synodic2 = 1/(1/period1 - 1/period3);
    anomaly2 = (synodic2/period1) % 1;
    alert(synodic1+","+synodic2);
    for (i = synodic2; i < limit; i+=synodic2){
        numb1 = i/synodic1 - (i/synodic1) % 1;
        numb2 = i/synodic2 - (i/synodic2) % 1;
        err1 = Math.abs((numb1 * anomaly1 - numb1 * synodic1/period3) % 1);
        err2 = Math.abs((numb2 * anomaly2 - numb2 * synodic2/period2) % 1);
        if (err1 > 1 - err1){
            err1 = 1 - err1;
        };
        if (err2 > 1 - err2){
            err2 = 1 - err2;
        };
        if ((err1 < newMargin) && (numb1 > 0)){
            results.push([numb1 * synodic1,err1]);
            newMargin = err1;
        };
        if ((err2 < newMargin) && (numb2 > 0)){
            results.push([numb2 * synodic2,err2]);
            newMargin = err2;
        };
    };
    return results;
};

For Jupiter, Saturn and Uranus, I get the following output:

Time,,,error

13.81170069444156,,,0.30449020900657225
39.71676854387252,,,0.12441143762575813
41.43510208332468,,,0.08652937298028318
139.00868990355383,,,0.06455996830984656
138.1170069444156,,,0.04490209006572288
179.55210902774027,,,0.041627282914560304
317.6691159721559,,,0.0032748071511630172
3991.581500693611,,,0.002329597100612091
4309.250616665767,,,0.0009452100505313865

The first of this periods is of no use, as the error in alignment is almost a third of an orbit. Note that the one you found (that is really impressive you did,actually) gives an error in the alignment of less than a percent. We have to look at periods more than a thousand years long to find any better alignment.

Be sure to feed this function with accurate orbital periods.