Mathematical Marginalia
Classical and Quaint Topics in Mathematics

Urmel's Twin Planet

In the children's novel Urmel fliegt ins All  [Kruse 2001] (Urmel goes to space ) the Urmel (of course accompanied by Prof.Habakuk Tibatong, Wutz and most of the other inhabitants of Titiwu) visits the planet Futura. This is a twin planet  of the earth, revolving on the same orbit around the sun, but always on exactly the opposite position relative to the sun. Regardless of the fact, that such a planet would cause perturbations of the orbits of Mercury and Venus, and thus could not stay undetected for a long time, the following question arises: Assume, that the twin planet had also a moon of the same size as our moon, revolving around its planet at the same distance as our moon. Shouldn't this moon be visible from time to time near the rim of the sun?

We try to answer this question by mental arithmetic, using only commonplace astronomical facts:

Under the assumption that the twin earth, as seen from the earth, is always exactly behind the midpoint of the solar disk, the problem is, to compute the angle of view both of the solar disk and of the twin earth-moon-system, when seen from the earth.

It's commonplace that the sun's distance from the earth is 8 light minutes, hence the twin planet is 16 light minutes apart. The moon is about 1 light second apart from earth, thus the diameter of the orbit of the twin moon around its twin earth is 2 light seconds. Hence the tangent of half the angle of view of the orbit of the twin moon is \[\tan\frac{\varphi}{2} = \frac{1''}{16'}= \frac{1}{960}\]

Admittedly, the diameter of the sun does not belong to the commonplace astronomical facts. But it is generally known that the disks of sun and moon are about the same size, when viewed from earth. The distance of the moon from the earth (about 400000 km) is commonplace, but its diameter of about 3500 km probably is not. However, you may remember that the diameter of the moon is roughly $1/3$ of the earth's diameter, so let's take the value $12000/3 = 4000.$ Now we can compute the tangent of half the angle of view of the moon disk (and of the sun disk): \[ \tan\frac{\psi}{2} = \frac{\frac{1}{2}4000}{400000} = \frac{1}{200} \]

For small angles it holds approximately $\tan\varphi \simeq \varphi$ (Small-angle approximation), hence we can say, that the apparent diameter of a hypothetical twin earth-moon-system would be roughly $1/4$ of the apparent diameter of the sun. Such a twin system would not  be visible from earth.

Of course, you could derive this result without any computation, if you consider the fact that the whole Earth-Moon-System fits readily into the Sun with its diameter of roughly 1.4 million kilometers. But I doubt that this fact is commonplace.

Whether a twin earth is compatible with Newton's laws, requires deeper investigation. On the one hand the axis from our earth to the twin earth should pass through the sun's midpoint, otherwise there would be periodically changing gravitational pulls between the two earth's, which would probably destabilize the whole system. On the other hand, such a geometry is not compatible with Kepler's second law, which states that the vector from the sun to the planet sweeps out equal areas in equal time intervals. Thus the planet runs faster in its perihelion and slower in its aphelion. As a consequence the mirror-planet must not run on the same path as the Earth, but the path must also be mirrored, in other words, the Sun is in one focal point of the earth's path and in the other focal point of Futura's path. If such a system can be stable for several billion years, taking into account the influence of Venus and Mercury, which obviously have no mirror partners, can be computed by mental arithmetic, only if one has the capabilities of Urmel's alien friend Neschnem-Kopf Otto.

But on a site, dedicated to Math and Java, it cannot be avoided to present a little Java program, which simulates the planet motion in our solar system. So you are encouraged to look at the Java program section, where you will find a program called Newton. The simulation of a planet system with a twin planet Futura may be started with a few clicks, and I will not spoil the party by whistle-blowing, how this system develops in the simulation.