Computational Physics Homework 1

2016-11-28 本文已影响0人墨兵笔阵

1.Abstract

EXERCISES

4.9. In this section we saw that orbits are unstable for any value of β that is not precisely 2 in (4.12). A related question, which we did not address (until now), is how unstable an orbit might be. That is, how long will it take for an unstable orbit to become obvious. The answer to this question depends on the nature of the orbit. If the initial velocity is chosen so as to make the orbit precisely circular, then the value of β in (4.12) will make absolutely no difference. Of course, in practice it is impossible to construct an orbit that is exactly circular, so the instabilities when β ≠ 2 will always be apparent given enough time. Even so, orbits that start out as nearly circular will remain almost stable for a longer period than those that are highly elliptical. Investigate this by studying orbits with the same value of β (say, β = 2.05) and comparing the behavior with different values of the ellipticity of the orbit. You should find that the orientation of orbits that are more nearly circular will rotate more slowly than those that are highly elliptical.
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4.10. Calculate the precession of the perihelion of Mercury, following the approach described in this section.

2.Background

The solar system

The Solar System is the gravitationally bound system comprising the Sun and the objects that orbit it, either directly or indirectly. Of those objects that orbit the Sun directly, the largest eight are the planets, with the remainder being significantly smaller objects, such as dwarf planets and small Solar System bodies. Of the objects that orbit the Sun indirectly, the moons, two are larger than the smallest planet, Mercury.

Planets

Precession of Mercury

Mercury in color

In 1859, the French mathematician and astronomer Urbain Le Verrier reported that the slow precession of Mercury's orbit around the Sun could not be completely explained by Newtonian mechanics and perturbations by the known planets. He suggested, among possible explanations, that another planet (or perhaps instead a series of smaller 'corpuscules') might exist in an orbit even closer to the Sun than that of Mercury, to account for this perturbation. (Other explanations considered included a slight oblateness of the Sun.) The success of the search for Neptune based on its perturbations of the orbit of Uranus led astronomers to place faith in this possible explanation, and the hypothetical planet was named Vulcan, but no such planet was ever found.

The perihelion precession of Mercury is 5,600 arcseconds (1.5556°) per century relative to Earth, or 574.10±0.65 arcseconds per century relative to the inertial ICRF. Newtonian mechanics, taking into account all the effects from the other planets, predicts a precession of 5,557 arcseconds (1.5436°) per century. In the early 20th century, Albert Einstein's general theory of relativity provided the explanation for the observed precession. The effect is small: just 42.98 arcseconds per century for Mercury; it therefore requires a little over twelve million orbits for a full excess turn. Similar, but much smaller, effects exist for other Solar System bodies: 8.62 arcseconds per century for Venus, 3.84 for Earth, 1.35 for Mars, and 10.05 for 1566 Icarus.