where , , , , , and . Here, , , and are the masses of the first planet, second planet, and star, respectively. It is assumed that . Hence, deduce that the general time variation of the osculating orbital elements , , , and is a linear combination of two normal modes of oscillation, which are characterized by
Demonstrate that in the limit , in which and , the first normal mode is such that and (assuming that ), whereas the second mode is such that and . Here, and . (Modified from Murray and Dermott 1999.)
where is the mass of the Sun, the radial distance of Mercury from the center of the Sun, the conserved angular momentum per unit mass of Mercury, and the velocity of light in vacuum. The second term on the right-hand side of the preceding expression comes from a small general relativistic correction to Newtonian gravity (Rindler 1977). Show that Mercury's equation of motion can be written in the standard form
where , and
is the disturbing function due to the general relativistic correction. Demonstrate that when the disturbing function is averaged over an orbital period it becomes
where and are the major radius and eccentricity, respectively, of Mercury's orbit. Hence, deduce from Lagrange's planetary equations that the general relativistic correction causes the argument of the perihelion of Mercury's orbit to precess at the rate
where is Mercury's mean orbital angular velocity. Finally, show that the preceding expression evaluates to .
where , is the terrestrial mass, , , are spherical coordinates that are centered on the Earth, and aligned with its axis of rotation, is the Earth's equatorial radius, and , (Yoder 1995). In the preceding expression, the term involving is caused by the Earth's small oblateness, and the term involving is caused by the Earth's slightly asymmetric mass distribution between its northern and southern hemispheres. Consider an artificial satellite in orbit around the Earth. Let , , , and be the orbital major radius, eccentricity, inclination (to the Earth's equatorial plane), and argument of the perigee, respectively. Furthermore, let be the unperturbed mean orbital angular velocity.
Demonstrate that, when averaged over an orbital period, the disturbing function due to the term takes the form
Hence, deduce that the term causes the eccentricity and inclination of the satellite orbit to evolve in time as
deduce that the variations in the orbital eccentricity and inclination induced by the term can be written
respectively, where .
where is a constant. Hence, deduce that the time required for a dust grain in an orbit of initial major radius and initial eccentricity to spiral into the Sun is