In Kepler's ellipse model, is the central mass at the center of the ellipse or at one focus?

An orbiting body sweeps equal areas in equal times. Is it moving faster near periapsis or apoapsis?

Two objects orbit the same star. One has semi-major axis \(4a\), and the other has semi-major axis \(a\). Find the ratio of their periods.

An elliptical orbit has periapsis distance \(7.0\times10^{10}\,\mathrm{m}\) and apoapsis distance \(1.9\times10^{11}\,\mathrm{m}\). Find its semi-major axis.

Find the period of an orbit with \(a=1.5\times10^{11}\,\mathrm{m}\) around a star of mass \(2.0\times10^{30}\,\mathrm{kg}\). Use \(G=6.67\times10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}\).

An object orbits with semi-major axis \(2.0\times10^{11}\,\mathrm{m}\) and period \(5.0\times10^7\,\mathrm{s}\). Estimate the central mass.

At periapsis \(r_p=4.0\times10^{10}\,\mathrm{m}\), and at apoapsis \(r_a=1.2\times10^{11}\,\mathrm{m}\). Use angular momentum conservation to find \(v_p/v_a\).

Two planets have the same semi-major axis, but one orbits a star with four times the mass of the other star. Compare their orbital periods.

An elliptical orbit around a fixed star has \(r_p=1.0\times10^{11}\,\mathrm{m}\) and \(r_a=3.0\times10^{11}\,\mathrm{m}\). Compare its period with a circular orbit of radius \(1.0\times10^{11}\,\mathrm{m}\) around the same star.

For an elliptical orbit with periapsis \(r_p\) and apoapsis \(r_a\), derive the period in terms of \(r_p\), \(r_a\), \(G\), and \(M\). Also derive \(v_p/v_a\), assuming the velocity is perpendicular to the radius at both apsides.