Kepler's Laws

Let us first consider applications of Kepler's third law in the solar system. If we let M₁ represent the Sun's mass and M₂ represent the mass of a planet or another object orbited the Sun, then if we adopt the Sun's mass (M₁ = 1.985 × 10³⁰ kg) as our unit of mass, the astronomical unit (a.u.; 1 a.u. = 149,597,871 km) as our unit of length, and the sidereal year (365.25636 mean solar days) as our unit of time, then (4π2/G) = 1, (M₁ + M₂) = 1 (we can neglect planet masses M₂ except those of the Jovian planets in the most precise calculations), and the formidable equation above is reduced to the simple algebraic equation P² = a³ where "P" is in sidereal years and "a" is in astronomical units for a planet, asteroid, or comet orbiting the Sun. Approximately the same equation can be found from the first equation if we let Earth be Planet B, since F_B =1 sidereal year and a_B is always close to 1 a.u. for Earth.

Let us return to the generalized form of Kepler's third law and apply it to planetary satellites; except for Earth-Moon and Pluto-Charon systems (these are considered "double planets"), one may neglect the satellite's mass (M₂=0). Then, solving the equation for M₁,

Measurements of a satellite's period of revolution (P) around a planet and of its mean distance "a" from the planet's center enable one to determine the planet's mass (M₁). This allowed accurate masses and mean densities to be found for Mars, Jupiter, Saturn, Uranus, and Neptune. The recent achievement of artificial satellites of Venus have enabled the mass and mean density of Venus to be accurately found. Also the total mass of the Pluto-Charon system has been determined.

Now we consider the use of Kepler's laws in stellar and galactic astronomy. The equation for Kepler's third law has allowed masses to be determined for double stars for which "P" and "a" have been determined. These are two of the orbital elements of a visual doublestar; they are determined from the doublestar's true orbit. Kepler's second law is used to select the true orbit from among the possible orbits that result from solutions for the true orbit using the doublestar's apparent orbit in the sky. The line joining the two stars must sweep out equal areas in the true and apparent orbits in equal time intervals (the time rate of the line's sweeping out area in the orbits must be constant). If the orbits of each star around their center of mass can be determined, then the masses of the individual stars can be determined from the sizes of these orbits. Such doublestars give us our only accurate information about the masses of stars other than the Sun, which is very important for our understanding of star structure and evolution.

In combination with data on the motions of the Sun, other stars, and interstellar gases, the equation for Kepler's third law gives estimates of the total mass in our Milky Way galaxy situated closer to its center than the stars and gas studied. If total mass (M₁ + M_s) is constant, the equation predicts that the orbital speeds of bodies decrease with increasing distance from the central mass; this is observed for planets in the solar system and planetary satellites. The recently discovered fact that the orbital speeds of stars and gas further from the center of the Milky Way than the Sun are about the same as the Sun's orbital speed and do not decrease with distance from the center indicates much of the Milky Way's mass is situated further from the center than the Sun and has led to a large upward revision of the Milky Way's total estimated mass. Similar estimates of the mass distributions and total masses of other galaxies can be made. The results allow estimates of the masses of clusters of galaxies; from this, estimates are made of the total mass and mean density of detectable matter in the observable part of our universe, which is important for cosmological studies.

When two bodies approach on a parabolic or hyperbolic orbit, if they do not collide at their closest distance (pericenter), they will then recede from each other indefinitely. For parabolic orbit, the relative velocity of the two bodies at an infinite distance apart (infinity) will be zero, and for a hyperbolic orbit their relative velocity will be positive at infinity (they will recede from each other forever).

The parabolic orbit is important in that a body of mass M₂ that is insignificant compared to the primary mass, M₁ (M₂=0) that moves along a parabolic orbit has just enough velocity to reach infinity; there it would have zero velocity relative to M₁. This velocity of a body on a parabolic orbit is sometimes called the parabolic velocity; more often it is called the "escape velocity." A body with less than escape velocity will move in an elliptical orbit around M₁; in the solar system a spacecraft has to reach velocity to orbit the Sun in interplanetary space. Some escape velocities from the surfaces of solar system bodies (ignoring atmospheric drag) are 2.4 km/sec for the Moon, 5 km/sec for Mars, 11.2 km/sec for Earth, 60 km/sec for the cloud layer of Jupiter. The escape velocity from Earth's orbit into interstellar space is 42 km/sec. The escape velocity from the Sun's photosphere is 617 km/sec, and the escape velocity from the photosphere of a white dwarf star with the same mass as the Sun and a photospheric radius equal to Earth's radius is 6,450 km/sec.

The last escape velocity is 0.0215 the vacuum velocity of light, 299,792.5 km/second, which is one of the most important physical constants and, according to the Theory of Relativity, is an upper limit to velocities in our part of the universe. This leads to the concept of a black hole, which may be defined as a volume of space where the escape velocity exceeds the vacuum velocity of light. A black hole is bounded by its Schwartzchild radius, inside which the extremely strong force of gravity prevents everything, including light, from escaping to the universe outside. Light and material bodies can fall into a black hole, but nothing can escape from it, and theory indicates that all we can learn about a black hole inside its Schwarzschild radius is its mass, net electrical charge, and its angular momentum. The Schwartzschild radii for the masses of the Sun and Earth are 2.95 km and 0.89, respectively. Black holes and observational searches for them have recently become very important in astrophysics and cosmology.

Hyperbolic orbits have become more important since 1959, when space technology had developed enough so that spacecraft could be flown past the Moon. Spacecraft follow hyperbolic orbits during flybys of the Moon, the planets, and of their satellites.