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Computing the Mass of A Celestial Binary





Astronomers love binary stars - two stars revolving around a common center of mass. That's because it's so easy to compute the combined mass of a binary star in solar masses, once you know the semi-major axis (mean distance between the two component stars) in astronomical units and its orbital period in Earth-years.


Simply use the equation below, courtesy of Johannes Kepler, where a = semi-major axis (mean distance) and p = orbital period.


Mass of binary system = a3/p2


Let's take the star 61 Cygni, a binary star in the constellation Cygnus the Swan. According to solstation, a = the mean distance (semi-major axis) between the component stars = 86.4 astronomical units, and p = orbital period = 722 Earth-years.


Here is the solution:

Mass of binary star = a3/p2

Mass of binary star = 86.4 x 86.4 x 86.4/722 x 722

Mass of binary star = 644,972/521,284

Mass of binary star = 1.237 solar masses


The star closer to the barycenter - center of mass - is the more massive star. The more massive star is 39.6 AU from the barycenter, and the less massive lodges 46.8 AU away. The more massive star has a mass of 0.67 solar (46.8/86.4 x 1.237 = 0.6700), whereas that of the less massive star is 0.567 solar (39.6/86.4 x 1.237 = 0.5669)


By the way, one solar mass = 1.9891 x 1030 kilograms


Computing the Mass of the Earth-Moon System


You can compute the mass of the "binary" planet - the Moon and Earth - in solar masses, with the same equation, whereby a = mean distance (semi-major axis) in astronomical units (AU) = 0.00257 AU (238,855 miles), and p = orbital period in Earth-years = 0.0748 Earth-years (27.322 days).

Mass of Earth & Moon = a3/p2

Mass of Earth & Moon = 0.00257 x 0.00257 x 0.00257/0.0748 x 0.0748

Mass of Earth & Moon = 0.000003 solar masses


Given that the mass of the Earth & Moon is 0.000003 solar masses, that makes the Sun about 333,333 times (1/0.000003 = 333,333) more massive than the Earth & Moon binary system. Because the barycenter of this double planet - the Earth & Moon - is buried within the Earth itself, Earth claims nearly 99% of the mass of the Earth & Moon system. The Moon is over 81 times farther out from the barycenter than the center of Earth is.


Image at right: More massive body and less massive body revolving around its barycenter or its common center of mass (red cross). Image credit: Wikipedia


By the way, Earth's mass = 5.97219 x 1024 kilograms


Computing the Mass of Jupiter


Because virtually all the mass in the vast Jupiter system in contained within the planet itself, we can use any one of Jupiter's moons to compute Jupiter's mass. Let's take the moon Io, whereby a = mean distance (semi-major axis) = 0.00282 AU (262,094 miles), and p = orbital period = 0.00484 Earth-years (1.769 Earth-days).

Mass of Jupiter = a3/p2

Mass of Jupiter = 0.00282 x 0.00282 x 0.00282/0.00484 x 0.00484

Mass of Jupiter = 0.0009573 solar masses


The figure of 0.0009573 solar masses might not look too impressive, since this means the Sun is some 1,045 times (1/0.0009573 = 1044.5826) more massive than Jupiter. But that makes Jupiter about 319 times the mass of Earth (333,333/1045 = 318.66)


Computing the Mass of Saturn


We may choose, for convenience, to figure out the mass of a solar system planet relative to Earth's mass. In this case, a = semi-major axis in terms of the Lunar Distance (LD) of 238,855 miles, and p = orbital period relative to the sidereal month of 27.322 days.


Let's compute Saturn's mass via Saturn's moon, Titan, given that a = semi-major axis = 3.18 LD (759,435/238,855 miles), and p = orbital period = 0.5836 sidereal month (15.945/27.322 days)

Mass of Saturn = a3/p2

Mass of Saturn = 3.18 x 3.18 x 3.18/0.5836 x 0.5836

Mass of Saturn = 94.42 Earthly Masses

Finis


copyright 2014 by Bruce McClure


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April 2014 Feature * June 2014 Feature