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Body Double: Understanding the Astrophysics of a Multi-Planet Binary Star System

A look at the newly discovered Kepler 47 exoplanetary system and the mathematical models that describe it



NASA/JPL-Caltech/T. Pyle

Last month astronomers announced the first discovery of multiple planets orbiting a pair of stars. Binary star systems, which comprise two stars orbiting around a common point, are very common—about half of sun-size stars are thought to be members of binary star systems. But it was only in 2011 that astronomers reported the first confirmed instance of a binary system with even a single planet. The newly discovered system, named Kepler 47, consists of a star about the size of the sun and a smaller companion, orbited by two planets.

The reason it has taken so long to find exoplanets around binary stars isn't because they are rare, but because it is hard for us to see them from an earthly vantage point. The data for the Kepler 47 discovery come from the Kepler space telescope, which orbits the sun, not Earth, so our planet doesn't block its view. The spacecraft's mission is to search for planets in the habitable zones around their suns, hopefully giving scientists a better idea of how many systems have planets that could support life.

Kepler detects exoplanets by measuring the tiny changes in a star's brightness caused by the passage of a body in front of it, à la the transit of Venus last witnessed in June. Thus, Kepler can only find planets whose orbits line up with its line of sight. Otherwise the planet will not alter the star's light from the telescope's point of view. Scientists thereby are limited in how many of these systems they can find; distant stars don't arrange themselves to provide a better view. Daniel Fabrycky, an astrophysicist at University of California, Santa Cruz, and one of the authors of the paper announcing the discovery, says, "the punch line is that there's nothing special about it; planets can form around binary stars, too."

Any description of celestial motion starts with classical physics—specifically, Newton's law of gravitation. This law describes the force exerted on one body by another. It says that the force is proportional to the masses of the bodies and inversely proportional to the square of the distance between them. Because force equals mass times acceleration, one can use these two different descriptions of force to write a system of differential equations that will describe the motions of any number of bodies. The problem of describing their motion is called the n-body problem, where n represents the number of bodies in a system. It is actually not too hard to derive the equations, once given the masses and velocities of a bunch of bodies. But only the 2-body problem yields an explicit answer. The solutions for other values of n have to be found through numerical methods, which lead to arbitrarily accurate approximations, rather than by finding universal solutions to the equations.

A binary star system with two orbiting planets is a 4-body problem. But, Fabrycky says, in the case of Kepler-47 the two planets are so much smaller than the two suns that their masses can be assumed to be zero. Thus, the 4-body problem becomes a 2-body one for the two stars, because the planets don't tug on them very much. The solution to this 2-body problem is for each star to be in an elliptical orbit around the two bodies' center of mass. These orbits are described as Keplerian because Johannes Kepler—the Renaissance astronomer for whom the spacecraft is named—first noticed that the planets in our solar system have elliptical, not circular, orbits.

(At left, a simulation of two similarly sized stars in elliptical orbit around their center of mass, the red cross. Source: Wikimedia Commons/Zhatt)

In the Kepler 47 system, however, the two stars' orbits of one another are nearly circular, and the stars are very close together. The length of their orbits is about 7.5 days, and they are only about 13 million kilometers apart. (For perspective, that is about a quarter of the average distance between the sun and Mercury.) The researchers determined these parameters using Doppler spectroscopy, which measures emitted light to determine how a smaller body exerts gravitational force on a larger body.

To model the planetary orbits, the researchers used numerical models to find the orbits that best fit the planetary transit data collected by Kepler. The inner planet takes about 49.5 days to orbit the stars and is only about 44 million kilometers away from them; the outer planet takes about 303 days and is about as far away from the pair as Earth is from the sun. That puts Kepler 47's outer planet in the habitable region, where liquid water could exist. With such a location, the outer planet might call to mind the Star Wars world Tatooine, but the researchers say that this planet is probably a gas giant, which is not a hospitable environment for Earth-like life. But Fabrycky notes that if you were in a hot air balloon looking through the outer planet's atmosphere, the dual sunset might look like something out of Star Wars, with the two suns positioned very close together at the horizon. And if the planet has a moon, there's a tiny chance that it could support life.

Ellipses are good approximations of the planets' paths, but they are not perfect because the gravitational forces acting on the worlds are not coming directly from the two stars' center of mass. Fabrycky says, "They're almost going on an ellipse, but they wiggle around it a little bit due to the stars dancing around each other." Over time these forces will cause the orbits of the planets—and thus the shape of this distant solar system to evolve.  Fabrycky says that it will take only about a century for the system to change substantially. (In our solar system, by contrast, those kinds of changes take tens of thousands of years.)

Going forward, astrophysicists are continuing to use the telescope to search for potentially habitable planets, applying Kepler's laws to describe their motion. Fabrycky expects that in the next few years researchers will be announcing many more exoplanetary discoveries in both single- and multiple-star systems. Astronomers hope that Kepler will help them find answers to big questions about our universe. Are all binary star systems planar? What different processes give rise to planets? And of course, is there anyone else out there?

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