Why
Back in ye olde days of exoplanet science (the 1990s), the surest way to demonstrate the discovery of a planet–as opposed to a star–was to measure its mass.  Planets have much lower masses than stars: Jupiter is about 1/1000 of the mass of the sun, and Earth is about 1/300,000 of the mass of the sun.  In the current canon of astronomical definitions, the most massive a planet can be is about thirteen times the mass of Jupiter.  Objects more massive than this fuse deuterium into lithium in their youth, and they are called brown dwarfs (or “failed stars,” because they are too low-mass to initiate the proton-proton fusion that powers real stars like the sun).

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An object on the star-planet continuum is classified by the mass of the object. From bottom right to top left: Jupiter, a giant planet, is about 300 times the mass of Earth. A brown dwarf, which fuses deuterium into lithium in its youth, is about 13-80 times the mass of Jupiter. A low-mass star, such as TRAPPIST-1, is about 80-200 times the mass of Jupiter, or 1/12-1/3 the mass of the sun.  The sun is roughly 300,000 times the mass of Earth. Image courtesy of NASA.

Nowadays, we know that small exoplanets are common and so a mass measurement is no longer critical for discovering a planet.  Nonetheless, a mass measurement is still one of the most powerful tools we have for confirming a planetary status.

Mass measurements not only help us find planets, but also help us understand their nature.  A planet’s mass is a fundamental physical parameter that conveys information about its composition, its gravitational interactions with other planets in the system, and its formation history.

How

In this post, I will cover two related methods for measuring planet mass: the radial velocity method and the astrometry method.  In a future post, I will discuss alternative techniques, including transit timing variations, microlensing, and direct imaging.

The Radial Velocity (Doppler) Method

Summary: We measure how much stars move toward and away from us over time based on the gravitational pull of their planets.

Planet masses measured this way: Several hundred

The most widely used technique for measuring exoplanet masses is to measure how the gravitational force of the planet pulls on the star.  As the planet orbits the star, the star also makes a miniature orbit that mirrors the planet’s.  In reality, both of these objects are orbiting the center of mass of the planetary system.  The physical principle at play is Newton’s Third Law of Motion: for every action there is an equal and opposite reaction.  The planet is pulling on the star as much as the star is pulling on the planet, but because the star is more massive it moves less.  This is illustrated in the diagram below.

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The physics of stellar orbits is analogous to a parent and child balancing on a see-saw.  On the see-saw, the parent and child balance the system by adjusting their positioning so that the center of mass is over the fulcrum.  In stellar orbits, the center of mass is the focus of both the planet and star’s orbit.  According to Newton’s Third Law of Motion, the mutual orbit of the star and planet do not affect the center of mass.  Therefore, the ratio of the planet mass to the stellar mass equals the ratio of the stellar orbit size to the planet orbit size, which also equals the ratio of the star’s speed to the planet’s speed.  By combining these relations with Kepler’s Laws, it is possible to solve for the mass of the planet, to the extent that the mass of the star is known.

How can we detect the tiny wobble of a star that corresponds to the orbit of a planet?  Many astronomers have tried to find planets by observing a star change its position on the sky, relative to the other stars, in a way that is consistent with a planet pulling the star.  This is actually maddeningly hard, as I will discuss below in the astrometry section.

What turns out to be much easier is to measure the motion of the star toward and away from our telescopes.  The forward-and-back speeds of things in space are so commonly measured that we have a special term for this quantity: radial velocity, an object’s speed in the radial (outward, along a radius) direction as defined by coordinates emanating from me, the observer.  An object moving away from me has a positive radial velocity, and an object moving toward me has a negative radial velocity.  If the object is also moving perpendicular to the radial direction, that motion does not contribute to the radial velocity.

We measure the radial velocity of the star by measuring the Doppler shift induced in the dark absorption lines of the star’s spectrum.  In 1995, Michel Mayor, Didier Queloz, Geoff Marcy, and Paul Butler used this method to detect and confirm 51 Pegasi b, a planet roughly half the mass of Jupiter and the first exoplanet found orbiting a sun-like star.

51Peg
The radial velocity of the star 51 Pegasi (vertical axis) changes over time (horizontal axis), completing one orbit every 4.23 days. The radial velocity is largest when the star is moving away from the observer and smallest when the star is moving toward the observer. Using the amplitude of the radial velocity signal, the orbital period of the planet, the mass of the star, and Kepler’s Laws, Marcy & Butler calculated that the minimum mass of the planet is 0.45 times the mass of Jupiter.*

It is easiest to measure the Doppler shifts due to a planet that is massive and close to its star, like 51 Pegasi b, since the gravitational pull of such a planet on the star is large.  51 Pegasi b moves its star fifty-five meters per second (the speed of an aggressive autobahn driver).  For comparison, Jupiter moves the sun twelve meters per second (the speed I drive in neighborhoods with children at play), and Earth moves the sun ten centimeters per second (the pace of a scurrying cockroach). The best spectrographs, which can currently measure motions of one meter per second, are not yet advanced enough to detect an Earth-mass planet in an Earth-like orbit around a solar-mass star.

 

Astrometry

Summary: We measure how much a star moves side to side in the sky due to the gravitational pull of a planet.

Planet masses measured this way: 0 (as of July 2017)

Astrometry means “measuring the positions of stars.”  It seems simple enough, but while astrometry is one of the oldest lines of inquiry in astronomy, it is also one of the trickiest.

We live in a galaxy full of moving stars.  However, the celestial sphere model, in which stars are imagined to be stuck on a rotating glass sphere, was popular for a long time because stars do not appear to move very much.  This is because although the stars move at impressive speeds, the stars are at great distances, and so their angular motions on the sky (also called proper motions) are tiny.

The stars’ proper motions are challenging to measure because any error in the measurement of the positions of stars will result in apparent motion of the stars.  If I observe that my star is further to the right today than it was yesterday, I might conclude that the star has moved to the right, when in reality my measurement process simply cannot resolve the small amount by which the star actually moved.  For thousands of years, measurement errors obscured the true motions of the stars, making it excruciatingly difficult to measure how stars move in the galaxy, let alone wobble due to a planet.

At the time of writing, exoplanets have not been detected via astrometry.  Peter van de Kamp claimed to find a whole system around Barnard’s Star this way, but later work falsified his claim and found that imperfections of his telescope had mimicked the stellar motion.

However, exoplanet detection via astrometry is about to get a makeover via the Gaia mission, which will measure unprecedentedly precise stellar positions from the vantage of atmosphere-free space.  Tracing the arcs of stellar motion on the sky, we will be able to determine the orbits and masses of their hidden planets.  Stay tuned!

The Future

The radial velocity method and astrometric method give complementary information about the star’s wobble.  The radial velocity describes the forward-back motion, and astrometry describes the side-to-side and up-down motion.  Combining these two types of measurements will allow us to describe the 3D motions of star in space, which translates to a description of the 3D motions of all their (detectable) planets!  Mapping the 3D orbits of planets will allow us to calculate accurate planet masses and to probe their possible formation histories in exquisite detail.

 

*Without additional orbital information, the radial velocity method only allows the computation of a minimum mass for a planet, msini.  This is because at the times of maximized and minimized radial velocity, the star might also have some component of its motion in another direction on the sky.

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