• Astrometric detection of planets
• Astrometry with interferometers
• Limits on ground-based astrometry
• Dual star astrometry

Astrometric Detection of Planets

A planet and star will orbit about their common center of mass. Because the mass of the planet is so small compared to the mass of the star, and the great distance to a typical nearby star, the reflex motion of the star is very small. Astrometry is the measurement of the angles between stars in the sky with very high precision. With sufficient accuracy, an astrometric instrument can measure the motion of a star caused by an orbiting planet.

While a Sun-Jupiter system at 10 parsec would have an astrometric signature of 1 mas p-p, most of the planets discovered to date have much smaller astrometric. The astrometric signature is fully determined by the mass of the star, the mass of the planet, the distance between the planet and the star, and the distance between the planet and our solar system. Below is a graph with planet mass on the Y axis and semi-major axis (orbital radius) on the X axis. The distance to the planetary system is assumed to be 10 parsec.The graph below shows the position of several planets that have been discovered with the radial velocity technique. The two slanted lines with 10, 50 microarcsec (uas) show the astrometric detection limits for astrometric instruments with 10 and 50 uas accuracy.

Astrometry with Interferometers

Astrometry, the measurement of the angle between stars can be done in a number of ways. Light from a star enters the interferometer through the two collecting apertures shown below. In order for the light to interfer, the optical path in the two arms must be equal. The paths are made equal by adjusting the position of mirrors in the optical delay line. The position of the mirrors in the delay line are measured with a laser interfrometer, with accuracies between ~10 picometers (pm) for space based interferometers and ~1 nanometer (nm) for ground based interferometers.

One interferometer, shown above measures the position of a star with respect to the baseline vector of the interferometer. Mathematically, the delay line position where stellar fringes are observed is related to the angular position of the star by X=S*B+c, where S is a unit vector to the star B is the baseline vector joining the two collectors and c is the zero point of the delay line. For the two dimensional case above, X=b*cos(theta) + c. b is the baseline length, theta is the angle between the baseline and the star.

An interferometer only measures the angle projected onto its baseline. In general stellar fringes must be observed at two or more baseline orientations to determine two angular coordinates of an astronomical object. For ground based interferometers, the baseline is fixed to the Earth and will rotate with the Earth. In space the interferometer/spacecraft must reorient the baseline to measure both angular coordinates.

Ground Based Limits to Astrometry

Conventional ground based telescopes with ~ 1 meter apertures measuring stars over a ~0.5 degree field of view are limited by atmospheric turbulence to a few milliarcsec(mas) accuracy for a ~ 1hr measurements. Atmospheric turbulence will cause the position of a star to fluctuate with time. However if two stars are sufficiently close together in the sky, the motion of the two stars will be almost identical because the light from the two stars traverse almost identical paths through the atmosphere.

For small telescopes (or interferometer) the amount of atmospheric inudced motion is independent of the size of the telescope. But when the size of the telescope or interferometer baseline grows to be larger than the separation of the two stellar beams at the top of the troposphere (~10km altitude) the behavior changes. For sufficiently large telescopes/interferometer, the angular motion of the stars becomes smaller for larger interferometer baselines.

As can be seen from the graph above, differential astrometric measurements with 10~30 uas are possible over a ~20 arcsec field of view with interferometers (or telescopes) ~ 100 meters in diameter at an excellent site like Mauna Kea. In early 1992 data was taken at the Mark III interferometer on Mt Wilson (show above) to validate the theoretical calculations. The Palomar Testbed is designed to observe two stars simultaneously to measure the angle between them with extremely high precision. But in addition to verifying the atmospheric limit for a 100m interferometer, there is an additional goal of identifying and removing instrumental errors as well. Lastly, the interferometer is to be used to conduct a modest survey of nearby stars, stars with known planets (from the radial velocity surveys) as well as new stars.

If one wishes to make measurements with significantly higher accuracy than 10 uas, the only solution is to go to space. The need to do this comes from the very small astrometric signatures of Earth like planets. While a Jupiter-Sun system has a signal of 1 mas, an Earth-Sun system has a very small 0.6 microarcsec signal. When an Earth-like planet is discovered, the only technique that can be used to unambiguously measure it's mass, is astrometry.

The very high astrometric accuracies possible with long baseline interferometer are possible only if two or more stars are observed simultaneously. This is in contrast to all other stellar interferometer built todate. The basic idea behind a dual star interferometer is quite simple and is shown below.

Each telescope at the ends of the interferometer uses a field splitter. Shown here is a mirror with a pinhole in the coating. Light from one star goes through the hole while light from the other star is reflected by the coating. Two separate beams of starlight are directed to two separate beam combiners. A laser metrology system "ties" the two interferometers together, so that active optics can be used to compensate for vibration and thermal drift of the equipment that are different between the two optical trains.