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# Galactic Distances through Parallax

Distance in astronomy is a problem. How does one measure it? Within the solar system we can measure distance by measuring the amount of time it takes for radio signals to travel to and from satellites, or the amount of time it takes for a radar signal reflected by a planet such as Venus to make the round trip from Earth. This allows us to precisely determine a physical length for the Earth's orbit, from which we can derive other lengths through geometry.

For instance, it is very easy to derive the distances of the planets from the Sun in terms of Earth's distance from the Sun by applying geometry to the observations of planetary motion against the stars. The path that a planet follows against the stars is determined by both its orbit and Earth's orbit around the Sun. For instance, Saturn's orbital motion produces a slow, almost constant motion along a great circle on the sky, but Earth's orbital motion superimposes on this a cyclic motion back and forth on the sky. The effect of Earth's motion is dependent on the size of Saturn's orbit relative to Earth's, so we can disentangle the effects of Earth's motion after observing Saturn for several years, and derive the size of Saturn's orbit relative to Earth's. In terms of Astronomical Units, the distances between the planets have been known for centuries. The distances in physical units, however, were very uncertain until the light travel times across the solar system could be derived.

This geometric technique, which works wonderfully for the Solar Systems, also works well for the nearest stars. By observing this parallax of the nearby stars against the distant stars over the course of a year, one can derive the distance of a star in units of AU. In astronomy, we have a unit specifically geared to this method: the parsec. This unit of measure is the distance of a star that moves by one arc second against the background when the observer moves by one Astronomical Unit. So over the course of a year, because the Earth moves by 2 AU, a star at one parsec moves in an ellipse with a major axis length of two arc second. The distance to a star is then simply found by measuring the semimajor axis over a year for that star in units of arc second, and inverting the value. As a practical matter, the closest star, Proxima Centauri, is at 1.3 parsecs, so all stars have a parallax that is less than an arc second.

The most ambitious experiment to measure stellar positions and motions was the Hipparcos mission conducted by ESA. From the sky surveys of the Hipparcos satellite, researchers produced a catalog of stars with parallaxes measure to an error of one milli-arc-second. This means that Hipparcos was able to measure the distance of a star to an accuracy better than 10% (standard deviation over parallax) if the star is within 100pc of Earth; in the catalog there are 20853 stars with distances measured to this accuracy. The Hipparcos catalog is complete for the full sky down to magnitude 7.3. The limiting magnitude is 13.0. This catalog is currently the best astrometric catalog available.

The stars in the Hipparcos are very close to Earth. The thickness of the galactic plane is about 100 parsecs, the distance of the Sun from the Galactic center is about 7.6 kpc, and the radius of the Galaxy is about 15kpc. We therefore have parallaxes for only a tiny percentage of the stars in the Galaxy.