Milky Way Galaxy

Distances Measures for Open Star Clusters

An attraction of open star clusters is that their distances from the Sun, and therefore the distances to hundreds of stars of the same heritage, can be derived through methods other than the measurement of annual parallaxes.   Open star clusters are distant system?only 10 of them are within 300 parsecs of the Sun?so deriving the distances to them from the parallaxes of their stars has been difficult.  Even with the best parallax measurements, the distances to stars are known to better than 10% only for stars within 100 parsecs of the Sun.  The nature of an open star cluster, however, provides alternative methods for determining distance.  These methods fall into two groups: distance from proper motions, and distance from stellar studies.

The proper motions of the stars in a cluster provide a means of deriving a cluster's distance from the Doppler shifts of stellar spectra.  The proper motion of a star is the rate at which the star moves across the sky; it is commonly measured in arc seconds per year.  Because the stars within an open cluster remain bound only if they have very small relative velocities, all stars within the cluster effectively have the velocity relative to the Sun.  The proper motions of the stars will therefore differ only through geometric effects.

The classical method for deriving a cluster's distance from the proper motions of the cluster's stars is to derive the convergent points?reminiscent of a vanishing point in art?for the cluster.  This is called the convergent point method.  The convergent points are the two point on the sky, separated by 180, that mark infinity for a cluster's motion.  All the stars in a cluster appear to travel from one convergent point to the other.  Think of how a star cluster would appear to us as it moved along a straight line.  Its path would be a great circle stretching from the initial convergent point to the final convergent point.  The cluster would begin as a dot at the starting convergent point.  As it moved away from the convergent  point, it would grow in size on the sky until it reached its maximum size when moving perpendicular to us.  It then would shrink in size as it moved towards its forward convergent point.  The path traced by any star in the cluster would be a great circle passing through the cluster's convergent points.  If one can determine the great circles for several stars in a cluster, then one can find the convergent points from the intersections of these circles.  The great circle for a star is derived from the star's proper motion.

The usefulness of the convergent points is that they tell us what fraction of a cluster's speed is directed along the line of sight.  If we can measure the velocity along the line of sight, then we know the velocity perpendicular to the line of sight.  This is just simple trigonometry: the velocity along the line of sight is v cos, where is the angle on the sky between the cluster and its forward convergent point, and v is the speed of the cluster, while the velocity perpendicular to the line of sight is v sin.  The velocity along the line of sight is found from the Doppler shift of emission and absorption lines in the spectra of cluster stars, which immediately gives us the perpendicular velocity if is known.  The distance to the cluster is then found by dividing the perpendicular velocity by the cluster's proper motion.

A variation on the convergent point method is to derive a convergent point from the doppler shifts of the stars' spectra rather than from the proper motions.  This is most easily visualized by imagining a cluster that is moving away from us along our line of sight.  The cluster stars with the largest redshifts would be those at the center of the cluster.  The cluster stars farthest from the center, and therefore farthest from the convergent point, would have the lowest redshifts.  This is simply a geometric effect: the radial velocity is proportional to cos.  The point of highest redshift therefore would give the forward convergent point for the cluster.  This effect would hold when the cluster is moving transverse to us: the Doppler shift of a star within the cluster would depend on the angle on the sky between the convergent point and the star.  By measuring how the doppler shift varies across a cluster, one can derive the convergent points.  The distance to the cluster is then derived from the angle between the cluster and the convergent point, the Doppler shift of stellar spectra, and the proper motion of the cluster.

You may be wondering why measuring the proper motion of a cluster is less troublesome than measuring the annual parallax.  Both involve measuring the positions of stars on the sky over time.  The difference is in the magnitude of the motion.  A star's annual parallax is induced by Earth's orbit around the Sun, and because the baseline is fixed, the size of the parallax is inversely-proportional to the distance to the star.  A star's proper motion, on the other hand, depends on the motion of a star relative to the Sun.  Because the average orbital velocity around the Galactic center is constant at about 210 km s^?1, the stars closer to the Galactic center orbit the Galaxy in less time than the Sun and its neighbors, while those farther out orbit the Galaxy in more time.  This shear in the rotation of the Galactic disk causes the velocity of the stars relative to the Solar neighborhood to change proportionally with distance from the Sun. So along a given line of sight, the stars at 100 parsecs move with twice the velocity relative to the solar neighborhood as the stars at 50 parsecs.  Because a star's proper motion is the velocity perpendicular to the line of sight divided by distance, the average proper motion is approximately constant with distance along a given line of sight.  For this reason, one can measure a proper motion when the annual parallax is unmeasurable.

The limitation on deriving a cluster's distance from stellar proper motions is not a disappearance of measurable proper motion, but rather the diminution of a cluster's size on the sky.  When the stars of a cluster are spread over many degrees, the great circles these stars travel on can be distinguished with great precision, and their crossing points, which are the two convergent points, are found to high precision.  When the cluster is tight on the sky, the great circles are very close, and the convergent points are found to lower precision, or cannot be found at all.  Because a cluster's size on the sky shrinks proportionally as distance increases, these methods fail if the distance becomes too great.  The open clusters are large enough physically to keep these methods from failing at distances where annual parallax measurements are impractical.

A second method of deriving the distance to an open cluster is to precisely measure the orbits of the binary stars within the cluster.  To accomplish this, the binary star systems must be visual binaries, meaning that one must be able to see each star on the sky?many binary systems are spectroscopic, meaning their stars are so close together that they appear as a single star on the sky, but their binary nature is apparent from the Doppler-shifted line in the system's spectrum.  With today's telescopic instruments, binary stars that are very close together can be resolved into two stars.  Open clusters contain many binary stars.  They tend to be concentrated at the cluster's center, because their masses are higher than the masses of individual stars.

One can derive a distance to a binary star by deriving a physical size for the orbit and comparing it to the size of the binary on the sky.  In principle, one needs Doppler shifts and precise positions over time for both stars in the system to derive a scale.  As a simple example, one can think of two stars in circular orbits around each other.  In this case, the stellar orbits projected on the sky would appear elliptical because of the orientation of the orbit relative to Earth; this apparent eccentricity would give us the orientation of the system.  The Doppler shifts of the stellar spectra would give us the velocities of the stars along the line of sight, and because we know the orientation of the system, we could derive the speed of the stars.  From the orbital speed and period, we could derive the physical size of each star's orbit.  Dividing these values by the angles traversed by each star, one would find a distance.  In general, the stars have eccentric orbits, so the derivation of a distance is in practice somewhat more involved than outlined for circular orbits.  The basic idea, however, persists: a physical size for a binary system is derived from the Doppler shifts of the stellar spectra and the period of the orbits, and these allow us to derive a distance from the proper motions of the binary stars.

One final method that is sometimes employed is to estimate a distance based on the Hertzsprung-Russell diagram for the cluster.  This is a method that relies on an understanding of stellar physics rather than on geometry.  The luminosity and color of a main-sequence star is principally set by its mass.  Low-mass stars are low-luminosity and red, and high-mass stars are luminous and blue.  This means that if one knows that a star is a main-sequence star, one can estimate its luminosity by measuring its color?the amount of blue light the star emits versus the amount of yellow light.  One can then derive a distance by measuring how bright the star is and comparing that value to the estimated luminosity.  Selecting the main sequence stars from a cluster is as simple as plotting their brightness against their color: the main sequence stars fall along a narrow curve on the plot.

Of course, deriving a distance from a star's color requires some knowledge of how luminosity depends on color.  This knowledge is acquired by studying stars that have distances determined through geometric methods.

There are some complications that arise in applying this method.  A main sequence star becomes more luminous as it age.  The chemical composition of the star also affects its luminosity.  One can compensate for these effects, but then the distance one derives becomes dependent on the models one uses.

Each of these methods has been used with great success over the past century.  How successful they are can be seen by comparing the distances they give for the Hyades cluster to the distance derived from recent measurements of annual parallax.  In the early 1990s, the stellar parallaxes of over 117,000 were measured by the ESA Hipparcos satellite, creating the currently most accurate catalog of stellar parallax.^[1]  These parallaxes are accurate to better than 1 milli-arc-second.  This precision is sufficient to give the three-dimensional placement of stars within the Hyades cluster, which is the open cluster closest to Earth.  The distance to the center of the Hyades cluster from the annual parallax is 46.340.27 parsecs.^[2]  The earliest distances to the Hyades open cluster came from the convergent point method, giving distances that were short of the annual parallax value by about 20% in the early part of the 20^th century.  By the end of the 20^th century, all of the methods discussed above were regularly giving values that are within 5% or better of the annual parallax value, which is quite respectable.^[2]