The Astrogators' Guide to
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In addition to the annual reciprocating motion on the sky due to parallax, nearby stars also display a uniform drift across the sky. Astronomers call that uniform drift proper motion and they attribute it to the differences between the motions of the stars and our sun in their orbits about the center of the galaxy. All stars also display a radial motion relative to our sun. Every astrogator who plots courses for interstellar ships, whether hyperdrive-equipped or not, must take these motions into account. Regardless of whether a starship travels through normal space or through hyperspace, the target star will not occupy the location that an astronomer would infer from direct observation at the time of the ship's departure.
As long as twenty-four centuries ago some astronomers suspected that what they called fixed stars might drift slowly across the sky relative to what we might call the average starscape, but certainty only came in 1718 when Edmund Halley determined that Sirius, Aldebaran, and Arcturus had moved somewhat over half a degree from the positions originally determined by Hipparcos some eighteen and a half centuries earlier. To know that the stars have radial motions, velocities toward or away from Earth, much less measure them, astronomers had to wait until near the middle of the Nineteenth Century, when they got acquainted with the Doppler shift.
Astronomers measure a star's motion in the directions perpendicular to our line of sight through the star by plotting the star's location on the sky, relative to some very distant object (such as a quasar), year by year, and then calculating the differences between the star's measured coordinates. That calculation gives us the annual change in the star's coordinates in seconds of arc per year in Declination (μδ) and in coordinate seconds per year in Right Ascension (μρ).
We need to take some care with Right Ascension because of the way in which astronomers define it. We measure Right Ascension from west to east along the projection of Earth's equator on the sky (or along lines parallel to the celestial equator). Starting at the First Point in Aries, the point where the center of the sun appears to cross the celestial equator (actually the celestial equator moves over the sun's center as Earth revolves on its orbit) on the first day of spring, we divide the celestial equator into twenty-four hours. That fact means that one second of Right Ascension, one coordinate second, does not equal one second of arc: the get arcseconds on the celestial equator we must multiply coordinate seconds by fifteen; that is, μα=15μρ.
Further, we must account for the fact that Right Ascension provides astronomers with a measure analogous to longitude on Earth. That fact means that, just as the length of a degree of longitude depends upon the latitude at which we measure it, so the angular length of one second of Right Ascension depends upon the Declination (degrees measured north or south of the celestial equator, the celestial analogue of latitude) at which we measure it. Thus, to correct a coordinate second of Right Ascension into the true angle on the sky we must multiply it by fifteen times the cosine of the Declination at which we make the measurement.
As usual, we calculate the total proper motion (μ) through the Pythagorean theorem,
We can get away with that equation instead of using the spherical trigonometric equivalent because the patch of sky on which we measure the proper motion of the fastest moving star (Barnard's star) the patch approximates very closely a flat plane. Smaller patches give us even closer approximations to a plane.
In addition to calculating the magnitude of the proper motion, we also want to calculate a description of the direction in which the star moves across the sky, which description astronomers call the position angle. We define the position angle (θ) as the angle, measured in the counterclockwise sense, from the line pointing from the star to the celestial north pole to the line pointing from the star in the direction of the star= s proper motion. That definition gives us
which gives us
If we use absolute values of the coordinates in that equation and then extract the arctangent, we get a position angle that we must modify in accordance with the quadrant indicated by the algebraic signs on the coordinates. If we have
then we have four possibilities:
We get proper motion in arcseconds per year, but we want to have it in terms of actual velocity. To calculate that number we simply convert arcseconds into radians and multiply the result by the distance D to the star;
with D expressed in either kilometers or AU. If you calculate the velocity in AU per year, you can convert that result into kilometers per second through the conversion,
1 AU/yr=4.740470441 km/sec.
That conversion gives us a number that we can combine directly with the remaining component of stellar motion.
In concept, measuring the radial motion of a star should come easier to astronomers than does the measurement of the star's proper motion, because in concept the measurement yields immediate results. All the astronomer needs to do consists of taking the spectrum of the star's light and then measuring the Doppler shift of one or more of the bright or dark lines relative to the locations of those lines appearing in a similar spectrum made from the light of glowing elements on Earth. Of course, in concept does not coincide with in practice. Many phenomena, especially magnetic activity in the star's photosphere, can interfere with the measurement. Over time, though, astronomers can eliminate the errors by averaging them out. They then get the radial velocity (vr) more or less directly in kilometers per second. Because the radial velocity represents a change in the star's distance from Sol, a negative value for vr means that the star moves progressively closer to Sol and a positive value means that the star moves progressively farther away from Sol. Now we can calculate the star's total speed through space relative to our sun,
That calculation enables another calculation, that of the angleφ between the star's velocity vector and the line extending from Sol to the star. Taking the total speed as the hypotenuse of a right triangle, we have
Now we can make another right triangle by extending a straight line through the star and parallel to its motion, then drawing a straight line perpendicular to that extended line and passing through Sol. The hypotenuse of that triangle coincides with the line connecting Sol and the star now and has length D. We then take the angle 90E -φ and calculate
L1 represents the distance between the star and Sol when the star will come (or came) closest to Sol and L2 represents the distance that the star will move (or has moved) between that closest approach and now. If we have vr<0, then the star's perihelion approach lies in the future and if vr>0, the star's closest approach to Sol occurred in the past. We calculate the amount of time between now and the closest approach simply by dividing the distance the star must move (or has moved) by the star's total speed and get
Since we have velocity in kilometers per second and distance in lightyears, we must use the conversion factor
1 km/sec=3.33571636x10-3 lightyear per millenium.
Thus we can obtain the information that we need about a star's motion in space in order to determine where it will lie when our ship arrives. We can also calculate one datum that has no relevance to interstellar astrogation.
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