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Physics LibreTexts

12.13 Stellar Motion

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Graphic representation of Wein's Law. Click here for original source URL.

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Doppler Shift. Click here for original source URL.

Stars move. Although it’s not noticeable over a human lifetime, the positions of the stars in the night sky (and the shapes of the constellations) change. Spectroscopy also lets us study this important stellar property. The Doppler shift gives a very simple way to detect part of a star's motion — its radial velocity, or motion along the line of sight. The Doppler shift occurs because waves from a source get bunched up if the source is moving toward us and stretched out if the source is moving away from us. The Doppler shift of a star is proportional to its velocity, expressed as a fraction of the velocity of light. You should be careful to clearly distinguish the Doppler shift from the shift in the thermal spectrum of an object that occurs when it gets hotter or colder, described by Wien's law. A Doppler shift is an indication of motion, not an indication of a change in the temperature of an object. Since the Doppler shift for stars is very small, usually well under 0.1%, we cannot not detect a change in the color of the star. Instead, we use the sharp spectral features as markers to measure the shift. 
 

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Absorption lines in the optical spectrum of a supercluster of distant galaxies (BAS11) (right), as compared to those in the optical spectrum of the Sun (left). Arrows indicating Redshift. Click here for original source URL.

A consistent blueshift or redshift of all of a star's spectral lines proves that the star is either moving toward or away from us. For example, if the star is receding at 0.1% the speed of light (or 300 kilometers per second), the light will be redshifted by 0.1% of its normal wavelength. A line normally found at wavelength 500.0 nanometers would appear at 500.5 nanometers (a nanometer is a billionth of a meter or 10-9 meters). Or, if the star is approaching at 0.02% of the speed of light (or 60 kilometers per second), the light will be blueshifted by 0.02% of its normal wavelength. In this case, a line normally found at 500.0 nanometers would appear at 499.9 nanometers. Stars near the Sun have radial velocities that are 10 to 20 kms; roughly half are redshifts and roughly half are blueshifts.
 

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Relation between proper motion and velocity components of an object. At emission, the object was at distance d from the Sun, and moved at angular rate μ radian/s, that is, μ = vt/ d with vt = velocity transverse to line of sight from the Sun. (The diagram illustrates an angle μ swept out in unit time at tangential velocity vt.). Click here for original source URL.

The other component of a star's motion is its tangential velocity — the motion perpendicular to the line of sight. It cannot be measured as simply as radial velocity. In order to measure tangential velocity, we must measure the distance of the star and its rate of angular motion across the sky, called proper motion. From the numbers given above, we can calculate the size of the proper motion. If 20 kilometers per second is a typical velocity for a star, then it will move 20 × 3600 × 24 × 365 = 6.3 × 108 kilometers in a year. The angle moved is given by the small angle equation. In seconds of arc, it is 206,265 (d/D), where d = 6.3 × 108 kilometers. We will take the distance to be 1 parsec or D = 3 × 1013 kilometers. So the angle is 1.3 × 1013 / 3 × 1013 = 0.4 seconds of arc per year. This is small but measurable. 
 

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A parsec is the distance from the Sunto an astronomical object which has aparallax angle of one arcsecond. (1 AU and 1 pc are not to scale (1 pc = 206265 AU)). Click here for original source URL.

Of course, 1 parsec is the distance of the nearest star so this is the largest proper motion possible. The angular motion will be smaller for slower or more distant stars: 0.04 seconds of arc per year for a star at 10 parsecs and 0.004 seconds of arc per year for a star at 100 parsecs. Also, stars nearby in the Milky Way have random motions in three dimensions. The proper motion will be small if the star happens to be moving almost entirely in the radial direction. On the other hand, we can compensate for a small proper motion with patience by making observations for longer than a year. As with a parallax, the ability to measure proper motion improves with observations from space. The Hipparcos satellite was able to measure the proper motion for 2.5 million stars, and Gaia will extend this to nearly a billion.

These measurements of stellar motion are often lumped together with parallax in a branch of astronomy called astrometry. Astrometry is the precise study of the positions and motions of stars. If both radial and tangential velocities are known, they can be combined to give the star's space velocity (its true speed) and direction of motion in three-dimensional space relative to the Sun. The two directions are separated by 90° since one is along the line of sight and the other is across the line of sight. The famous theorem of Pythagoras allows us to combine the lengths of the sides of a right-angled triangle (a2 + b2 = c2). It also allows us to combine the two perpendicular components of velocity (va2 + vb2 = vc2, where va is the radial velocity, vb is the tangential velocity, and vc is the true space velocity). Space velocities of most stars near the Sun are a few tens of kilometers per second and are directed nearly randomly