2 - STELLAR MASSES AND RADII
Stellar Radii
Another important parameter to determine is the size of a star. Timing the duration of the ingresses and egresses of eclipsing binary stars provides this information. There is a small cottage industry in the determination of stellar radii of such stars, and it can be done using small telescopes with lots of available time. I will not discuss the details here (see your textbook).
Other ways to get radii are Michelson interferometry, intensity interferometry, lunar occulations, and speckle interferometry.
Lunar Occultations
If a star is located within 5 degrees of the ecliptic, eventually it will be occulted by the Moon. Precise timing of the occulation (where diffraction fringes are usually measured) yields the angular size of the source. If the distance is known from parallax measurements, the radius can be calculated.
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Michelson Inerferometry
Coherently combining the light from a star using a Michelson interferometer will produce an interference pattern for the same reason that a double-slit produces one. The interference/diffraction pattern could yield angular separations of binaries and angular diameters of stars, but is subject to the destructive effects of atmospheric turbulence, or “seeing”. Tried in the 1920’s, it yielded a few positive results.
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Keck Interferometer |
CHARA Interferometer |
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Spacer Interferometry Mission (SIM) |
Terrestrial Planet Finder |
Intensity (Hanbury Brown) Interferometry
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Because of the wave/particle duality of light, there is a weird way to measure the “interference” pattern of the light from a distant “slit” (star) or “pair of slits” (binary separation) by counting photons from the source at different locations simultaneously. While phase information is lost when localizing the position of the photons, the detection events will be correlated in time. During the 1970’s Hanbury Brown measured the angular diameters of a few dozen bright stars using this method.
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Speckle Interferometry
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The blurred image of a star viewed through the earth’s atmosphere is actually the superposition of many near-diffraction-limited images. With a sufficiently bright stars and a good telescope, you can actually see that the blurred image of a close binary star breaks up into hundreds of pairs of bright dots (I have seen it myself). Taking a series of rapid images and combining them can yield a nearly diffraction-limited image of the binary, or an image of a very large star.
Adaptive Optics Imaging
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Likewise, it is now possible, with image sensors and computer-controlled feedback circuits, to “bend” the optics of a telescope to compensate for the wavefront distortion that density fluctuations produce in the atmosphere produce. This if sufficiently accurate, a near-diffraction-limited image will result. This is all the rage right now. In principle, the image distortions include x-y jitter, called “tip-tilt”, as well as higher-order terms. In sites of good seeing, tip-tilt correction will often get you 70% of the total correction, without higher-order terms. Tip-tilt is also the major source of seeing blur for small telescopes. Amateur TT correctors are now available for $3k.
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Go to Space
The Hubble Space Telescope is diffraction-limited, and can image the disks of sufficiently large (angular size) stars. The advantage over AO is that the entire field of view is sharp, not just a small patch of the sky. The disadvantage is that the HST is not a particularly large telescope, by today’s standards.
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Because the luminosity of a (spherical) star is just ,
the relative radii of a star compared to that of the Sun is simply:
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Stellar Masses
As we will see later on, the mass of a star determines its structure and fate. Consequently, determining the masses of stars is a critical, if unglamorous, activity of some very dedicated astronomers.
The determination of stellar masses derives from the observations of binary, or double stars. There are many types of binaries, some of which are more useful than others.
Optical Doubles these are simply two stars in the line
of sight with no physical relationship. The double star Mizar and Alcor in Ursa
Major is a famous example.
Visual Binaries Two physically related stars orbiting
one another that can be resolved independently. Mizar itself is a true visual
binary. Given time and adequate angular resolution, the mutual orbital motions
can be determined.
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Astrometric Binaries Physically the same as a visual binary,
except that one member is too faint to be detected.
Eclipsing Binaries here, the plane of the orbit is so close
to the line of sight that the stars pass in front of one another as they orbit
their mutual center of mass. Timing the eclipse durations yields information
about the radii of the stars.
Spectrum Binaries An unresolved binary where the spectra
of both stars are visible, but no orbital motion is detected through the
Doppler effect.
Spectroscopic Binaries
Here, the orbital velocity and orbital
inclination provide a Doppler shift large enough to be detected. A double-line
spectroscopic binary exhibits the spectra of both stars, while in a single-line
spectroscopic binary, one star is sufficiently brighter than the other that
only one component's spectrum is detected.
Masses from Visual Binaries
Suppose one could measure the motion of both stars in a visual binary about their center of mass.
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For a pair of stars with masses m1 and m2, located a1 and a2 from the center of mass:
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Because both of the stars are at approximately the same distance from the observer (us), this means
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Here and
are the observed angular values of the semimajor axes, measured in the same
units (radians, arcsec).
The orbital motion of the stars is also governed by Kepler’s 3rd Law:
where P is the period in seconds, a = a1 + a2, and G is the gravitational constant. If we use P in years, a in AU, and m’s in solar masses (“Astronomer’s Units”), this reduces to
The above is true strictly if the orbit is in the plane of the sky, i.e., its orbital inclination i is 0° . What we see is the projection of the orbit, so that
or, because
Thus, the determination of stellar masses requires very precise measurements of the distance and angular separation of the two stars, as well as “precise estimates” of the inclination. The periods are also often measured in decades or centuries. While the masses of stars are perhaps the single most important parameter to be determined, the masses of only a few hundred stars have actually been obtained!
Masses from Spectroscopic Binaries
If only the spectra of an unresolved binary is measurable, it is still possible to get some information of the masses of the stars. Because we only see the radial component of the velocity
what we measure is and life is fairly simple.
Similarly,
which is fine, as long as you know i pretty well.
In the case of a single-line spectroscopic binary, we are not so lucky. If we can only measure the light of star 1, all we get is
where all the stuff on the left side of the equation is called the “mass function”, a term invented to let you know that you really don’t get the masses, but some mysterious function related to it. It is useful for statistical studies, but you cannot get specific information on individual masses from it.
Mass-Luminosity Relation
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Mass-Luminosity relation for main sequence stars
Relative Temperatures
The total drop in observed brightness in an eclipsing binary is proportional to the product of the surface brightness (per unit solid angle) and solid angle eclipsed. The relative change, which is what is actually measured, will be the ratio of the net change to the total brightness. Because the angular size blocked in a total eclipse (for example) is that of the smaller star, regardless of whether it is in front or in back, this factor divides out. If B0 is the total brightness, Bp the brightness at primary eclipse, and Bs the brightness at secondary eclipse, then
