Temperature     Distances     Apparent Magnitude     Absolute Magnitude/Luminosity      Radius     Mass      Motions      Composition

How do we know... A Star's Temperature?

Surface Temperature
We can determine a star's surface temperature from the wavelength of its peak radiation emissions. (Wavelength also corresponds to color). Very hot stars emit radiation most strongly at short wavelengths; cooler stars radiate most strongly at longer wavelengths. Astronomers measure a star's brightness at each of many wavelengths to find the wavelength of peak emission, which in turn gives us the star's temperature using Wien's Law: the temperature (in degrees Kelvin) is equal to 3,000,000 divided by the wavelength (in nanometers) of maximum emission.

For example, Sirius emits most strongly at about 300 nm (in the ultraviolet range), giving it a surface temperature of about 10,000K; Betelgeuse radiates most strongly in the infrared at about 1000 nm, corresponding to a surface temperature of about 3,000K.

Though we can't see either ultraviolet or infrared, where these stars radiate most strongly - the band of visible light in the electromagnetic spectrum is very narrow compared to the range of ultraviolet or infrared - the light we do see from red giant Betelgeuse is noticeably red/orange. Similarly, the light we see from extremely massive, hot "blue supergiant" stars such as Rigel is noticeably bluish.

Peak emissions in the visible light range, 380 nm to 740nm, correspond to temperatures of about 4000K to 7900K, which includes F, G, and K-class stars; our star, the Sun, is a G-class.

How do we know... Distances to Astronomical Objects?

Nearby Stars (within 200 light-years): Parallax
Parallax is the apparent shift in an object's position due to a change in the observer's position. It's the same phenomenon that allows us to use our stereovision to judge distance. Because our eyes are a few inches apart, each sees a slightly different image. Try holding your thumb up with your arm fully extended and look at it first with one eye and then the other - it appears to shift position against the background. The brain uses the different views to judge the distance to the object.

Astronomers employ the same principle, using the diameter of the Earth's orbit as a baseline (like the distance between our eyes) to triangulate distances to stars. Photographs of a star taken at opposite points in our orbit (6 months apart) are compared. A nearby star will appear to have moved slightly against the background stars. The tiny angular shift of the star relative to a known distant object (such as a galaxy) provides the apex angle of a right triangle whose base is the radius of the Earth's orbit (1 AU). The distance to the star is then calculated by trigonometry (1 AU divided by the tangent of 1/2 of the angular shift). A picture will help: check out the explanation on NASA's "Star Child" web site or Professor Richard Pogge's animation.

If distance is measured in parsecs (parallax-seconds) and parallax is measured in seconds of arc, we get the tidy equation: d = 1/p: distance becomes simply the reciprocal of parallax, making it very easy to determine one from the other. The trick is measuring the parallax with any kind of accuracy, as the angular shifts of stars are extremely small in the best of cases.

With ground-based telescopes, astronomers were able to determine distances to stars up to about 60 light-years away - the distances to only a few dozen stars were known to an accuracy of 1%. More precise information on star positions was collected by the European Space Agency satellite Hipparcos from 1989-1993, giving us reasonably accurate distances for stars up to 200 light-years away. We now have distances accurate to 1% for 400 stars, and accurate to 5% for over 7,000 stars. (Check out the Hipparcos website: astro.estec.esa.nl/Hipparcos - the satellite is still up there, though it is no longer in use).

B. Distant Stars: Comparing Apparent Magnitude to Absolute Magnitude
[I'm working on this one...]
There is a lovely, straightforward mathematical relationship among a star's apparent magnitude, absolute magnitude (or luminosity), and distance. Given any two, we can determine the other.

How do we measure... How Bright Objects Appear?

Apparent Magnitude
Our system of ranking naked-eye stars in 6 categories of relative brightness (to an observer on Earth) goes back at least to the Alexandrian astronomer Ptolemy (d. 151 AD). Naming the brightest stars in the sky "first magnitude" and the dimmest "sixth magnitude" made perfect sense at the time, though nowadays, with our precise measurements of apparent magnitude, it seems a little odd that dim stars have very high magnitude numbers and bright stars have negative magnitudes.

The rough-estimate visual categories were not significantly improved upon until the early 1800's, when William Herschel came up with a method of obtaining mathematical comparisons of pairs of stars. He would point two telescopes at different stars, then cover up the aperture of the telescope aimed at the brighter of the pair until the two stars appeared equally bright. The ratio of the uncovered portion of the lenses is equal to the ratio of the magnitudes of the two stars. (For example, if star A appears as bright as B when telescope A only has 1/4 the aperture of telescope B, then star A is 4 times brighter than B if viewed at the same aperture.)

These comparative measurements allowed astronomers to standardize the old naked-eye scale to the modern system, in which stars of magnitude 1.0 are 100 times as bright as stars of magnitude 6.0 (five steps), and to extend the scale in both directions. Each step on the scale is approximately a decrease of 2.5 times in brightness - a first magnitude star is about 2.5 times as bright as a second magnitude star, and so on. The Sun has an apparent magnitude of -26.5. The limit for large ground-based telescopes is about magnitude 24, and the limit for the Hubble Space Telescope is magnitude 29.

(At Haggart, with suburban light pollution, our naked-eye magnitude limit is 4 or 5, and the limit for all of our telescopes is probably about 13. The Catskills Astronomy Club website provides a "Limiting Magnitude Calculator" for telescopes - www.geocities.com/catskills_astronomy_club/calculator.htm.)

Note that there are no naked-eye "standard candle" stars at exact steps of the scale; however, Vega and Arcturus are closest to 0 and Antares and Spica are closest to 1.

How do we know... How Bright Stars Are, Really?

Absolute Magnitude
Apparent magnitude, of course, depends both on a star's actual brightness and its distance from us. In order to use the familiar apparent magnitude scale to compare the true brightness of stars, astronomers define absolute magnitude as the apparent magnitude a star would have if it were at a standard distance of 10 parsecs from us.

Luminosity
A star's luminosity is the total amount of radiation (across the whole electromagnetic spectrum) that it radiates per second. It is often given in units of the Sun's luminosity (4 x 10²⁶ watts). An absolute magnitude of 0 corresponds to a luminosity of 100 solar units; an absolute magnitude of 5 is equivalent to a luminosity of 1 solar unit. (The Sun, at 10 parsecs away - about the distance to Pollux - would be a faint fifth-magnitude star.)

A star's apparent brightness depends on its xand its An absolute magnitude of 0 corresponds to a luminosity of 100 solar units; an absolute magnitude of 5 is equivalent to a luminosity of 1 solar unit. (The Sun, at 10 parsecs away - about the distance to Pollux - would be a faint fifth-magnitude star.)

A star's apparent brightness depends on its actual luminosity and its distance from us. Specifically, apparent brightness is directly proportional to luminosity and inversely proportional to the square of its distance; this mathematical relationship is the "Inverse-Square Law." Hence, if we know distance and apparent brightness, we can calculate luminosity.

How do we know... a Star's Radius / Diameter?

Stefan-Boltzmann Law: calculate from luminosity and temperature
A star's luminosity is proportional to its temperature (determined from peak wavelength of emission) and radius. This makes intuitive sense: the higher the temperature, or the greater the surface area, the brighter the star. This equation links luminosity, temperature and radius so that if we know any two, we can calculate the third.

Angular Size - Interferometry
An object's radius can be calculated if its distance and angular size are known. This works great for the Moon and the Sun, but the angular size of stars is too small to be measurable by any telescope. It can, however, be measured by interferometry - using two widely-separated telescopes, and then combining their images by computer. This has been done for a few dozen nearby stars and a few giant stars.

How do we know... a Star's Mass?

A star's mass is a very important property - a star's mass at formation determines its temperature, luminosity, longevity and ultimate fate. Unfortunately, it is very hard to measure.

Binary Stars: Kepler's Third Law of Orbital Motion
Stellar mass can be directly determined only for binary stars whose orbital motions can be measured. For such stars, Kepler's Third Law, which relates the mass, mean distance, and period of revolution of a pair of orbiting bodes, can be used to calculate each star's mass. Specifically, the sum of the masses (in solar masses) of 2 orbiting bodies is equal to their mean distance (in AU) cubed, divided by their period of revolution (in years). Such direct determinations of mass have been made for only a few hundred stars.

Mass-Luminosity Equation:
Luminosity and mass turn out to be directly proportional. A star's luminosity (in solar luminosity) is equal to its mass (in solar masses) to the power 3.5: L = M^3.5.

How do we know... Where Stars Are Going?

"Proper Motion"
All stars are in motion relative to one another, as we all spin along in the Galaxy. Their motion with respect to us is termed their "proper motion" - to distinguish it from nightly and yearly apparent motions due to the Earth's rotation and revolution. Proper motion is, of course, very slow from our point of view, and hard to detect. Halley, back in 1718, compared his own measurements to those of Greek astronomers 2000 years earlier, and found that Aldebaran, Sirius, and Arcturus had each moved about half a degree from their earlier postions in the celestial sphere. In modern times, comparisons of stellar positions measured decades apart, and recent measurements by satellite Hipparcos, have given us reasonably accurate information on proper motion for over 100,000 stars.
For a nice illustration of how the Big Dipper will change due to the proper motion of its stars over the next 100,000 years, see the "Proper Motion" page of the "Your Sky" website: www.fourmilab.ch/yoursky/help/proper.html

Where is the Sun headed? - "Solar Motion"
Toward Hercules and away from Columba (opposite Hercules in the celestial sphere).
    We know this from measurements of "solar motion" - the fact that stars we are moving toward should appear to disperse in front of us, while stars we are moving away from will appear to converge. (As portrayed in the viewscreens in Star Trek and such.)

How do we know... What Stars Are Made Of?

[under construction...]

Formulas

Mathematical relationships link various properties, and unknown property can be calculated if we know all the other linked properties.

Sources:

Arny, Thomas. 2004. Explorations: An Introduction to Astronomy. McGraw-Hill.

Fix, John D. 2004. Astronomy: Journey to the Cosmic Frontier. McGraw-Hill.

Mitton, Simon and Jacqueline Mitton. 1995. The Young Oxford Book of Astronomy.