Determining the Properties of the Stars This set of notes by Nick Strobel covers: The properties of stars--their distances, luminosities, compositions, velocities, masses, radii, and how we determine those properties from afar. I will also discuss the spectral classification of stars and the Hertzsprung-Russell (color-magnitude) diagram. These notes will be in outline form to aid in distinguishing various concepts. As a way to condense the text down I'll often use phrases instead of complete sentences. The vocabulary terms are italicized. Contents Stars--What Are They Like? Distances Inverse Square Law of Light Brightness Magnitude System Star Colors (Temperature) Stellar Composition Velocity of Stars Doppler Effect Stellar Masses Stellar Radius Types of Stars and HR diagram Temperature dependence Spectral Types Hertzsprung-Russell diagram Color-Magnitude diagram Spectroscopic Parallax Spectral Type to Distance Cluster Main Sequence Analogy Stars--What Are They Like? A. Distances We use trigonometric parallax to measure the distances of the nearby stars. Look at nearby object from different vantage points and it appears to shift against more distant background objects. The farther apart the vantage points are, the greater the shift. The farther away the object is, the less it appears to shift. Since the shifts so small, we use arc seconds as the unit of their angular shift. The distances are quoted in parsecs (abbreviated with ``pc''). One parsec = 206,265 Astronomical Units (A.U.= mean distance between the Earth and the Sun or km). In terms of light years, one parsec = 3.26 light years. We can measure shifts to 1/50 arc seconds 50 parsecs in distance. http://userzweb.lightspeed.net/astronomy/starprop/starprop.html (1 of 7) [5/25/1999 11:44:45 AM]
B. Inverse Square Law of Light Brightness Inverse Square Law of Light Brightness: Energy from any light source radiates out in radial direction so concentric spheres (centered on the light source) have the same amount of energy pass through them every second. As light moves outward it spreads out to pass through each square centimeter of those spheres. Amount of energy passing through sphere-1 surface = amount of energy passing through sphere-2 surface. Surface area of sphere =, where is the radius of the sphere. Amount of energy passing through sphere surface = flux surface area. So sphere-1 flux = sphere-2 flux flux-1 = flux-2. The inverse square law! The flux is the amount of energy reaching each square centimeter of a detector (eg., your eye, CCD, piece of the sphere) every second. C. Magnitude System We specify the brightness of stars with the magnitude system. The magnitude system is based on how the human eye perceives differences in brightnesses so it is logarithmic. Also fainter things have larger, more positive magnitudes. It's screwy, but it's tradition! (Song from Fiddler on the Roof could be played here.) The apparent brightness of a star observed from the Earth is called the apparent magnitude. The apparent magnitude is a measure of the star's flux. If the star was at 10 pc distance from us, then its apparent magnitude would equal to its absolute magnitude. The absolute magnitude is a measure of the star's luminosity--amount of energy radiated by the star every second. If know apparent magnitude and absolute magnitude, you can find the star's distance. If know apparent magnitude and distance, you can find the star's luminosity. Some apparent magnitudes: Sun = -26.8, Moon = -12.6, Venus = -4.4, Sirius = -1.4, Vega = 0.00, faintest naked eye star =+6.5, brightest quasar = +12.8, faintest object = +27 - +28. Most famous apparently bright stars are intrinsically bright (luminous). Most nearby stars are intrinsically faint. Assume we live in typical patch of galaxy (Copernican principle) most stars are puny emitters of light. Faintest stars have absolute magnitudes = 19, brightest stars have absolute magnitudes = -8 huge range in luminosity! ``Distance modulus'' = app. mag. - abs. mag. = less than a fainter star's absolute magnitude!. So the ratio of two stars luminosities is. Remember the more luminous star has an absolute magnitude that is Star App.Mag. Distance(pc) Abs.Mag. Luminosity(rel. to Sun) Sirius -1.4 2.7 1.4 23 Arcturus 0 11.0-0.2 100 Vega 0 8.1 0.5 52 Spica 1 80.0-3.4 1900 Barnard's http://userzweb.lightspeed.net/astronomy/starprop/starprop.html (2 of 7) [5/25/1999 11:44:45 AM]
Star 9.5 1.8 13.3 1/2500 Proxima Centauri 11.0 1.3 15.5 1/19000 D. Star Colors (Temperature) The color of stars depends on their temperature--hotter stars are bluer and cooler stars are redder. Use different filters--allow only a narrow range of wavelengths (colors) through. By sampling the star's spectrum at two different wavelength ranges (``bands''), we can determine if the spectrum is that for a hot, warm, cool, or cold star. Hot stars have temperatures around 60,000 K while cold stars have temperatures around 3,000 K. See the magnitudes page for more details. E. Stellar Composition We determine the composition of stars through spectroscopy--breaking the starlight up into individual colors and noting what absorption (or sometimes, emission) lines are present. See absorption lines similar to sun-hydrogen and Helium with traces of other elements. From these absorption lines we learn some important things beside the star's composition: 1. Structure of stars: hot dense body producing continuous spectrum, covered by cooler thin gas. 2. The physics we use on Earth works everywhere else in Universe! Hydrogen spectrum is same in Sun, stars, distant galaxies and quasars. All absorption lines seen in celestial objects can be seen in laboratories on Earth. Charge and mass of electron and proton are same as electrons and protons everywhere we look. Physical laws are the same everywhere! 3. Since light has a finite speed, the light we receive from far away sources tell us how they were long ago. See spectra that can be explained with terrestrial physical laws. Physical laws are the same throughout time! F. Velocity of Stars We determine the velocity of stars by using the doppler effect. Motion causes wavelength shift such that ( is the speed of light). Amount of shift depends on the speed of the object relative to observer. Red-shift object is moving away; blue-shift object is moving toward you. This only gives us speed along the line-of-sight. To get the tangential speed, we need to measure angular speed of star across the sky and determine the star's distance. Then where is a conversion factor that will take care of the conversion from arc seconds and parsecs to kilometers/second. http://userzweb.lightspeed.net/astronomy/starprop/starprop.html (3 of 7) [5/25/1999 11:44:45 AM]
G. Stellar Masses To determine the masses of stars we use spectroscopic binaries and Kepler's third law. Since stars have about the same mass (within a factor of 20), they both orbit around a center of mass that is significantly different from one of the star's center. The center of mass is the point where (mass star 1) (C.M. distance 1) = (mass star 2) (C.M. distance 2) or the point they would be balanced upon if the stars were on a stellar seesaw. Heavy star closer to C.M. than light star. Kepler's 3rd law: (mass 1 + mass 2) = if use solar masses, A.U. for the distance from the C.M. and years for the orbital period. Distance a = C.M.-distance-1 + C.M.-distance-2. C.M.-distance = velocity orbit period / so use doppler shifts of spectral lines. Remember these rules: 1. Stars stay on opposite side of C.M. from each other. 2. Heavy star moves slower than light star. 3. C.M. is also point where mass1 velocity1 = mass2 velocity2 H. Stellar Radius To determine the radii of stars we use eclipsing binaries. Light curve--plot of brightness vs. time. Example of circular orbit seen edge-on with small hot star and large cool star: Types of Stars and HR diagram http://userzweb.lightspeed.net/astronomy/starprop/starprop.html (4 of 7) [5/25/1999 11:44:45 AM]
A. Temperature dependence Strength and wavelength of absorption lines does vary between stars. Some stars have strong (dark) Hydrogen lines, other stars have no Hydrogen lines but strong Calcium and Sodium lines. Are their abundances different? No. Temperature effects, in particular, the temperature of the photospheres. 1. Example: Hydrogen atom with energy orbits (remember Bohr model for atom?). To absorb photon of certain energy, electron needs to be at right energy level. If Hydrogen atoms heated to high temperatures, the atomic collisions can ionize the Hydrogen atoms no absorption. If the star's temperature is too low, then there are not many electrons in 2nd energy level--most in ground state because there are not that many atomic collisions. To produce absorption lines in visible spectrum, need electrons in 2nd energy level. 2. Hydrogen lines strong for temperature = 4,000-12,000 K. Helium lines strong for 15,000-30,000 K. Calcium lines strong for 3000-6000 K. 3. The strength of lines is sensitive to temperature. Cross-referencing elements' line strengths gives accurate temperature (within 20-50 K). Some stars have peaks of continuous spectrum outside of visible range so use spectral lines. Stars not perfect thermal radiators so continuum spectrum gives only a rough temperature (within a few hundred Kelvin). B. Spectral Types The spectral types were based on Hydrogen (absorption) line strength. A-type is strongest, B-type next strongest, F-type next, etc. Originally there was the whole alphabet of types, based on Hydrogen line strengths, but then astronomers discovered that the lines depended on temperature. After some rearranging and merging of some classes we now have OBAFGKM classes ordered by temperature. Each class is subdivided into 10 intervals, e.g., G2 or F5, with 0 hotter than 1, 1 hotter than 2, etc. Color Class Temperature Prominent Lines bluest O 40,000 ionized Helium bluish B 18,000 neutral Helium, neutral Hydrogen blue-white A 10,000 neutral Hydrogen white F 7,000 neutral Hydrogen, ionized Calcium yellow-white G 5,500 neutral Hydrogen, strongest ionized Calcium orange K 4,000 neutral metals (Calcium, iron), ionized Calcium red M 3,000 molecules and neutral metals C. Hertzsprung-Russell diagram Hertzsprung-Russell diagram. Look for correlations between stellar properties. Hertzsprung and Russell independently found a surprising correlation between temperature and luminosity for 90% of the stars. Also called a color-magnitude diagram. Diagonal strip is main-sequence. Luminous ones are easier to observe but rarer; faint ones are harder to see but more common. Also see correlation between mass and luminosity: Luminosity =. The H-R diagram below is for all stars visible to the naked eye (down to apparent magnitude = +5) plus all stars within 25 parsecs. http://userzweb.lightspeed.net/astronomy/starprop/starprop.html (5 of 7) [5/25/1999 11:44:45 AM]
Spectroscopic Parallax A. Spectral Type to Distance Use correlation between main sequence luminosity and temperature (spectral type) to get distances. Steps: 1. Determine spectral type (from spectroscopy) and measure star's flux. 2. Use calibrated main sequence to get luminosity (Hyades cluster in Taurus constellation is calibrator). 3. Use Inverse Square Law for Brightness to get distance. B. Cluster Main Sequence Can use an entire color-magnitude diagram for an individual cluster compared with a calibration cluster's color-magnitude diagram (of known distance) in the same way! Make adjustments for cluster's age and composition differences in stars--called ``main sequence fitting''. C. Analogy Analogy: Determine that far-away light bulb is a 100 Watt light bulb. Measure the number of photons passing through 1 square cm every second reaching your detector ( Watts/cm ). Bulb's luminosity is 100 Watts. Use Inverse Square Law to find out bulb's distance. Return to lecture notes home page last updated 27 Oct 95 http://userzweb.lightspeed.net/astronomy/starprop/starprop.html (6 of 7) [5/25/1999 11:44:45 AM]
Nick Strobel -- Email: strobel@astro.washington.edu (206) 543-1979 University of Washington Astronomy Box 351580 Seattle, WA 98195-1580 http://userzweb.lightspeed.net/astronomy/starprop/starprop.html (7 of 7) [5/25/1999 11:44:45 AM]