Properties of Stars http://youtu.be/zaOPsmlJyw8 1 Lecture Learning goals: ! Explain what is meant by the inverse-square law and apply it to the measurements of light at different distances. ! Define apparent and absolute magnitudes. ! Explain what a “parsec” is and why it is convenient for astronomers in estimating distances and luminosities of stars. !Describe the methods used to determine the temperature, luminosity, and radius of a star. !State the “goldilocks” analogy for strengths of hydrogen lines in the spectra of stars OBAFGKM. 2 Inverse square law for light Explain what is meant by the inverse-square law and apply it to the measurements of light ! 3 at different distances. Brightness How the star looks to US HERE ON EARTH. L apparent brightness = 4πD2 Each of these light bulbs will appear to be the same brightness. € 1000 times farther away " 100 Watt 1000 Watt 1 Watts € 10 times farther away " 2 x farther away, 1/4 as bright 3 x farther away, 1/9 as bright ! Explain what is meant by the inverse-square law and apply it to the measurements4 of light at different distances. Apparent Magnitude # Every 5 magnitudes difference means 100 x difference in brightness # One magnitude difference is 2.512 times in brightness. (2.5125 = 100) 5 !Define apparent and absolute magnitudes. When you see only “magnitude,” that means APPARENT magnitude. 1. The magnitude (m) of star A is 1, the magnitude (m) of star B is 6. How many times brighter is A than B? a) 5 b) 10 c) 100 d) 1000 2. m of star C is 12, m of star D is 2: How many times brighter is star D than star C? (Or, equally stated, how many times dimmer is star C than star D?) a) 10 b) 24 c) 100 d) 10,000 • The Sun is the brightest star in the sky, with an apparent magnitude of about -26.5 Sirius is next in line, with an apparent magnitude of -1.5; how many times brighter is the Sun than Sirius? a) 25 b) 28 c) 100,000 d) 10,000,000,000 6 !Define apparent and absolute magnitudes. Brightness: Magnitude Some useful definitions: ▪ Brightness of a star is measured by logarithmic magnitude. => Brighter objects have a smaller magnitude. ▪ Apparent magnitude: how bright the star appears • to us in the sky. m ▪ Absolute magnitude: how bright the star would be at a distance of 10 parsecs from us. M d is in parsecs !Define apparent and absolute magnitudes. PARSEC: Parallax ARc SECond A star having a parallax of 1 arc second is 1 parsec away 1 parsec (pc) = 3.26 light years 1 kiloparsec (1 kpc) = 1000 pc; 1 megaparsec (1 Mpc) = 1,000,000 pc Baseline is 1 Astronomical Unit ! Explain what a “parsec” is and why it is convenient for astronomers in estimating8 distances and luminosities of stars. Continuous Spectrum - created by thermal radiator (blackbody radiation) Hotter stars look more blue-white than cooler stars because hot stars emit most of their light at shorter wavelengths. Wien’s Law Star field in Sagittarius 2.9 ×106 λ = nm peak T 2.9 ×106 T = K λpeak € 9 ! Describe the methods used to determine the temperature, luminosity, and radius of a star. http://stardate.org/radio/program/delta-lyrae 10 ˃˅˰˄˥The Temperature, Size, and Composition of Stars ˆ˂ˉ Math Tools 13.2 Estimating Sizes of Stars In Chapter 5 you learned that according to the Stefan- Now turn this question around and ask, How large must a Boltzmann law, the amount of energy radiated each second star of a given temperature be to have a total luminosity of L? by each square meter of the surface of a star is equal to the The star’s luminosity (L) and temperature (T) are quantities constant σ multiplied by the surface temperature of the star that we can measure, and the star’s radius (R) is what we want raised to the fourth power. Written as an equation, this rela- to know. We can rearrange the previous equation, moving the tionship says: things that we know how to measure (temperature and lumi- nosity) to the right-hand side of the equation, and the thing Energy radiated each 4 2 = σT that we would like to know (the radius of the star) to the left ( second by 1 m of surface ) side. After a couple of steps of algebra, we find: To find the total amount of light radiated each second by the _____ ____L ___1 star, we need to multiply the radiation per second from each R = × 2 4πσ T square meter by the number of square meters of the star’s √ surface: Again, the right-hand side of the equation contains only things that we know or can measure. The constants 4, π, and Energy radiated Energy radiated each Surface are always the same. We can find L, the luminosity of the = 2 × σ ( each second ) ( second by 1 m of surface ) ( area ) star, by combining measurements of the star’s brightness and parallax (although only for those nearer stars with known The left-hand item in this equation—the total energy emitted parallax). T is the surface temperature of the star, which can by the star per second (in units of joules per second, watts)—is = be measured from its color. From the relationship of these the star’s luminosity, L. The middle item—the energy radi- measurements we now know something new: the size of ated by each square meter of the star in a second (in units the star. We refer to this last equation as the luminosity- of joules per square meter per second, or J/m2/s)—can be temperature-radius relationship for stars. replaced with the T4 factor from the Stefan-Boltzmann law. σ Often we compare two stars and the constants all cancel The remaining item—the number of square meters cover- out, leaving L, T, and R: ing the surface of the star—is the surface area of a sphere, ______ 2 4 2 2 2 A 4 R (in units of square meters, or m ), where R is Lstar 1 R star 1 T star 1 Rstar 1 Lstar 1 T star 2 sphere _____ _____ _____ _____ _____ _____ = π 2 4 or 2 L = R × T R = L × T the radius of the star. star 2 star 2 star 2 star 2 √ star 2 star 1 If we replace the words in the equation with the appro- Suppose we compare the Sun to the second brightest priate mathematical expressions for the Stefan-Boltzmann star in the constellation of Orion, a red star called Betel- law and the area of a sphere, our equation for the luminos- geuse (see the chapter-opening photo). From its spectrum, ity of a star looks like this: we know that Betelgeuse’s surface temperature T is about J 3500 K. Its distance is about 200 parsecs, and from that Luminosity __ T4 ____J 4 R2(m2) ( s ) = σ ( m2s ) × π and its brightness, its luminosity is estimated to be 140,000 times that of the Sun. What can we say about the size of Be- Combining gives: telgeuse? Using the preceding equation, we can determine 2 4 the following: L = 4πR σT J/s (W) _________ 2 RBetelgeuse LBetelgeuse T Sun Because the constants (4, , and ) do not change, the lu- _______ _______ ________ π σ 2 2 4 R = L × T minosity of a star is proportional only to R T . Make a star 3 Sun √ Sun Betelgeuse times as large, and its surface area becomes 32 9 times as _________ = 140,000 5,8002 large. There is 9 times as much area to radiate, so there is 9 _______ _______ 374 2.7 1,030 = √ 1 × 3,5002 = × = times as much radiation. Make a star twice as hot, and each Understanding Our Universe, 1st Edition ! Describe the methods used to determine the temperature, luminosity, and radiusCopyright of a © 2012star. W. W. Norton & Company 4 square meter of the star’s surface radiates 2 = 16 times as Betelgeuse has a radius over 1,000 times larger than that of much energy. Larger, hotter stars are more luminous than the Sun. As you will see in Section 13.4, such stars are called smaller, cooler stars. supergiants. Kay4e_ch13.indd 407 1/4/13 8:26 AM Tutorial: Stefan-Boltzmann Law (Adapted from Lecture-Tutorials for Introductory Astronomy, ©CAPER Team, Preliminary Edition, 2002) Part I: Size, Temperature, and Luminosity Beginning astronomy students often think that they must come up with totally new and different explanations for things that go on in other places of the Universe besides here on Earth. This definitely is not the case. If you are not venturing near a black hole nor attempting to go nearly as fast as the speed of light, the physics we know here on Earth applies to the rest of the solar system, the Galaxy, and the Universe. In this tutorial you will be led through the steps to understanding the Stefan-Boltzmann Law: The amount of energy put out per second (the number of watts) is proportional to the surface area of the th sphere (4 pi times the radius squared) and the temperature raised to the 4 power. (The σ represents the 2 4 Stefan-Boltzmann constant, 5.67 ×10−8 W·m-2·K-4.) L = 4" r ! T You are comparing the ability of a grouping of electric hot plates (shown below as burners on a “stove top”) of different sizes and temperatures to bring identical pots of water to a boil.