The Properties of Stars

Transcription

1 10/7/011 The Proerties of Stars Distance, Luminosity, and Radius Trigonometric Parallax To measure the distance to star X, hotograh it on two dates searated by 6 months. In the figure, A and are the ositions of Earth on these two dates. In 6 months, the angular osition of X relative to the background stars changes by an amount θ. X Half of θ is the arallax angle. θ S A Consider triangle SX: S = 1 AU, SX = d = the unknown distance, and XS = = the arallax angle measured from the hotograhs. Photograh on Date 1 Photograh on Date 1

2 10/7/011 SX can be solved for d. Since the angle is small for all stars, the small angle formula can be used. The result is 06,65 d = where is in arcseconds and d is in AU s. 1 AU S d X The distances to stars are so large that the AU is not a convenient unit for stellar distances. The distance unit most commonly used is the arsec. 1 arsec = 1 c = 06,65 AU = 3.6 light years. In these units, the distance formula is 1 d = Examle 1 Procyon has a arallax of 0.9". Calculate the distance to Procyon (a) in arsecs and (b) in light years.. 1 d = = 3.7 c d = ly = 11.3ly 0.88 Energy Flux = = Area Time Luminosity Area flux = F = "brightness" L = "luminosity" = L π R W F =σt σ= mk F 8 L= πr σt F = L π d Examle (CP11.9b) Suose a star has the same aarent brightness as Alha Centauri A ( W/m ) but is located at a distance of 00 light years How far away is it? = L π d F 8 d = 00 ly F =.7 10 W / m d = 00 ly ( ) ly = m d = m = m ( )( m) ( ) L = πd F = π.7 10 W / m = W

3 10/7/011 Absolute Magnitude and Luminosity The aarent magnitude of a star is defined by the equation m m FA =.5log F A where m A is the magnitude of star A, m is the magnitude of star, F A is the flux at Earth from star A, and F is the flux at Earth from star. The flux at Earth tells how bi bright a star aears to the detector. t The luminosity of a star is the total electromagnetic energy it emits in a unit of time. Two stars can have the same luminosity and very different aarent magnitudes. Alha Centauri A and the Sun, for examle, are both sectral class G stars and have about the same luminosity L = W. On the other hand, the aarent magnitude of Alha Centauri A is 0.1 and the aarent magnitude of the Sun is 6.8. The difference is due to the fact that the Sun is much closer to us than Alha Centauri A is. In order to make absolute brightness comarisons among stars, we calculate how bright they would aear if they were all at the same distance from the Sun. The standard distance used is 10 c. The absolute magnitude M of a star is defined as the aarent magnitude it would have if it were at the standard distance of 10 arsecs. Radii of Stars F = The Stefan-oltzman Law: energy emitted by one square meter in one second =. σt L = energy emitted from the entire stellar surface in one second at all wavelengths. Surface area of a shere = A =πr. L = πr σt = luminosity of the Sun W = 1watt = 1Joule er second 6 L = LSun = W radius of the Sun = R = km = m T = surface temerature of the Sun = 5800 K L = π R σ T L πr σt = L πr σt L RT R T = = L R T R T R L = T R L T 3

4 10/7/011 Examle 3 Vega s luminosity is 66.7 times the luminosity of the Sun, and its surface temerature is 9900 K. What is its radius? L R T R 9900 = 66.7 = L R T R 5800 R 66.7 = ( 1.797) R R 66.7 = 8.88 R 66.7 R = 8.88 R R 7.86 = R R R = = 5 6 R =.80R = km= km Deendence of Stellar Sectra on Temerature (Sectral Classification) 0,000 K 0,000 K 10,000 K 7500 K 5500 K 500 K 3000 K

5 10/7/011 Stellar Sectra Sectral Classes Tye Hydrogen almer Line Strength Aroximate Surface Temerature Main Characteristics Examles Singly ionized helium emission or absortion O Weak > 5,000 K 10 Lacertra lines. Strong ultraviolet continuum. Medium 11,000-5,000 Neutral helium absortion lines. A Strong 7,500-11,000 Hydrogen lines at maximum strength for A0 stars, decreasing thereafter. F Medium 6,000-7,500 Metallic lines become noticeable. G Weak 5,000-6,000 K M Very Weak Very Weak 3,500-5,000 < 3,500 Solar-tye sectra. Absortion lines of neutral metallic atoms and ions (e.g. singly-ionized calcium) grow in strength. Metallic lines dominate. Weak blue continuum. Molecular bands of titanium oxide noticeable. Rigel Sica Sirius Vega Canous Procyon Sun Caella Arcturus Aldebaran etelgeuse 5

6 10/7/011 Star Colors It is useful to measure the brightness of a star through colored filters. One standard set (called the UV system) consists of filters that transmit a narrow range around 350 nm (ultraviolet), 35 nm (blue), and 555nm (green = visual). The ratio F V /F deends on the temerature of the star. This can be understood by referring to the continuous sectrum of stars with different temeratures. F F V F V F Notice that F V is greater than F for the cooler star, but F is greater than F V for the hotter star. FV 300 K Star: FV > F >1 15,000 K Star: FV < F F 6