Deducing Temperatures and Luminosities of Stars (and other objects ) Review: Electromagnetic Radiation Gamma Rays X Rays Ultraviolet (UV) Visible Light Infrared (IR) Increasing energy Microwaves Radio waves 10-15 m 10-9 m 10-6 m 10-4 m 10 - m 10 3 m Increasing wavelength EM radiation is the combination of time- and space- varying electric + magnetic fields that convey energy. Physicists often speak of the particle-wave duality of EM radiation. Light can be considered as either particles (photons) or as waves, depending on how it is measured Includes all of the above varieties -- the only distinction between (for example) X-rays and radio waves is the wavelength. Electromagnetic Fields Sinusoidal Fields BOTH the electric field E and the magnetic field B have sinusoidal shape Direction of Travel Wavelength of Sinusoidal Function Frequency ν of Sinusoidal Wave time Wavelength is the distance between any two identical points on a sinusoidal wave. 1 unit of time (e.g., 1 second) Frequency: the number of wave cycles per unit of time that are registered at a given point in space. (referred to by Greek letter ν [nu]) ν is inversely proportional to wavelength 1
Units of Frequency meters c second cycles = ν meters second cycle cycle 1 = 1 "Hertz" (Hz) second Wavelength and Frequency Relation Wavelength is proportional to the wave velocity v. Wavelength is inversely proportional to frequency. e.g., AM radio wave has long wavelength (~00 m), therefore it has low frequency (~1000 KHz range). If EM wave is not in vacuum, the equation becomes v ν = c where v = and n is the "refractive index" n Light as a Particle: Photons Photons are little packets of energy. Each photon s energy is proportional to its frequency. Specifically, energy of each photon energy is E = hν Energy = (Planck s constant) (frequency of photon) h 6.65 10-34 Joule-seconds = 6.65 10-7 Erg-seconds Planck s Radiation Law Every opaque object at temperature T > 0-K (a human, a planet, a star) radiates a characteristic spectrum of EM radiation spectrum = intensity of radiation as a function of wavelength spectrum depends only on temperature of the object This type of spectrum is called blackbody radiation http://scienceworld.wolfram.com/physics/plancklaw.html Planck s Radiation Law Wavelength of MAXIMUM emission max is characteristic of temperature T Wavelength max as T Sidebar: The Actual Equation ( ) BT = hc 1 5 hc kt e 1 Complicated!!!! h = Planck s constant = 6.63 10-34 Joule - seconds k = Boltzmann s constant = 1.38 10-3 Joules -K -1 c = velocity of light = 3 10 +8 meter - seconds -1 max http://scienceworld.wolfram.com/physics/plancklaw.html
Temperature dependence of blackbody radiation As temperature T of an object increases: Peak of blackbody spectrum (Planck function) moves to shorter wavelengths (higher energies) Each unit area of object emits more energy (more photons) at all wavelengths Wien s Displacement Law Can calculate where the peak of the blackbody spectrum will lie for a given temperature from Wien s Law: 3.898 10 max [ meters] = T K [ ] (recall that human vision ranges from 400 to 700 nm, or 0.4 to 0.7 microns) Colors of Stars Star Color is related to temperature If star s temperature is 5000 K, the wavelength of the maximum of the spectrum is: 3.898 10 max = m 0.579µ m= 579nm 5000 (in the visible region of the spectrum) Colors of Stars If T << 5000 K (say, 000 K), the wavelength of the maximum of the spectrum is: 3.898 10 max = m 1.45µ m= 1450nm 000 (in the near infrared region of the spectrum) The visible light from this star appears reddish Why are Cool Stars Red? Less light in blue Star appears reddish 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1. 1.3 (µm) max Colors of Stars If temperature >> 5000-K (say, 15,000-K), wavelength of maximum brightness is: 3.898 10 max = m 0.193µ m= 193nm 15000 Ultraviolet region of the spectrum Star emits more blue light than red appears bluish Visible Region 3
Why are Hotter Stars Blue? More light in blue Star appears bluish Betelgeuse: 3,000 K (a red supergiant) Betelguese and Rigel in Orion 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 max (µm) Rigel: 30,000 K (a blue supergiant) Visible Region Blackbody curves for stars at temperatures of Betelgeuse and Rigel Stellar Luminosity Sum of all light emitted over all wavelengths is the luminosity brightness per unit surface area luminosity is proportional to T 4 : L = σ T 4 σ Joules m-sec-k 8 5.67 10, Stefan-Boltzmann constant 4 L can be measured in watts often expressed in units of Sun s luminosity L Sun L measures star s intrinsic brightness, rather than apparent brightness seen from Earth Stellar Luminosity Hotter Stars Hotter stars emit more light per unit area of its surface at all wavelengths T 4 -law means that small increase in temperature T produces BIG increase in luminosity L Slightly hotter stars are much brighter (per unit surface area) Two stars with Same Diameter but Different T Hotter Star emits MUCH more light per unit area much brighter 4
Stars with Same Temperature and Different Diameters Area of star increases with radius ( R, where R is star s radius) Measured brightness increases with surface area If two stars have same T but different luminosities (per unit surface area), then the MORE luminous star must be LARGER. How do we know that Betelgeuse is much, much bigger than Rigel? Rigel is about 10 times hotter than Betelgeuse Measured from its color Rigel gives off 10 4 (=10,000) times more energy per unit surface area than Betelgeuse But the two stars have equal total luminosities Betelguese must be about 10 (=100) times larger in radius than Rigel to ensure that emits same amount of light over entire surface So far we haven t considered stellar distances... Two otherwise identical stars (same radius, same temperature same luminosity) will still appear vastly different in brightness if their distances from Earth are different Reason: intensity of light inversely proportional to the square of the distance the light has to travel Light waves from point sources are surfaces of expanding spheres Sidebar: Absolute Magnitude Recall definition of stellar brightness as magnitude m m.5 log F = 10 F0 F, F 0 are the photon numbers received per second from object and reference, respectively. Sidebar: Absolute Magnitude Absolute Magnitude M is the magnitude measured at a Standard Distance Standard Distance is 10 pc 33 light years Allows luminosities to be directly compared Absolute magnitude of sun +5 (pretty faint) ( 10 pc) ( ) F M =.5 log10 + m F earth Sidebar: Absolute Magnitude Apply Inverse Square Law Measured brightness decreases as square of distance ( ) ( ) 1 = = F 10pc 10 pc distance F earth 1 10 pc distance 5
Simpler Equation for Absolute Magnitude distance M =.5 log10 + m 10pc distance = 5 log10 + m 10pc Stellar Brightness Differences are Tools, not Problems If we can determine that stars are identical, then their relative brightness translates to relative distances Example: Sun vs. α Cen spectra are very similar temperatures, radii almost identical (T follows from Planck function, radius R can be deduced by other means) luminosities about equal difference in apparent magnitudes translates to relative distances Can check using the parallax distance to α Cen Plot Brightness and Temperature on Hertzsprung-Russell Diagram H-R Diagram 1911: E. Hertzsprung (Denmark) compared star luminosity with color for several clusters 1913: Henry Norris Russell (U.S.) did same for stars in solar neighborhood http://zebu.uoregon.edu/~soper/stars/hrdiagram.html Hertzsprung-Russell Diagram Clusters on H-R Diagram n.b., NOT like open clusters or globular clusters Rather are groupings of stars with similar properties Similar to a histogram 90% of stars on Main Sequence 10% are White Dwarfs <1% are Giants http://www.anzwers.org/free/universe/hr.html 6
H-R Diagram Hertzsprung-Russell Diagram Vertical Axis luminosity of star could be measured as power, e.g., watts or in absolute magnitude Lstar or in units of Sun's luminosity: L Sun H-R Diagram Horizontal Axis surface temperature Sometimes measured in Kelvins. T traditionally increases to the LEFT Normally T given as a ``ratio scale' Sometimes use Spectral Class OBAFGKM Oh, Be A Fine Girl, Kiss Me Could also use luminosities measured through color filters Standard Astronomical Filter Set 5 Bessel Filters with approximately equal passbands : 100 nm U: ultraviolet, max 350 nm B: blue, max 450 nm V: visible (= green ), max 550 nm R: red, max 650 nm I: infrared, max 750 nm sometimes II, farther infrared, max 850 nm Transmittance (%) Filter Transmittances 100 Visible Light R U B V 50 0 Wavelength (nm) II I 00 300 400 500 600 700 800 900 1000 1100 Measure of Color If image of a star is: Bright when viewed through blue filter Fainter through visible Fainter yet in red Star is BLUISH and hotter L(star) / L(Sun) 0.3 0.4 0.5 0.6 0.7 0.8 (µm) Visible Region 7
Measure of Color If image of a star is: Faintest when viewed through blue filter Somewhat brighter through visible Brightest in red Star is REDDISH and cooler L(star) / L(Sun) 0.3 0.4 0.5 0.6 0.7 0.8 (µm) How to Measure Color of Star Measure brightness of stellar images taken through colored filters used to be measured from photographic plates now done photoelectrically or from CCD images Compute Color Indices Blue Visible (B V) Ultraviolet Blue (U B) Plot (U V) vs. (B V) Visible Region 8