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Luminosity and Colour of Stars Michael Balogh Stellar Evolution (PHYS 375) The physics of stars A star begins simply as a roughly spherical ball of (mostly) hydrogen gas, responding only to gravity and it’s own pressure. To understand how this simple system behaves, however, requires an understanding of: X-ray ultraviolet infrared radio 1. 2. 3. 4. 5. 6. 7. Fluid mechanics Electromagnetism Thermodynamics Special relativity Chemistry Nuclear physics Quantum mechanics Course Outline Part I (lectures 1-5) Basic properties of stars and electromagnetic radiation Stellar classification Measurements of distance, masses, etc. Part II (lectures 6-13) Chemical composition of stars (interpretation of spectra) Stellar structure (interiors and atmospheres) Energy production and transport Part III (lectures 14-22) Stellar evolution (formation, evolution, and death) White dwarfs, neutron stars, black holes The nature of stars Betelgeuse • Stars have a variety of brightnesses and colours • Betelgeuse is a red giant, and one of the largest stars known • Rigel is one of the brightest stars in the sky; blue-white in colour Rigel Apparent brightness of stars The apparent brightness of stars depends on both: • their intrinsic luminosity • their distance from us Their colour is independent of distance The five brightest stars Star name The five nearest stars Relative Distance brightness (light years) Star name Relative Distance brightness (light years) Proxima Centauri 0.0000063 4.2 Sirius 1 8.5 Canopus 0.49 98 Alpha Centauri 0.23 4.2 Alpha Centauri 0.23 4.2 0.000040 5.9 Vega 0.24 26 Barnard’s star Wolf 359 0.000001 7.5 Arcturus 0.25 36 0.00025 8.1 Capella 0.24 45 Lalande 21185 The Astronomical Unit Astronomical distance scale: Basic unit is the Astronomical Unit (AU), defined as the semimajor axis of Earth’s orbit How do we measure this? Relative distances of planets from sun can be determined from Kepler’s third law: 2 3 P a 2 E.g. given Pearth, Pmars: PEarth aEarth PMars aMars 1AU = 1.49597978994×108 km 3 Parallax p d 1 AU The “parallax” is the apparent shift in position of a nearby star, relative to background stars, as Earth moves around the Sun in it’s orbit This defines the unit 1 parsec = 206265 AU = 3.09×1013 km ~ 3.26 light years Measuring Parallax The star with the largest parallax is Proxima Centauri, with p=0.772 arcsec. What is its distance? These small angles are very difficult to measure from the ground; the atmosphere tends to blur images on scales of ~1 arcsec. It is possible to measure parallax angles smaller than this, but only down to ~0.02 arcsec (corresponding to a distance of 1/0.02 = 50 pc). Until recently, accurate parallaxes were only available for a few hundred very nearby stars. A star field with 1” seeing Hipparcos The Hipparcos satellite (launched 1989) collected parallax data from space, over 3 years 120,000 stars with 0.001 arcsec precision astrometry More than 1 million stars with 0.03 arcsec precision The distance limit corresponding to 0.001 arcsec is 1 kpc (1000 pc). Since the Earth is ~8 kpc from the Galactic centre it is clear that this method is only useful for stars in the immediate solar neighbourhood. Parallax: summary 1. 2. 3. 4. A fundamental, geometric measurement of distance Can be measured directly Limited to nearby stars Is used to calibrate other, more indirect distance indicators. Ultimately even our estimates of distances to the most remote galaxies rests on a reliable measure of parallax to the nearest stars Break The electromagnetic spectrum Different filters transmit light of different wavelengths. Common astronomy filters are named: U B V R I • The Earth’s atmosphere blocks most wavelengths of incident radiation very effectively. It is only transparent to visual light (obviously) and radio wavelengths. • Observations at other wavelengths have to be made from space. Blackbodies The energy radiated from a surface element dA is given by: B (T )d dA cos d B (T )d dA cos sin dd Units of B(T): W/m2/m/sr Blackbodies The energy radiated from a surface element dA is given by: B (T )d dA cos d B (T )d dA cos sin dd Units of B(T): W/m2/m/sr Energy quantization leads to a prediction for the spectrum of blackbody radiation: c B (T ) u (T ) 4 2hc 2 hckT e 1 5 Planck’s law Calculate the luminosity of a spherical blackbody: Each surface element dA emits radiation isotropically Integrate over sphere (A) and all solid angles () L d 2 / 2 B d dA cos sin dd 0 0 A AB d Properties of blackbody radiation 1. The wavelength at which radiation emission from a blackbody peaks decreases with increasing temperature, as given by Wien’s law: max T 0.290 cm K 2. The total energy emitted (luminosity) by a blackbody with area A increases with temperature (Stefan-Boltzmann equation) This defines the effective temperature of a star with radius R and luminosity L L 4R 2Te4 Examples The sun has a luminosity L=3.826×1026 W and a radius R=6.96×108 m. What is the effective temperature? At what wavelength is most of the energy radiated? max T 0.290 cm K L 4R 2Te4 Example Why does the green sun look yellow? The human eye does not detect all wavelengths of light equally Examples Spica is one of the hottest stars in the sky, with an effective temperature 25400 K. The peak of its spectrum is therefore at 114 nm, in the far ultraviolet, well below the limit of human vision. We can still see it, however, because it emits some light at longer wavelengths max T 0.290 cm K L 4R 2Te4 Apparent magnitudes The magnitude system expresses fluxes in a given waveband X, on a relative, logarithmic scale: f m X mref 2.5 log f ref Note the negative sign means brighter objects have lower magnitudes Scale is chosen so that a factor 100 in brightness corresponds to 5 magnitudes (historical) The magnitude scale f m X mref 2.5 log f ref One common system is to measure relative to Vega By definition, Vega has m=0 in all bands. Note this does not mean Vega is equally bright at all wavelengths! Setting mref=0 in the equation above gives: mX 2.5 log f 2.5 log fVega, X 2.5 log f m0, X • Colour is defined as the relative flux between two different wavebands, usually written as a difference in magnitudes Apparent magnitudes The faintest (deepest) telescope image taken so far is the Hubble Ultra-Deep Field. At m=29, this reaches more than 1 billion times fainter than what we can see with the naked eye. f m X mref 2.5 log f ref Object Apparent mag Sun -26.5 Full moon -12.5 Venus -4.0 Jupiter -3.0 Sirius -1.4 Polaris 2.0 Eye limit 6.0 Pluto 15.0 Reasonable telescope limit (8-m telescope, 4 hour integration) 28 Deepest image ever taken (Hubble UDF) 29 10( 296) / 2.5 1046 / 5 109