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Fundamental Astrophysics Course Lecture 1 - Overview

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Fundamental Astrophysics Course Lecture 1 - Overview Fundamental Astrophysics Course Lecture 1 - Overview • An Introduc3on to Stars • Observed Features of Star Formaon • Overview of the Star Formaon Sequence • The InterStellar Medium • Mean Molecular Weight • The Angular Momentum Issue • The Virial Theorem • Linearized Hydrodynamical Equaons 10:10 pm An Introduc3on to Stars 10:10 pm Star field image 10:10 pm The Sun is the best studied star • Stellar interiors not directly observable. • Solar neutrinos emiLed in core and detectable. • Helioseismology - vibraons of solar surface can be used to probe density structure • Models of stellar interiors constructed and predic3ons of these models are tested by comparison with observed proper3es of individual stars. 10:10 pm The building blocks in the universe:stars corona photosphere Stellar core chromosphere 10:10 pm Star clusters Star clusters are very useful to understand stars and their evolu3on: • Stars all at same distance • Stars are dynamically bound • Stars have same age • Stars have same chemical composi3on Can contain 103 – 106 stars NGC3603 from Hubble Space Telescope 10:10 pm Globular clusters are more massive star clusters 10:10 pm Observable properties of stars Basic parameters to compare theory and observations: • Stellar mass (M) • Luminosity (L) ∞ – The total energy radiated per second i.e. power (in W): L = L dλ ∫0 λ – If F is the flux (W·m−2) and L is the luminosity (W) or where F is the flux (erg·s−1·cm−2) and L is the luminosity (erg·s−1) and D the distance to the star: L F = 2 • Radius (R) 4πD € • Effective temperature (Te) – The temperature of a black body of the same radius as the star that would radiate the €same amount of energy. Thus 2 4 L= 4πR σ Te where σ is the Stefan-Boltzmann constant (5.67× 10-8 Wm-2K-4) 10:10 pm A few fundamental values 30 • Solar mass (M) = 1.989x10 kg ~ 2x1030 kg 26 • Solar luminosity (L) = 3.839x10 W (3.939x1026W accoun3ng for solar neutrinos) 8 8 • Solar radius (R) = 6.955x10 m ~ 7x10 m (432 000 miles) 2 4 • L= 4 π R σ Te => Te4 = 1.114x1015 => Te (Sun) = 5 777 K 10:10 pm σ = 5.67x10-8 W m-2 K44 Stellar radii • Angular diameter of sun at distance of 10pc: -9 -3 θ= 2 R / 10pc = 5× 10 radians = 10 arcsec • Compare with Hubble resolu3on of ~0.05 arcsec ⇒ very difficult to measure R directly • Radii of ~ 1000 stars measured with techniques such as interferometry and eclipsing binaries. 10:10 pm • 1 parsec = 206 265 AU = 3.26 light-year = 3.0856x1016 m = 1.92x1013 miles • Θ ~ tanθ for small angles • Θ = 2R / 10 pc = (2 x 7x108) / (10 x 3.0856x1016) = 4.537x10-9rd = 9x10-4 arcsec 10:10 pm The Hertzsprung-Russell diagram M, R, L and Te do not vary independently. Two major relaonships: 1. L with T 2. L with M The first is known as the Hertzsprung- Russell (HR) diagram or its observed counterpart, the colour-magnitude diagram (CMD). 10:10 pm Colour-Magnitude Diagrams (CMD) 2 3 Measuring accurate Te for ~10 or 10 stars is an intensive task – spectra needed and model atmospheres Magnitudes of stars are measured at different wavelengths: standard system is UBVRI Band U B V R I λ/nm 365 445 551 658 806 W/nm 66 94 88 138 149 10:10 pm Other filters exist over the en3re electromagne3c spectrum and not only in the op3cal range 10:10 pm Magnitudes and Colours (γ Orionis) (α Orionis) 10:10 pm Magnitudes and Colours Various calibraons* can be used to provide the colour relaon: B-V = f (Te) or any color = mag1 – mag2 = f (Te) relaon •Note that the observed (B - V) must be corrected for the interstellar ex3nc3on to get (B - V)0 10:10 pm Extinction •Ex3nc3on: Absorp3on and scaering of electromagne3c radiaon by dust and gas between the observed object and the observer •The exncon due to dust is not equally effecve at all wavelengths. The shorter the wavelength, the higher the ex3nc3on - blue light is affected more strongly than red light. •Therefore , stars behind a lot of dust look redder than they really are. This is called interstellar reddening. 10:10 pm UV and Dust Emission in the Far-IR at all scales UV/op3cal Far-infrared 10:10 pm UV and Dust Emission in the Far-IR at all scales 10:10 pm Absolute magnitude and bolometric magnitude • Absolute Magnitude M (≠ from mass) defined as apparent magnitude of a star if it were placed at a distance of 10 pc m – M = 5 log(d/10) – 5 where d is in pc • Magnitudes are measured in some wavelength band e.g. UBV. To compare with theory it is more useful to determine bolometric magnitude – defined as absolute magnitude that would be measured by a detector sensitive to all wavelengths. We define the bolometric correction to be: BC = Mbol – Mv Bolometric luminosity is then: Mbol – Mbol = -2.5 log L/L 10:10 pm Bolometric corrections for Main-Sequence Stars From Allen’s Astrophysical Quan33es (4th edion) 10:10 pm 2. The Mass-luminosity relation For main-sequence stars, there is a Mass-luminosity relation. L ∝ Mn Where n ~ 3.5. Slope changes at extremes, less steep for low and high mass stars. This implies that the main- sequence (MS) on the HRD is a function of mass i.e. from bottom to top of main-sequence, stars increase in mass We must understand the M-L relaon and L-Te relaon theore3cally. Models must reproduce observaons 10:10 pm Over a wide mass range 10:10 pm Age and metallicity There are two other fundamental properties of stars that we can measure: age and chemical composition Composition is parameterised with X, Y, Z ≡ mass fraction of H, He and all other elements e.g. X = 0.747 ; Y = 0.236 ; Z = 0.017 Note: Z often referred to as metallicity (while z is redshift) Ideally, astronomers would prefer studying stars of same age and chemical composition to keep these parameters constant and determine how models reproduce the other observables => star clusters. 10:10 pm Globular cluster example Selecon of Open clusters Age (yrs) 2.0x106 branch Horizontal 6.5x106 2.8x107 Turn off 1.6x108 1.9x109 • In clusters, t and Z must be same for all stars 7.1x109 • Hence differences must be due to Minit • Stellar evolution assumes that the 10 differences in cluster stars are due mainly to 2.9x10 initial mass: Minit • Cluster HR (or colour-magnitude) diagrams are quite similar: the age defines the overall appearance 10:10 pm Globular vs. Open clusters Globular Open • MS turn-off points in similar • MS turn off point varies position. Giant branch joining massively, faintest is MS consistent with globulars • Horizontal branch from giant • Maximum luminosity of MS branch to above the MS turn- stars can get to Mv ≈ -10 off point • Very massive stars found in • Old stellar populations (at these clusters least in the MW) The differences are interpreted due to age. Open clusters lie in the disk of the Milky Way and have large range of ages. In the Milky Way, globulars are all ancient, with the oldest tracing the earliest stages of the formaon of Milky Way (~ 12× 109 yrs) 10:10 pm Summary • Definition : a star burn hydrogen in its core • Characteristics: o 0.08 < M / Msun < a few 100 -4 6 o 10 < L / Lsun < 10 o L ~ M3.5 (strong dependence) o Lifetime τ = M / L ~ M / M3.5 = M-2.5 • Four fundamental observables used to parameterise stars and compare with models M, R, L, Te • M and R can be measured directly in small numbers of stars • Age and chemical composition also shape the HR diagram • Stellar clusters are crucial laboratories to study stars and their evolution: all stars at same distance, same t, and initial Z • We will develop models to attempt to reproduce the M, R, L, Te relationships and understand HR diagrams 10:10 pm Mass funcons • The stellar masses are one of the most important factor in determining their evolu3on, so when studying a stellar populaon, we are interested in es3mang their masses. • An important informaon is the number of stars per unit mass which is called a mass funcon. • We define the mass funcon Φ(M) such that Φ(M) dM is the number of stars with masses between M and M + dM. • With this definion, the total number of stars with masses between Mlow and Mup is: Mup N(M ,M ) = Φ(M)dM low up ∫Mlow • By deriving both sides, we get: dN = Φ(M) € dM • Φ is the derivave of the number of stars with respect to mass, i.e. the number of stars dN within some mass interval dM. € 10:10 pm Total mass of stars between Mlow and Mup • If we are interested in the total mass of the stars between Mlow and Mup in a given system rather than the number of such stars, we must integrate Φ 3mes the mass per star. Thus the total mass of stars with masses between Mlow and Mup is: Mup M (M ,M ) = MΦ(M)dM * low up ∫Mlow • or equivalently: y = ln(m) dM dy/dm = 1/m * = MΦ(M) = ξ(M) dy = dm/m dM d[ln(m)] = dm/m € • ξ (M) gives the number of star per ln(M), rather than per number in mass. • Physical explana.on: – Suppose that€ Φ(M) = Cte constant: there are as many stars from 1 − 2 M as there are from 2 − 3 M as there are from 3 − 4 M , etc.
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