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The Sun: Its Structure and Energy Generation

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The Sun: Its Structure and Energy Generation Sun, Stars and Planets J C Pickering 2014 Part 1: The Sun: its structure and energy generation Lecture 1. Basic properties of the Sun – an introduction 1.1 Setting the scene… 1.2 Definition of a star 1.3 Properties of the Sun 1.4 How are these properties quantified? 1.1 Setting the scene Earth- Sun distance = 1 AU (Astronomical Unit), or 8 minutes for light to reach us Distances: (in light years) Nearest star Proxima Centauri 4.2 Sirius 8.6 Orion Nebula 1500 Galactic Centre 28,000 Galactic Diameter 100,000 Andromeda galaxy 2,300,000 (biggest galaxy in our Local Group of 30 galaxies 6 million lys across) Virgo Cluster 50,000,000 (2000 galaxies, nearest other galactic cluster, and centre of our Local Supercluster – diameter 100,000,000) Most Distant Observed Object ~13,000,000,000 Number of stars and galaxies… Galaxies contain anything from between 100,000 up to 3,000,000,000,000 (3 thousand billion) stars! Our Galaxy contains 100 billion stars. Number of galaxies estimated in observable Universe – 200 - 300 billion… So, it is vital we understand the physics of stars! How do we go about describing the internal structure of stars and their evolution? Evolution of stars is very slow; for a typical ordinary star like our Sun a small change in the Sun’s properties would make the Earth uninhabitable for us, but we have been on Earth for hundreds of thousands of years. Geologists believe the Earth’s crust has been solid for several thousand million years, and that the Sun’s luminosity cannot have changed significantly during this time span. So how can we make progress in our understanding of stars and their evolution… ? What are the fundamental things we know about the Sun? and what is the Sun? 1.2 Definition of a star A star is a self-gravitating mass of gas that radiates energy. 1.3 Properties of a star we need to know about Before we can talk about evolution and internal structure we need to know the basic properties of a star: it has a Mass, which means there is a Pressure, and a gas pressure means that we want to know about the temperature of the gas. The star radiates heat which can be quantified by Luminosity. Mass pressure temperature heat luminosity The Sun – is our closest star and has global properties: 30 mass M = 1.99 x 10 kg (one solar Mass) 8 radius R 6.96 x 10 m (solar radius) = 26 luminosity L = 3.83 x 10 W (solar luminosity) 11 Sun-Earth mean distance = 1 Astronomical Unit (AU) = 1.50 x 10 m 1 1.4 How are these quantities determined? 1.4.1 Distance: 2 3 rd Kepler’s 3 law P µ D P is planet’s orbital period D is the planet’s orbital semimajor axis) - gives relative scale of the solar system but not the absolute scale; then use accurate measurement of distance to a planet to set the absolute scale e.g. radar-ranging to Venus. (Earlier methods of distance measurement: transit observations; Greek astronomy) 1.4.2 Radius of the Sun: Measure the angular size of the Sun, and knowing the Earth-Sun distance ⇒ radius Using the small angle approximation, in radians, θ/2 = r/d where θ = angular size of the Sun, r = Sun’s radius, d = Earth-Sun distance 1.4.3 Mass of the Sun: Knowing the orbital motions of planets and their distance ⇒ G M to high precision. [G = the gravitational constant, Ms mass of star, mp mass of planet, d star-planet separation, ω angular velocity, P orbital period] 1.4.4 Luminosity and flux: A star’s Luminosity, L, is the power (W) emitted by the entire surface of a star. At a distance d from the star, its luminosity is found by measuring flux density F (Wm-2): At distance d from a star, its luminosity is spread over a sphere of area 4 π d2. We assume that the star radiates uniformly in all directions. So at any point on the sphere the flux density is given by: inverse-square law: F = L / (4πππd2 ) ( d = 1 A.U. ) For the Sun we can measure flux (energy incident from the Sun per unit time per unit area) at the Sun-Earth distance, known as the solar “constant” = 1368 W m-2 d and from this we can calculate solar luminosity. F 2 TO DO: calculate the solar luminosity using the solar constant 1368 Wm-2. TO DO: what would be the “solar constant” as measured on: Also, we can compare d the solar flux at the Earth with 2 (i) Mercury the solar flux at other planets: (ii) Pluto ? for example to estimate the d surface temperature of other 1 planets. F 1 F 2 1.4.5 Surface temperature and radiation: Reminder: Black-body Radiation. A Black body: - is in thermodynamic equilibrium: photons and matter have the same temperature. Reach this through collisions/interactions, mean free path (mfp) is short, ie are opaque. - The light within the source is more likely to interact with the material of the source than to escape, it will only escape after considerable interaction with material within the source. So a common feature of BB sources is that they are opaque. At a given T, any body in thermodynamic equilibrium will show a black-body (≈ continuum) spectrum. For a star, the BB spectrum is a good approximation up to a point only – where the stellar atmosphere’s mfp increases, a line spectrum is formed (and this can tell us about the star’s composition, etc). Many astronomical sources produce continuous spectra with reasonably good approximation to BB form. Planck’s law : Wien’s law: Stefan’s law: 3 Radiation from the surface of stars can be approximated to that of black-body radiation: For black body, flux F per unit area of emitter F = σσσ T4 (Stefan’s law) where σ = Stefan-Boltzmann constant The Effective temperature is defined such that a black body of temperature Teff with the same radius as the star would radiate the same total amount of energy. Define effective temperature of surface of star by: L = 4 π R2 σ (Teff)4 where R is the radius of the star. TO DO: For the Sun, using values of L and R from 1.3 calculate the effective temperature of the Sun L = 4 π R2 σ (Teff)4 Teff = K 1.4.6 Chemical composition of the Sun: We believe this to be similar to typical composition in the universe. The composition of a sample of material at any depth within the Sun can be defined using 3 simple parameters: hydrogen mass fraction X = mass of hydrogen in sample/mass of sample, and similarly helium mass fraction Y, and metallicity Z = mass of other elements in sample/mass of sample, Hydrogen ~73% by mass X Helium ~25% Y Heavier elements ~2% Z (O, C, N, Ne, Fe, … in order of abundance) How do we know this? : Observational data: solar spectrum, meteorites 1.4.7 Age of the Sun: Only known indirectly: radioactive dating of rocks; computed from evolutionary models of the Sun. ~ 4.6 x 109 years LECTURE KEY POINTS: • Definition of a star • Sun’s global properties: mass, radius, luminosity • Astronomical unit AU, solar units • Luminosity, inverse square law, flux density • Definition of effective temperature; Stefan’s law • Sun’s chemical composition and age Learning outcomes include: • be able to give definitions of a star, astronomical unit, effective temperature • be able to explain what is meant by luminosity, flux, and a black body source. • have an awareness of how the Sun’s properties are measured, more details to follow later in the course as indicated during the lecture. • know Stefan’s law, and the definition of effective temperature such that you are able to write down the equations and use them To Do: - Problem Sheet 1 - Question 1 - Check you understand what a black body is - calculate the solar luminosity using the measured solar constant - Quick calculations: solar flux at Mercury and Pluto - Calculate the effective temperature of the Sun 4 Sun Stars and Planets J C Pickering 2014 Part 1 continued: The Sun: its structure and energy generation Lecture 2: Modelling the Stellar interior Aims: derive equations governing the physical properties of a star and examine their validity 2.1 Hydrostatic equilibrium 2.2 Equation of mass continuity 2.3 How good an approximation is hydrostatic equilibrium? 2.4 How long would it take a star to collapse if pressure forces were negligible? 2.5 Mean density of the Sun 2.6 Very simple estimate of the mean pressure of the Sun 2.7 Estimate of minimum central pressure of the Sun 2.1 Hydrostatic equilibrium A star is held together by the force of gravitation, the attraction exerted on each part of the star by all other parts. If this force was the only important one then the star would rapidly shrink – but this attractive gravitational force is resisted by the pressure of the stellar material in the same way that the kinetic energy of the molecules, or equivalently the pressure of the Earth’s atmosphere, prevents the atmosphere from collapsing to the surface of the Earth. These two forces, gravitational attraction and thermal pressure, play key roles in determining stellar structure. 2.1.1. Assumptions (a) assume stars are spherical and symmetrical about their centres (b) The stellar properties change so slowly with time - neglect the rate of change with time of these properties.
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