Principles of Astrophysics and Cosmology Welcome Back to PHYS 3368 Henry Norris Russell October 25, 1877– February, 18 1957 Principles of Astrophysics & Cosmology - Professor Jodi Cooley Announcements - Reading Assignments: Chapter 3.3 - 3.12. - Problem Set 4 is due Wednesday, February 18th, 2015. - Next lab is Monday, February 23rd. Be sure to report to FOSC 032 that day. Principles of Astrophysics & Cosmology - Professor Jodi Cooley Goals for this Class 1. Calculate how long it takes a photon to travel from the center of the sun and emerge at its surface. 2. Estimate the luminosity of the sun. 3. Derive equations for the gas and radiation pressure inside a star. Principles of Astrophysics & Cosmology - Professor Jodi Cooley basicastro4 October 26, 2006 STELLAR PHYSICS 37 A density profile, ρ(r), that falls with radius will give a somewhat more negative 2 value of Egr. Taking a characteristic Egr GM /r, the mean pressure in the Sun is ∼− 2 2 1 GM GM 15 2 9 P¯ ⊙ = ⊙ 10 dyne cm− = 10 Atm. (3.30) ⊙ 4 3 4 ∼ 3 3 πr r 4πr ≈ ⊙ ⊙ ⊙ To find a typical temperature, which we will call the virial temperature, let us assume again a classical nonrelativistic ideal gas, with particles of mean mass m¯ . Equation 3.27 then applies, and 3 1 GM 2 1 GM Nm¯ NkT ⊙ = ⊙ . (3.31) From Last Time:2 vir ∼ 2 r 2 r ⊙ ⊙ The mass of an electron is negligibly small, only 1/2000 compared to the mass Mean mass m of a proton. For an ionized hydrogen gas, consisting≈ of an equal number of protons and electrons, the mean mass m¯ , m + m m m¯ = e p = H , (3.32) 2 2 is therefore close to one-half the mass of the proton or exactly one-half of the 24 hydrogen atom, m =1.7 10− g. The typical thermal energy is then H × 8 33 24 TheGM meanmH mass6 .is7 1/210 the− cgs mass2 of the10 hydrogeng 1.7 atom.10− g kTvir ⊙ = × × × × × (3.33) ∼Since,6r the me << mp, the mean6 mass7 10 of10 hydrogencm is ⊙ × × close to 1/2 the mass of a10 proton. =5.4 10− erg = 0.34 keV. × 16 1 5 1 With k =1.4 10− erg K− =8.6 10− eV K− , this gives a virial tempera- ture of about 4× 106 K. As we will see,× at temperatures of this order of magnitude, nuclear reactions× can take place, and thus replenish the thermal energy that the star radiates away, halting the gravitational collapse (if only temporarily). Of course, in reality, just like P (r), the density ρ(r) and the temperature T (r) are also functions of radius and they grow toward the center of a star. To find them,Principles we of Astrophysics will need & Cosmology to define - Professor additionalJodi Cooley equations. We will see that the equation of hydrostatic equilibrium is one of four coupled differential equations that determine stellar structure. 3.2 MASS CONTINUITY In the hydrostatic equilibrium equation (Eq. 3.19), we have M(r) and ρ(r), which are easily related to each other: dM(r)=ρ(r)4πr2dr, (3.34) or dM(r) =4πr2ρ(r). (3.35) dr Although this is, in essence, merely the definition of density, in the context of stellar structure this equation is often referred to as the equation of mass continuity or the equation of mass conservation. basicastro4 October 26, 2006 38 CHAPTER 3 Figure 3.2 A volume element in a field of targets as viewed in perspective (left). The target number density is n, and each target presents a cross section σ. From the base of the cylindrical volume, ndx targets per unit area are seen in projection (right). A straight line along the length of the volume will therefore intercept, on average, nσdx targets. 3.3 RADIATIVE ENERGY TRANSPORT The radial gradient in P (r) that supports a star is produced by a gradient in ρ(r) and T (r). In much of the volume of most stars, T (r) is determined by the rate at which radiative energy flows in and out through every radius, i.e., the luminosity L(r). To find theRadiative equationbasicastro4basicastro4 that October determines 26, OctoberEnergy 2006 26,T 2006(r), we Transport will need to study some of the basics of “radiative transfer”, the passage of radiation through matter. In some of the volumeThe of radial some gradient stars, in thepressure energy that transportsupports a star mechanism is produced that by a dominatesgradient in is con- vection,38 ratherdensity than and temperature. radiative transport. Pressure, density We will andCHAPTER discuss temperature 3 convection are functions is Section of radius. 3.12. 38 CHAPTER 3 Energy transport by means of conduction plays a role only in dense stellar remnants – white dwarfs and neutron stars – which are discussed in Chapter 4. σ = effective cross Photons in stars can be absorbed or scattered out of a beamsection via interactions for absorption with molecules, with atoms (either neutral or ions), and with electrons.or scattering. If a photon traverses a path dx filled with “targets” with a number density n (i.e., the number of targets per unit volume), then the projected number of targets per unit area lying in the pathFigure of 3.2 the A volume photon element in is a fieldndx of targets(see as viewed Fig. in perspective3.2). If (left). each The target target poses an effective “cross A numberphoton density transverses is n, and each target a presentspath adx, cross section σ. FromProjected the base of number of targets section”2 Figureσthefor cylindrical 3.2 absorption A volume volume, elementndx targets inor a fieldper scattering, unit of area targets are as seen viewed in then projection in perspectiveper the (right). unit fraction A (left). area The lying of target the in the area path covered by filledstraight with linenumber along “targets” densitythe length is ofn ,with the and volume each a target will therefore presents intercept, a cross section on average,σ. From the base of targets is σnumberndxnσdx targets..the Thus,density. cylindrical the volume, numberndx targets of per targets unit area are thatof seen the will in projection photon typically (right). is given A be intersectedas ndx. by a straight line along the length of the volume will therefore intercept, on average, straight3.3 RADIATIVE line traversing ENERGYnσdx targets. TRANSPORT the path dx, or in other words, the number of interactions the photonFor undergoes, a straight willline along be the length of the path The radial3.3 gradient RADIATIVE in P (r) that ENERGY supports TRANSPORT a star is produced by a gradient in ρ(r) and T (r). In much of the volume of# most of stars, interactionsT (r) is determined by = then rateσdx. at (3.36) which radiative energy flows in and out through every radius, i.e., the luminosity The radial gradient in P (r) that supports a star is produced by a gradient in ρ(r) L(r). To find the equation that determines T (r), we will need to study some of the EquationPrinciples 3.36 of Astrophysics defines & Cosmology the - Professor concept Jodi Cooley of cross section. (Cross section can be defined basics ofand “radiativeT (r). In transfer”, much of the the passage volume of radiation of most stars,throughT matter.(r) is determined In some of by the rate at equivalentlythe volumewhich of as some radiative the stars, ratio energy the energy flows between transport in and mechanism out the through interaction that every dominates radius, is i.e., rate con- the per luminosity target particle and the vection,L rather(r). To than find radiative the equation transport. that We determines will discussT convection(r), we will is Section need to 3.12. study some of the incomingEnergy transportbasics flux of of by “radiative means projectiles.) of conduction transfer”, plays the Settingpassage a role only of inradiation the dense left-hand stellar through remnants matter. side In some equal of to 1, the typical distance– white athe dwarfs photon volume and neutron of will some stars travel stars, – which the between energy are discussed transport interactions in Chapter mechanism 4. that is dominates called is the con- “mean free path”: Photonsvection, in stars rather can be than absorbed radiative or scattered transport. out We of a will beam discuss via interactions convection with is Section 3.12. molecules,Energy with transport atoms (either by means neutral of or conduction ions), and plays with electrons. a role1 only If in a dense photon stellar remnants traverses– a white path dx dwarfsfilled withand neutron “targets” stars with a – number which aredensityl = discussedn (i.e.,. the in Chapter number of 4. (3.37) targets per unit volume), then the projected number of targetsn perσ unit area lying in Photons in stars can be absorbed or scattered out of a beam via interactions with the path of the photon is ndx (see Fig. 3.2). If each target poses an effective “cross section”molecules,2 σ for absorption with atoms or scattering, (either then neutral the fraction or ions), of and the area with covered electrons. by If a photon traverses a path dx filled with “targets” with a number density n (i.e., the number of 2 targets is σndx. Thus, the number of targets that will typically be intersected by a Thestraight crosstargets line section traversing per unit of the a volume), particle, path dx, then or which in the other projected has words, units thenumber number of area, of targets of interactions quantifies per unit area the lying degree in to which the particle is liable tothe take photonthe part undergoes, path in of a the particular will photon be is ndx interaction(see Fig.