Static Stellar Structure

Size: px
Start display at page:

Download "Static Stellar Structure"

Transcription

1

2 Most of the Life of A Star is Spent in Equilibrium Evolutionary Changes are generally slow and can usually be handled in a quasistationary manner We generally assume: Hydrostatic Equilibrium Thermodynamic Equilibrium The Equation of Hydrodynamic Equilibrium dr Gmr () ρ P = ρ dt r r

3 Limits on Hydrostatic Equilibrium If the system is not Moving - accelerating in reality - then d r/dt = 0 and then one recovers the equation of hydrostatic equilibrium: If P/ P/ r r ~ 0 then which is just the freefall condition for which the time scale is t ff (GM/R 3 ) -/ Gm() r ρ P dp = = r r dr dr dt Gmr () = r 3

4 Dominant Pressure Gradient When the pressure gradient dp/dr dominates one gets (r/t) ~ P/ρ This implies that the fluid elements must move at the local sonic velocity: c s = P/ P/ ρ. When hydrostatic equilibrium applies V << c s t e >> t ff where t e is the evolutionary time scale 4

5 Hydrostatic Equilibrium Consider a spherical star Shell of radius r, thickness dr and density ρ(r) Gravitional Force: (Gm(r)/r ) 4πr4 ρ(r)dr Pressure Force: 4r dp where dp is the pressure difference across dr Equate the two: 4πr dp = (Gm(r)/r ) 4πr4 ρ(r)dr r dp = Gm(r) ρ(r)dr dp/dr = -ρ(r)(gm(r)/r ) The - sign takes care of the fact that the pressure decreases outward. 5

6 Mass Continuity m(r) ) = mass within a shell = mr () = 4 πr ρ( r ) dr 0 This is a first order differential equation which needs boundary conditions We choose P c = the central pressure. Let us derive another form of the hydrostatic equation using the mass continuity equation. Express the mass continuity equation as a differential: dm/dr dr = 4πr4 ρ(r). Now divide the hydrostatic equation by the mass- continuity equation to get: dp/dm = Gm/4πr 4 (m) r 6

7 The Hydrostatic Equation in Mass dp/dm = Gm/4πr 4 (m) Coordinates The independent variable is m r is treated as a function of m The limits on m are: 0 at r = 0 M at r = R (this is the boundary condition on the mass equation itself). Why? Radius can be difficult to define Mass is fixed. 7

8 The Central Pressure Consider the quantity: P + Gm(r) /8πr 4 Take the derivative with respect to r: d Gm() r dp Gm() r dm Gm() r P dr + 8πr = + dr 4πr dr πr But the first two terms are equal and opposite so the derivative is -Gm /r 5. Since the derivative is negative it must decrease outwards. At the center m /r 4 0 and P = P c. At r = R P = 0 therefore P c > GM /8πR 4 8

9 The Virial Theorem dp mg = 4 dm 4 π r( m) 3 dp mg 4π r = dm r d 3 dr mg (4 πrp) 4πr3P = dm dm r M 3 3 M M P mg (4 π r P) 0 dm = dm 0 ρ 0 r Remember: 4π r rdr = dm 9

10 The Virial Theorem 3 The term (4 π rp) M 0 is 0: r(0) = 0 and P(M) = 0 Remember that we are considering P, ρ,, and r as variables of m For a non-relativistic gas: 3P/ = * Thermal energy per unit mass. M 3 Pdm = U for the entire star 0 ρ GM dm =Ω r the gravitional binding energy 0

11 The Virial Theorem -U = Ω U + Ω = 0 Virial Theorem Note that E = U + Ω or that E+U = 0 This is only true if quasistatic. If hydrodynamic then there is a modification of the Virial Theorem that will work.

12 The Importance of the Virial Theorem Let us collapse a star due to pressure imbalance: This will release - Ω If hydrostatic equilibrium is to be maintained the thermal energy must change by: U U = -/ Ω This leaves / Ω to be lost from star Normally it is radiated

13 What Happens? Star gets hotter Energy is radiated into space System becomes more tightly bound: E decreases Note that the contraction leads to H burning (as long as the mass is greater than the critical mass). 3

14 An Atmospheric Use of Pressure We use a different form of the equation of hydrostatic equilibrium in an atmosphere. The atmosphere s s thickness is small compared to the radius of the star (or the mass of the atmosphere is small compared to the mass of the star) For the Sun the photosphere depth is measured in the 00's of km whereas the solar radius is 700,00 km. The photosphere mass is about % of the Sun. 4

15 Atmospheric Pressure Geometry: Plane Parallel dp/dr = -Gm(r) Gm(r)ρ/r (Hydrostatic Equation) R r and m(r) ) = M. We use h measured with respect to some arbitrary 0 level. dp/dh = - gρ where g = acceleration of gravity. For the Sun log(g) ) = 4.4 (units are cgs) Assume a constant T in the atmosphere. P = nkt and we use n = ρ/μm H (μ is the mean molecular weight) so P = ρkt/ kt/μm H 5

16 Atmospheric Pressure Continued P = ρkt/ kt/μm H dp = -g ρ dh dp = -g P(μm H /kt) ) dh Or dp/p = -g( g(μm H /kt) ) dh Integrate : P = P e 0 ρ = ρ e 0 gmh μh kt gmh μh kt Where: P 0 = P and ρ 0 = ρ at h = 0. 6

17 Scale Heights H (the scale height) is defined as kt / μm H g It defines the scale length by which P decreases by a factor of e. In non-isothermal atmospheres scale heights are still of importance: H - (dp/dh) - P = -(dh/d(lnp)) 7

18 Simple Models To do this right we need data on energy generation and energy transfer but: The linear model: ρ = ρ c (-r/r) Polytropic Model: P = Kρ γ K and γ independent of r γ not necessarily c p /c v 8

19 The Linear Model r ρ = ρc( ) R dp Gm() r r = ρ ( ) c dr r R Defining Equation Put in the Hydrostatic Equation Now we need to deal with m(r) 9

20 An Expression for m(r) dm r = 4 πr ρc( ) dr R 3 4π r ρc = 4πr ρc R 3 4 4π r ρc πr ρc mr () = 3 R 3 π R ρc Total Mass M = m( R) = 3 4r 3r So m() r = M R R Equation of Continuity Integrate from 0 to r 0

21 Linear Model We want a particular (M,R) ρ c = 3M/πR 3 and take P c = P(ρ c ) Substitute m(r) into the hydrostatic equation: dr r 3 R R 3 4 dp πg 4r r r = ρ c 4r 7r r = πgρc + 3 3R R r 7r r So P = Pc πgρc + 3 9R 4R where P is the central pressure. c

22 Central Pressure At the surface r = R and P = 0 4 R c 5 G PR ( ) = P = 0 c 36 c 6 5 G R 5 GR 9M P = = c π R P c = 5GM 4 R π ρ π ρ π π

23 The Pressure and Temperature Structure The pressure at any radius is then: Pr () = G R R 5R 5R 3 4 π r r r ρ c 3 4 Ignoring Radiation Pressure: T μmh P = kρ 5π Gμm r 9r 9r 36 k R 5R 5R 3 H = ρcr

24 Polytropes P = Kρ γ which can also be written as P = Kρ K (n+)/n N Polytropic Index Consider Adiabatic Convective Equilibrium Completely convective Comes to equilibrium No radiation pressure Then P = Kρ γ where γ = 5/3 (for an ideal monoatomic gas) ==> n =.5 4

25 Gas and Radiation Pressure P g = (N 0 k/μ)ρt T = βp P r = /3 a T 4 = (- β)p P = P P g /β = P r /(-β) /β(n 0 k/μ)ρt = /(-β) ) /3 a T 4 Solve for T: T 3 = (3(- β)/aβ) ) (N 0 k/μ)ρ T = ((3(- β)/aβ) ) (N 0 k/μ)) )) /3 ρ /3 5

26 The Pressure Equation P = Nk 0 μ ρt β β 3 Nk = μ a β True for each point in the star If β f(r), i.e. is a constant then P = KρK 4/3 n = 3 polytrope or γ = 4/3 ρ 6

27 The Eddington Standard Model n = 3 For n = 3 ρ T 3 The general result is: ρ T n Let us proceed to the Lane-Emden Equation ρ λφ n λ is a scaling parameter identified with ρ c - the central density φ n is normalized to at the center. Eqn 0: P = Kρ K (n+)/n = Kλ K (n+)/n φ n+ Eqn : Hydrostatic Eqn: dp/dr = -Gρm(r)/r Eqn : Mass Continuity: dm/dr dr = 4πr4 ρ 7

28 Working Onward Solve for m( r): m( r) = d r dp Put m() r in : 4πGρ = r dr ρ dr dp dr n+ n n = Kλ ( n+ ) φ r dp ρgdr dφ dr φ n Kλ ( n+ ) φ = 4πGλφ n+ d r n n d n r dr dr λφ 8

29 Simplify n+ d r n dφ n n ( ) 4 Kλ n+ φ n = πgλφ r dr λφ dr d n dφ n Kλ ( n+ ) r 4πGλφ r dr dr = n d n dφ n Kλ ( n+ ) r φ 4 πg r = dr dr n n Kλ ( n+ ) Define a 4π G r ξ so adξ = dr a d dφ n So : ξ ξ dξ dξ = φ 9

30 The Lane-Emden Equation Boundary Conditions: ξ d = ξ d ξ d dφ ξ φ n Center ξξ = 0; φ = (the function); dφ/d ξ = 0 Solutions for n = 0,, 5 exist and are of the form: φ(ξ) = C 0 + C ξ + C 4 ξ = - (/6) ξ + (n/0) ξ for ξ = (n>0) For n < 5 the solution decreases monotonically and φ 0 at some value ξ which represents the value of the boundary level. 30

31 General Properties For each n with a specified K there exists a family of solutions which is specified by the central density. For the standard model: K 4 3 Nk 0 3 β = 4 We want Radius, m(r), or m(ξ) Central Pressure, Density Central Temperature μ a β 3

32 Radius and Mass R = aξ ( n+ ) K = 4π G aξ M( ξ) = 4πr ρdr 0 ξ 3 n 0 λ n n = 4πa λφ ξ dξ d dφ ξ ξ dξ = dξ d dφ ξ π λ ξ ξ dξ dξ ξ 3 M ( ) = 4 a d 0 ξ n φ ξ 3 dφ M( ξ) = 4πa λ d 0 ξ dξ 3 dφ = 4πa λξ dξ Now substitute for a and evaluate at ξ = ξ ( boundary) M 3 3 n ( n+ ) K n dφ = 4π λ ξ 4πG dξ ξ= ξ 3

33 The Mean to Central Density ρ c = λ 3 3 n ( n+ ) K dφ n 4π λ ξ 4πG dξ ξ = ξ 3 3 3( n) R 4 ( n+ ) K n 3 π λ ξ M ρ = = 4 π 3 3 4π G 3 dφ ρ = λ ξ dξ ρ 3 dφ = ρc ξ dξ ξ= ξ ξ= ξ 33

34 The Central Pressure At the center ξ = 0 and φ = so P c = KλK n+/n This is because P = KρK n+/n = Kλ K (n+)/n φ n+ Now take the radius equation: ( n+ ) K R = 4π G λ n n n n ( n + ) ξ = Kλ 4π G 4π GR So Kλ n n = n + ) ξ ξ 34

35 Central Pressure c n n n n c P = Kλ λ = Kλ ρ 4πGR ξ = ρ ( n+ ) ξ 3 ( dφ/ dξ ) ξ But M = πr ρ so M = π R ρ 3 9 P c ξ M 6 4πGR 9 = ( n+ ) ξ 3 ( dφ/ dξ) 6π R ξ GM = 4 4 πr ( n+ ) ( dφ/ dξ ) ξ 35

36 Central Temperature Nk P T P = 0 ρ = β g c c c c which means T c = β μ c 0 P Nk c μ ρ c 36

Summary of stellar equations

Summary of stellar equations Chapter 8 Summary of stellar equations Two equations governing hydrostatic equilibrium, dm dr = 4πr2 ρ(r) Mass conservation dp dr = Gm(r) r 2 ρ Hydrostatic equilibrium, three equations for luminosity and

More information

CHAPTER 16. Hydrostatic Equilibrium & Stellar Structure

CHAPTER 16. Hydrostatic Equilibrium & Stellar Structure CHAPTER 16 Hydrostatic Equilibrium & Stellar Structure Hydrostatic Equilibrium: A fluid is said to be in hydrostatic equilibrium (HE) when it is at rest. This occurs when external forces such as gravity

More information

Free-Fall Timescale of Sun

Free-Fall Timescale of Sun Free-Fall Timescale of Sun Free-fall timescale: The time it would take a star (or cloud) to collapse to a point if there was no outward pressure to counteract gravity. We can calculate the free-fall timescale

More information

3 Hydrostatic Equilibrium

3 Hydrostatic Equilibrium 3 Hydrostatic Equilibrium Reading: Shu, ch 5, ch 8 31 Timescales and Quasi-Hydrostatic Equilibrium Consider a gas obeying the Euler equations: Dρ Dt = ρ u, D u Dt = g 1 ρ P, Dɛ Dt = P ρ u + Γ Λ ρ Suppose

More information

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 11 Polytropes and the Lane Emden Equation

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 11 Polytropes and the Lane Emden Equation Physics 576 Stellar Astrophysics Prof. James Buckley Lecture 11 Polytropes and the Lane Emden Equation Reading/Homework Assignment Makeup class tomorrow morning at 9:00AM Read sections 2.5 to 2.9 in Rose

More information

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 10

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 10 Physics 576 Stellar Astrophysics Prof. James Buckley Lecture 10 Reading/Homework Assignment Makeup class tomorrow morning at 9:00AM Read sections 2.5 to 2.9 in Rose over spring break! Stellar Structure

More information

Fundamental Stellar Parameters. Radiative Transfer. Stellar Atmospheres

Fundamental Stellar Parameters. Radiative Transfer. Stellar Atmospheres Fundamental Stellar Parameters Radiative Transfer Stellar Atmospheres Equations of Stellar Structure Basic Principles Equations of Hydrostatic Equilibrium and Mass Conservation Central Pressure, Virial

More information

Physics 556 Stellar Astrophysics Prof. James Buckley

Physics 556 Stellar Astrophysics Prof. James Buckley hysics 556 Stellar Astrophysics rof. James Buckley Lecture 8 Convection and the Lane Emden Equations for Stellar Structure Reading/Homework Assignment Read sections 2.5 to 2.9 in Rose over spring break!

More information

SPA7023P/SPA7023U/ASTM109 Stellar Structure and Evolution Duration: 2.5 hours

SPA7023P/SPA7023U/ASTM109 Stellar Structure and Evolution Duration: 2.5 hours MSc/MSci Examination Day 28th April 2015 18:30 21:00 SPA7023P/SPA7023U/ASTM109 Stellar Structure and Evolution Duration: 2.5 hours YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL

More information

Physics 556 Stellar Astrophysics Prof. James Buckley. Lecture 9 Energy Production and Scaling Laws

Physics 556 Stellar Astrophysics Prof. James Buckley. Lecture 9 Energy Production and Scaling Laws Physics 556 Stellar Astrophysics Prof. James Buckley Lecture 9 Energy Production and Scaling Laws Equations of Stellar Structure Hydrostatic Equilibrium : dp Mass Continuity : dm(r) dr (r) dr =4πr 2 ρ(r)

More information

2. Equations of Stellar Structure

2. Equations of Stellar Structure 2. Equations of Stellar Structure We already discussed that the structure of stars is basically governed by three simple laws, namely hyostatic equilibrium, energy transport and energy generation. In this

More information

Phases of Stellar Evolution

Phases of Stellar Evolution Phases of Stellar Evolution Phases of Stellar Evolution Pre-Main Sequence Main Sequence Post-Main Sequence The Primary definition is thus what is the Main Sequence Locus of core H burning Burning Process

More information

Ay123 Set 1 solutions

Ay123 Set 1 solutions Ay13 Set 1 solutions Mia de los Reyes October 18 1. The scale of the Sun a Using the angular radius of the Sun and the radiant flux received at the top of the Earth s atmosphere, calculate the effective

More information

dr + Gm dr Gm2 2πr 5 = Gm2 2πr 5 < 0 (2) Q(r 0) P c, Q(r R) GM 2 /8πR 4. M and R are total mass and radius. Central pressure P c (Milne inequality):

dr + Gm dr Gm2 2πr 5 = Gm2 2πr 5 < 0 (2) Q(r 0) P c, Q(r R) GM 2 /8πR 4. M and R are total mass and radius. Central pressure P c (Milne inequality): 1 Stellar Structure Hydrostatic Equilibrium Spherically symmetric Newtonian equation of hydrostatics: dp/dr = Gmρ/r 2, dm/dr = 4πρr 2. 1) mr) is mass enclosed within radius r. Conditions at stellar centers

More information

Star Formation and Protostars

Star Formation and Protostars Stellar Objects: Star Formation and Protostars 1 Star Formation and Protostars 1 Preliminaries Objects on the way to become stars, but extract energy primarily from gravitational contraction are called

More information

Equations of Stellar Structure

Equations of Stellar Structure Equations of Stellar Structure Stellar structure and evolution can be calculated via a series of differential equations involving mass, pressure, temperature, and density. For simplicity, we will assume

More information

The Equations of Stellar structure

The Equations of Stellar structure PART IV: STARS I Section 1 The Equations of Stellar structure 1.1 Hyostatic Equilibrium da P P + dp r Consider a volume element, density ρ, thickness and area da, at distance r from a central point. The

More information

Examination, course FY2450 Astrophysics Wednesday 23 rd May, 2012 Time:

Examination, course FY2450 Astrophysics Wednesday 23 rd May, 2012 Time: Page 1 of 18 The Norwegian University of Science and Technology Department of Physics Contact person Name: Robert Hibbins Tel: 93551, mobile: 94 82 08 34 Examination, course FY2450 Astrophysics Wednesday

More information

M J v2 s. Gm (r) P dm = 3. For an isothermal gas, using the fact that m r, this is GM 2 R = 3N ot M 4πR 3 P o, P o = 3N ot M

M J v2 s. Gm (r) P dm = 3. For an isothermal gas, using the fact that m r, this is GM 2 R = 3N ot M 4πR 3 P o, P o = 3N ot M 1 Stellar Birth Criterion for stability neglect magnetic fields. Critical mass is called the Jean s mass M J, where a cloud s potential energy equals its kinetic energy. If spherical with uniform temperature,

More information

Polytropic Stars. c 2

Polytropic Stars. c 2 PH217: Aug-Dec 23 1 Polytropic Stars Stars are self gravitating globes of gas in kept in hyostatic equilibrium by internal pressure support. The hyostatic equilibrium condition, as mentioned earlier, is:

More information

Lecture 2. Relativistic Stars. Jolien Creighton. University of Wisconsin Milwaukee. July 16, 2012

Lecture 2. Relativistic Stars. Jolien Creighton. University of Wisconsin Milwaukee. July 16, 2012 Lecture 2 Relativistic Stars Jolien Creighton University of Wisconsin Milwaukee July 16, 2012 Equation of state of cold degenerate matter Non-relativistic degeneracy Relativistic degeneracy Chandrasekhar

More information

Ay 1 Lecture 8. Stellar Structure and the Sun

Ay 1 Lecture 8. Stellar Structure and the Sun Ay 1 Lecture 8 Stellar Structure and the Sun 8.1 Stellar Structure Basics How Stars Work Hydrostatic Equilibrium: gas and radiation pressure balance the gravity Thermal Equilibrium: Energy generated =

More information

ASTM109 Stellar Structure and Evolution Duration: 2.5 hours

ASTM109 Stellar Structure and Evolution Duration: 2.5 hours MSc Examination Day 15th May 2014 14:30 17:00 ASTM109 Stellar Structure and Evolution Duration: 2.5 hours YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY

More information

9.1 Introduction. 9.2 Static Models STELLAR MODELS

9.1 Introduction. 9.2 Static Models STELLAR MODELS M. Pettini: Structure and Evolution of Stars Lecture 9 STELLAR MODELS 9.1 Introduction Stars are complex physical systems, but not too complex to be modelled numerically and, with some simplifying assumptions,

More information

Tides in Higher-Dimensional Newtonian Gravity

Tides in Higher-Dimensional Newtonian Gravity Tides in Higher-Dimensional Newtonian Gravity Philippe Landry Department of Physics University of Guelph 23 rd Midwest Relativity Meeting October 25, 2013 Tides: A Familiar Example Gravitational interactions

More information

dp dr = ρgm r 2 (1) dm dt dr = 3

dp dr = ρgm r 2 (1) dm dt dr = 3 Week 5: Simple equilibrium configurations for stars Follows on closely from chapter 5 of Prialnik. Having understood gas, radiation and nuclear physics at the level necessary to have an understanding of

More information

Introduction. Stellar Objects: Introduction 1. Why should we care about star astrophysics?

Introduction. Stellar Objects: Introduction 1. Why should we care about star astrophysics? Stellar Objects: Introduction 1 Introduction Why should we care about star astrophysics? stars are a major constituent of the visible universe understanding how stars work is probably the earliest major

More information

EVOLUTION OF STARS: A DETAILED PICTURE

EVOLUTION OF STARS: A DETAILED PICTURE EVOLUTION OF STARS: A DETAILED PICTURE PRE-MAIN SEQUENCE PHASE CH 9: 9.1 All questions 9.1, 9.2, 9.3, 9.4 at the end of this chapter are advised PRE-PROTOSTELLAR PHASE SELF -GRAVITATIONAL COLL APSE p 6

More information

ASTR 1040 Recitation: Stellar Structure

ASTR 1040 Recitation: Stellar Structure ASTR 1040 Recitation: Stellar Structure Ryan Orvedahl Department of Astrophysical and Planetary Sciences University of Colorado at Boulder Boulder, CO 80309 ryan.orvedahl@colorado.edu February 12, 2014

More information

The Virial Theorem for Stars

The Virial Theorem for Stars The Virial Theorem for Stars Stars are excellent examples of systems in virial equilibrium. To see this, let us make two assumptions: 1) Stars are in hydrostatic equilibrium 2) Stars are made up of ideal

More information

Astronomy 112: The Physics of Stars. Class 9 Notes: Polytropes

Astronomy 112: The Physics of Stars. Class 9 Notes: Polytropes Astronomy 112: The Physics of Stars Class 9 Notes: Polytropes With our discussion of nuclear reaction rates last time, we have mostly completed our survey of the microphysical properties of stellar matter

More information

Examination paper for FY2450 Astrophysics

Examination paper for FY2450 Astrophysics 1 Department of Physics Examination paper for FY2450 Astrophysics Academic contact during examination: Rob Hibbins Phone: 94820834 Examination date: 01-06-2015 Examination time: 09:00 13:00 Permitted examination

More information

Ay101 Set 1 solutions

Ay101 Set 1 solutions Ay11 Set 1 solutions Ge Chen Jan. 1 19 1. The scale of the Sun a 3 points Venus has an orbital period of 5 days. Using Kepler s laws, what is its semi-major axis in units of AU Start with Kepler s third

More information

Gasdynamical and radiative processes, gaseous halos

Gasdynamical and radiative processes, gaseous halos Gasdynamical and radiative processes, gaseous halos Houjun Mo March 19, 2004 Since luminous objects, such as galaxies, are believed to form through the collapse of baryonic gas, it is important to understand

More information

The Interiors of the Stars

The Interiors of the Stars The Interiors of the Stars Hydrostatic Equilibrium Stellar interiors, to a good first approximation, may be understood using basic physics. The fundamental operating assumption here is that the star is

More information

Lecture 5-2: Polytropes. Literature: MWW chapter 19

Lecture 5-2: Polytropes. Literature: MWW chapter 19 Lecture 5-2: Polytropes Literature: MWW chapter 9!" Preamble The 4 equatios of stellar structure divide ito two groups: Mass ad mometum describig the mechaical structure ad thermal equilibrium ad eergy

More information

Stellar Structure & Evolution

Stellar Structure & Evolution Stellar Structure & Evolution Sommersemester 2007 Ralf Klessen Zentrum für Astronomie der Universität Heidelberg Institut für Theoretische Astrophysik rklessen@ita.uni-heidelberg.de Albert-Überle-Straße

More information

Astro Instructors: Jim Cordes & Shami Chatterjee.

Astro Instructors: Jim Cordes & Shami Chatterjee. Astro 2299 The Search for Life in the Universe Lecture 8 Last time: Formation and function of stars This time (and probably next): The Sun, hydrogen fusion Virial theorem and internal temperatures of stars

More information

Basic concepts in stellar physics

Basic concepts in stellar physics Basic concepts in stellar physics Anthony Brown brown@strw.leidenuniv.nl 1.2.21 Abstract. These notes provide extra material on the basics of stellar physics. The aim is to provide an overview of the nature

More information

Stellar Interiors. Hydrostatic Equilibrium. PHY stellar-structures - J. Hedberg

Stellar Interiors. Hydrostatic Equilibrium. PHY stellar-structures - J. Hedberg Stellar Interiors. Hydrostatic Equilibrium 2. Mass continuity 3. Equation of State. The pressure integral 4. Stellar Energy Sources. Where does it come from? 5. Intro to Nuclear Reactions. Fission 2. Fusion

More information

read 9.4-end 9.8(HW#6), 9.9(HW#7), 9.11(HW#8) We are proceding to Chap 10 stellar old age

read 9.4-end 9.8(HW#6), 9.9(HW#7), 9.11(HW#8) We are proceding to Chap 10 stellar old age HW PREVIEW read 9.4-end Questions 9.9(HW#4), 9(HW#4) 9.14(HW#5), 9.8(HW#6), 9.9(HW#7), 9.11(HW#8) We are proceding to Chap 10 stellar old age Chap 11 The death of high h mass stars Contraction of Giant

More information

AST1100 Lecture Notes

AST1100 Lecture Notes AST1100 Lecture Notes 20: Stellar evolution: The giant stage 1 Energy transport in stars and the life time on the main sequence How long does the star remain on the main sequence? It will depend on the

More information

UNIVERSITY OF SOUTHAMPTON

UNIVERSITY OF SOUTHAMPTON UNIVERSITY OF SOUTHAMPTON PHYS3010W1 SEMESTER 2 EXAMINATION 2014-2015 STELLAR EVOLUTION: MODEL ANSWERS Duration: 120 MINS (2 hours) This paper contains 8 questions. Answer all questions in Section A and

More information

1.1 An introduction to the cgs units

1.1 An introduction to the cgs units Chapter 1 Introduction 1.1 An introduction to the cgs units The cgs units, instead of the MKS units, are commonly used by astrophysicists. We will therefore follow the convention to adopt the former throughout

More information

Topics ASTR 3730: Fall 2003

Topics ASTR 3730: Fall 2003 Topics Qualitative questions: might cover any of the main topics (since 2nd midterm: star formation, extrasolar planets, supernovae / neutron stars, black holes). Quantitative questions: worthwhile to

More information

General Relativity and Compact Objects Neutron Stars and Black Holes

General Relativity and Compact Objects Neutron Stars and Black Holes 1 General Relativity and Compact Objects Neutron Stars and Black Holes We confine attention to spherically symmetric configurations. The metric for the static case can generally be written ds 2 = e λ(r)

More information

Ordinary Differential Equations

Ordinary Differential Equations Ordinary Differential Equations Part 4 Massimo Ricotti ricotti@astro.umd.edu University of Maryland Ordinary Differential Equations p. 1/23 Two-point Boundary Value Problems NRiC 17. BCs specified at two

More information

THIRD-YEAR ASTROPHYSICS

THIRD-YEAR ASTROPHYSICS THIRD-YEAR ASTROPHYSICS Problem Set: Stellar Structure and Evolution (Dr Ph Podsiadlowski, Michaelmas Term 2006) 1 Measuring Stellar Parameters Sirius is a visual binary with a period of 4994 yr Its measured

More information

11.1 Local Thermodynamic Equilibrium. 1. the electron and ion velocity distributions are Maxwellian,

11.1 Local Thermodynamic Equilibrium. 1. the electron and ion velocity distributions are Maxwellian, Section 11 LTE Basic treatments of stellar atmospheres adopt, as a starting point, the assumptions of Local Thermodynamic Equilibrium (LTE), and hydrostatic equilibrium. The former deals with the microscopic

More information

2. Basic Assumptions for Stellar Atmospheres

2. Basic Assumptions for Stellar Atmospheres 2. Basic Assumptions for Stellar Atmospheres 1. geometry, stationarity 2. conservation of momentum, mass 3. conservation of energy 4. Local Thermodynamic Equilibrium 1 1. Geometry Stars as gaseous spheres!

More information

P M 2 R 4. (3) To determine the luminosity, we now turn to the radiative diffusion equation,

P M 2 R 4. (3) To determine the luminosity, we now turn to the radiative diffusion equation, Astronomy 715 Final Exam Solutions Question 1 (i). The equation of hydrostatic equilibrium is dp dr GM r r 2 ρ. (1) This corresponds to the scaling P M R ρ, (2) R2 where P and rho represent the central

More information

Stellar structure Conservation of mass Conservation of energy Equation of hydrostatic equilibrium Equation of energy transport Equation of state

Stellar structure Conservation of mass Conservation of energy Equation of hydrostatic equilibrium Equation of energy transport Equation of state Stellar structure For an isolated, static, spherically symmetric star, four basic laws / equations needed to describe structure: Conservation of mass Conservation of energy (at each radius, the change

More information

Stellar Winds: Mechanisms and Dynamics

Stellar Winds: Mechanisms and Dynamics Astrofysikalisk dynamik, VT 010 Stellar Winds: Mechanisms and Dynamics Lecture Notes Susanne Höfner Department of Physics and Astronomy Uppsala University 1 Most stars have a stellar wind, i.e. and outflow

More information

AST111 PROBLEM SET 6 SOLUTIONS

AST111 PROBLEM SET 6 SOLUTIONS AST111 PROBLEM SET 6 SOLUTIONS Homework problems 1. Ideal gases in pressure balance Consider a parcel of molecular hydrogen at a temperature of 100 K in proximity to a parcel of ionized hydrogen at a temperature

More information

2. Basic assumptions for stellar atmospheres

2. Basic assumptions for stellar atmospheres . Basic assumptions for stellar atmospheres 1. geometry, stationarity. conservation of momentum, mass 3. conservation of energy 4. Local Thermodynamic Equilibrium 1 1. Geometry Stars as gaseous spheres

More information

Overview of Astronomical Concepts III. Stellar Atmospheres; Spectroscopy. PHY 688, Lecture 5 Stanimir Metchev

Overview of Astronomical Concepts III. Stellar Atmospheres; Spectroscopy. PHY 688, Lecture 5 Stanimir Metchev Overview of Astronomical Concepts III. Stellar Atmospheres; Spectroscopy PHY 688, Lecture 5 Stanimir Metchev Outline Review of previous lecture Stellar atmospheres spectral lines line profiles; broadening

More information

dt 2 = 0, we find that: K = 1 2 Ω (2)

dt 2 = 0, we find that: K = 1 2 Ω (2) 1 1. irial Theorem Last semester, we derived the irial theorem from essentially considering a series of particles which attract each other through gravitation. The result was that d = K + Ω (1) dt where

More information

The structure and evolution of stars. Introduction and recap

The structure and evolution of stars. Introduction and recap The structure and evolution of stars Lecture 3: The equations of stellar structure 1 Introduction and recap For our stars which are isolated, static, and spherically symmetric there are four basic equations

More information

Lecture 7: Stellar evolution I: Low-mass stars

Lecture 7: Stellar evolution I: Low-mass stars Lecture 7: Stellar evolution I: Low-mass stars Senior Astrophysics 2018-03-21 Senior Astrophysics Lecture 7: Stellar evolution I: Low-mass stars 2018-03-21 1 / 37 Outline 1 Scaling relations 2 Stellar

More information

ASTR3007/4007/6007, Class 2: Stellar Masses; the Virial Theorem

ASTR3007/4007/6007, Class 2: Stellar Masses; the Virial Theorem ASTR7/47/67, Class : Stellar Masses; the Virial Theorem 4 February In the first class we discussed stars light output and the things than can be derived directly from it luminosity, temperature, and composition.

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Quiz 2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Quiz 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.282 April 16, 2003 Quiz 2 Name SOLUTIONS (please print) Last First 1. Work any 7 of the 10 problems - indicate clearly which 7 you want

More information

How the mass relationship with a White Dwarf Star

How the mass relationship with a White Dwarf Star White dwarf star Page 1 Daiwei Li Dimitrios Psaltis PHYS 305 November 30, 2018 How the mass relationship with a White Dwarf Star Introduction. A white dwarf star (also known as a degenerate dwarf star)

More information

Convection. If luminosity is transported by radiation, then it must obey

Convection. If luminosity is transported by radiation, then it must obey Convection If luminosity is transported by radiation, then it must obey L r = 16πacr 2 T 3 3ρκ R In a steady state, the energy transported per time at radius r must be equal to the energy generation rate

More information

ρ dr = GM r r 2 Let s assume that µ(r) is known, either as an arbitrary input state or on the basis of pre-computed evolutionary history.

ρ dr = GM r r 2 Let s assume that µ(r) is known, either as an arbitrary input state or on the basis of pre-computed evolutionary history. Stellar Interiors Modeling This is where we take everything we ve covered so far and fold it into models of spherically symmetric, time-steady stellar structure. (We ll emphasize insight over rigorous

More information

Convection When the radial flux of energy is carried by radiation, we derived an expression for the temperature gradient: dt dr = - 3

Convection When the radial flux of energy is carried by radiation, we derived an expression for the temperature gradient: dt dr = - 3 Convection When the radial flux of energy is carried by radiation, we derived an expression for the temperature gradient: dt dr = - 3 4ac kr L T 3 4pr 2 Large luminosity and / or a large opacity k implies

More information

AST1100 Lecture Notes

AST1100 Lecture Notes AST1100 Lecture Notes 20: Stellar evolution: The giant stage 1 Energy transport in stars and the life time on the main sequence How long does the star remain on the main sequence? It will depend on the

More information

Nuclear Reactions and Solar Neutrinos ASTR 2110 Sarazin. Davis Solar Neutrino Experiment

Nuclear Reactions and Solar Neutrinos ASTR 2110 Sarazin. Davis Solar Neutrino Experiment Nuclear Reactions and Solar Neutrinos ASTR 2110 Sarazin Davis Solar Neutrino Experiment Hydrogen Burning Want 4H = 4p è 4 He = (2p,2n) p è n, only by weak interaction Much slower than pure fusion protons

More information

VII. Hydrodynamic theory of stellar winds

VII. Hydrodynamic theory of stellar winds VII. Hydrodynamic theory of stellar winds observations winds exist everywhere in the HRD hydrodynamic theory needed to describe stellar atmospheres with winds Unified Model Atmospheres: - based on the

More information

Mass-Radius Relation: Hydrogen Burning Stars

Mass-Radius Relation: Hydrogen Burning Stars Mass-Radius Relation: Hydrogen Burning Stars Alexis Vizzerra, Samantha Andrews, and Sean Cunningham University of Arizona, Tucson AZ 85721, USA Abstract. The purpose if this work is to show the mass-radius

More information

Astro 210 Lecture 5 Sept. 6, 2013

Astro 210 Lecture 5 Sept. 6, 2013 Astro 210 Lecture 5 Sept. 6, 2013 Announcements: PS 1 due now PS 2 available, due next Friday Sept 13 1 Last time: star distances Q: what s the gold plated way to measure distance? update: 2013 www: discovery

More information

2. Basic assumptions for stellar atmospheres

2. Basic assumptions for stellar atmospheres . Basic assumptions for stellar atmospheres 1. geometry, stationarity. conservation of momentum, mass 3. conservation of energy 4. Local Thermodynamic Equilibrium 1 1. Geometry Stars as gaseous spheres

More information

Part II. Cosmology. Year

Part II. Cosmology. Year Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 2, Section I 9B 26 (a) Consider a homogeneous and isotropic universe with a uniform distribution of galaxies.

More information

AST1100 Lecture Notes

AST1100 Lecture Notes AST1100 Lecture Notes 5 The virial theorem 1 The virial theorem We have seen that we can solve the equation of motion for the two-body problem analytically and thus obtain expressions describing the future

More information

Stellar Interiors - Hydrostatic Equilibrium and Ignition on the Main Sequence.

Stellar Interiors - Hydrostatic Equilibrium and Ignition on the Main Sequence. Stellar Interiors - Hydrostatic Equilibrium and Ignition on the Main Sequence http://apod.nasa.gov/apod/astropix.html Outline of today s lecture Hydrostatic equilibrium: balancing gravity and pressure

More information

Stellar Models ASTR 2110 Sarazin

Stellar Models ASTR 2110 Sarazin Stellar Models ASTR 2110 Sarazin Jansky Lecture Tuesday, October 24 7 pm Room 101, Nau Hall Bernie Fanaroff Observing the Universe From Africa Trip to Conference Away on conference in the Netherlands

More information

Chapter 7 Neutron Stars

Chapter 7 Neutron Stars Chapter 7 Neutron Stars 7.1 White dwarfs We consider an old star, below the mass necessary for a supernova, that exhausts its fuel and begins to cool and contract. At a sufficiently low temperature the

More information

Stellar Dynamics and Structure of Galaxies

Stellar Dynamics and Structure of Galaxies Stellar Dynamics and Structure of Galaxies in a given potential Vasily Belokurov vasily@ast.cam.ac.uk Institute of Astronomy Lent Term 2016 1 / 59 1 Collisions Model requirements 2 in spherical 3 4 Orbital

More information

1.1 Motivation. 1.2 The H-R diagram

1.1 Motivation. 1.2 The H-R diagram 1.1 Motivation Observational: How do we explain stellar properties as demonstrated, e.g. by the H-R diagram? Theoretical: How does an isolated, self-gravitating body of gas behave? Aims: Identify and understand

More information

Static Spherically-Symmetric Stellar Structure in General Relativity

Static Spherically-Symmetric Stellar Structure in General Relativity Static Spherically-Symmetric Stellar Structure in General Relativity Christian D. Ott TAPIR, California Institute of Technology cott@tapir.caltech.edu July 24, 2013 1 Introduction Neutron stars and, to

More information

Chapter 16. White Dwarfs

Chapter 16. White Dwarfs Chapter 16 White Dwarfs The end product of stellar evolution depends on the mass of the initial configuration. Observational data and theoretical calculations indicate that stars with mass M < 4M after

More information

Convection and Other Instabilities. Prialnik 6 Glatzmair and Krumholz 11 Pols 3, 10

Convection and Other Instabilities. Prialnik 6 Glatzmair and Krumholz 11 Pols 3, 10 Convection and Other Instabilities Prialnik 6 Glatzmair and Krumholz 11 Pols 3, 10 What is convection? A highly efficient way of transporting energy in the stellar interior Highly idealized schematic.

More information

Physics 160: Stellar Astrophysics. Midterm Exam. 27 October 2011 INSTRUCTIONS READ ME!

Physics 160: Stellar Astrophysics. Midterm Exam. 27 October 2011 INSTRUCTIONS READ ME! Physics 160: Stellar Astrophysics 27 October 2011 Name: S O L U T I O N S Student ID #: INSTRUCTIONS READ ME! 1. There are 4 questions on the exam; complete at least 3 of them. 2. You have 80 minutes to

More information

The Chaplygin dark star

The Chaplygin dark star The Chaplygin dark star O. Bertolami, J. Páramos Instituto Superior Técnico, Departamento de Física, Av. Rovisco Pais 1, 149-1 Lisboa, Portugal and E-mail addresses: orfeu@cosmos.ist.utl.pt; x jorge@fisica.ist.utl.pt

More information

Physics 161 Homework 7 - Solutions Wednesday November 16, 2011

Physics 161 Homework 7 - Solutions Wednesday November 16, 2011 Physics 161 Homework 7 - s Wednesday November 16, 2011 Make sure your name is on every page, and please box your final answer Because we will be giving partial credit, be sure to attempt all the problems,

More information

dp dr = GM c = κl 4πcr 2

dp dr = GM c = κl 4πcr 2 RED GIANTS There is a large variety of stellar models which have a distinct core envelope structure. While any main sequence star, or any white dwarf, may be well approximated with a single polytropic

More information

PHAS3135 The Physics of Stars

PHAS3135 The Physics of Stars PHAS3135 The Physics of Stars Exam 2013 (Zane/Howarth) Answer ALL SIX questions from Section A, and ANY TWO questions from Section B The numbers in square brackets in the right-hand margin indicate the

More information

2. Basic assumptions for stellar atmospheres

2. Basic assumptions for stellar atmospheres . Basic assumptions for stellar atmospheres 1. geometry, stationarity. conservation of momentum, mass 3. conservation of energy 4. Local Thermodynamic Equilibrium 1 1. Geometry Stars as gaseous spheres

More information

11/19/08. Gravitational equilibrium: The outward push of pressure balances the inward pull of gravity. Weight of upper layers compresses lower layers

11/19/08. Gravitational equilibrium: The outward push of pressure balances the inward pull of gravity. Weight of upper layers compresses lower layers Gravitational equilibrium: The outward push of pressure balances the inward pull of gravity Weight of upper layers compresses lower layers Gravitational equilibrium: Energy provided by fusion maintains

More information

Stellar structure and evolution. Pierre Hily-Blant April 25, IPAG

Stellar structure and evolution. Pierre Hily-Blant April 25, IPAG Stellar structure and evolution Pierre Hily-Blant 2017-18 April 25, 2018 IPAG pierre.hily-blant@univ-grenoble-alpes.fr, OSUG-D/306 10 Protostars and Pre-Main-Sequence Stars 10.1. Introduction 10 Protostars

More information

Stars Lab 2: Polytropic Stellar Models

Stars Lab 2: Polytropic Stellar Models Monash University Stars Lab 2: Polytropic Stellar Models School of Mathematical Sciences ASP3012 2009 1 Introduction In the last lab you learned how to use numerical methods to solve various ordinary differential

More information

Substellar Interiors. PHY 688, Lecture 13

Substellar Interiors. PHY 688, Lecture 13 Substellar Interiors PHY 688, Lecture 13 Outline Review of previous lecture curve of growth: dependence of absorption line strength on abundance metallicity; subdwarfs Substellar interiors equation of

More information

Atomic Physics 3 ASTR 2110 Sarazin

Atomic Physics 3 ASTR 2110 Sarazin Atomic Physics 3 ASTR 2110 Sarazin Homework #5 Due Wednesday, October 4 due to fall break Test #1 Monday, October 9, 11-11:50 am Ruffner G006 (classroom) You may not consult the text, your notes, or any

More information

A Guide to the Next Few Lectures!

A Guide to the Next Few Lectures! Dynamics and how to use the orbits of stars to do interesting things chapter 3 of S+G- parts of Ch 11 of MWB (Mo, van den Bosch, White) READ S&G Ch 3 sec 3.1, 3.2, 3.4 we are skipping over epicycles 1

More information

Lecture 1. Overview Time Scales, Temperature-density Scalings, Critical Masses

Lecture 1. Overview Time Scales, Temperature-density Scalings, Critical Masses Lecture 1 Overview Time Scales, Temperature-density Scalings, Critical Masses I. Preliminaries The life of any star is a continual struggle between the force of gravity, seeking to reduce the star to a

More information

Lecture 1. Overview Time Scales, Temperature-density Scalings, Critical Masses. I. Preliminaries

Lecture 1. Overview Time Scales, Temperature-density Scalings, Critical Masses. I. Preliminaries I. Preliminaries Lecture 1 Overview Time Scales, Temperature-density Scalings, Critical Masses The life of any star is a continual struggle between the force of gravity, seeking to reduce the star to a

More information

Low Mass Star Forma-on: The T Tauri Stars

Low Mass Star Forma-on: The T Tauri Stars Low Mass Star Forma-on: The T Tauri Stars Cloud Collapse Thermally- supported non- rota-ng cloud Inside- out collapse R=c s t (c s : sound speed) m acc =m dot t Gm acc m H /R = m H v 2 (by VT); v=c 2 s

More information

Atmosphere, Ocean and Climate Dynamics Answers to Chapter 3

Atmosphere, Ocean and Climate Dynamics Answers to Chapter 3 Atmosphere, Ocean and Climate Dynamics Answers to Chapter 3 1. Use the hydrostatic equation to show that the mass of a vertical column of air of unit cross-section, extendin from the round to reat heiht,

More information

Stellar Interiors ASTR 2110 Sarazin. Interior of Sun

Stellar Interiors ASTR 2110 Sarazin. Interior of Sun Stellar Interiors ASTR 2110 Sarazin Interior of Sun Test #1 Monday, October 9, 11-11:50 am Ruffner G006 (classroom) You may not consult the text, your notes, or any other materials or any person Bring

More information

Neutron Star) Lecture 22

Neutron Star) Lecture 22 Neutron Star) Lecture 22 1 Neutron star A neutron star is a stellar object held together by gravity but kept from collapsing by electromagnetic (atomic) and strong (nuclear including Pauli exclusion) forces.

More information

AY202a Galaxies & Dynamics Lecture 7: Jeans Law, Virial Theorem Structure of E Galaxies

AY202a Galaxies & Dynamics Lecture 7: Jeans Law, Virial Theorem Structure of E Galaxies AY202a Galaxies & Dynamics Lecture 7: Jeans Law, Virial Theorem Structure of E Galaxies Jean s Law Star/Galaxy Formation is most simply defined as the process of going from hydrostatic equilibrium to gravitational

More information