Ay123 Set 1 solutions

Size: px
Start display at page:

Download "Ay123 Set 1 solutions"

Transcription

1 Ay13 Set 1 solutions Mia de los Reyes October 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 temperature of the Sun. First, relate the angular radius of the Sun θ to the physical radius R: R a sin θ, 1 where a is the distance from the Earth to the Sun. Note that you can also use the small-angle approximation sin θ θ. You ll get basically the same answer. Then, from conservation of energy i.e., inverse square law: L πa f And from the Stefan-Boltzmann equation: L πr σ B T eff 3 Set equations and 3 equal, plug in equation 1 for R and solve for T eff : Plugging in given values yields T eff 5775K. πa f πr σ B T eff a f a sin θ σ B Teff 5 1/ f T eff σ B sin 6 θ b Using Venus orbital period of 5 days and Kepler s laws, what is the semi-major axis of Venus in AU? Start with Kepler s third law: Ω GM + M a 3 7 First, consider the Earth-Sun system, assuming that M E M : Ω E GM a 3 E 8 Now, Ω E can be calculated from known values Earth s period is 1 year and a E is known it s 1 AU, so can solve for M : M Ω E a3 E G Then consider the Venus-Sun system, again assuming that M V M : 9 Ω V GM a 3 V 1 1

2 Then solve for a V, substituting equation 9 for M and converting angular velocities to periods P π Ω : a V 1/3 GM 11 Ω V Ω E a 3 1/3 E 1 PV Ω V P E /3 a E 13 Plug in known values P E days, a E 1 AU and given values P V 5 days, and find that a V.7 AU. c At conjunction with the Sun, it takes astronomers on Earth 76s to detect the radio waves that reflect off Venus. Assuming circular orbits for the Earth and Venus, compute the distance between in 1 AU. Note: conjunction Venus is directly between Earth and the Sun. Call the distance between Earth and Venus d V. The light takes time t to travel distance d V : From the previous problem, we can compute d V d V ct.1 11 cm 1 in terms of AU: d V a E a V 1.7 AU.76 AU 15 Then combine equations 1 and 15 to convert AU to cm: 1 AU cm. d Use your results above to compute the absolute mass, radius, and luminosity of the Sun in cgs. Inverse square law: L πa E f Plug in a E 1 AU cm and f from problem 1a. Find that L erg/s. Kepler s third law: M Ω E a3 E G Plug in known values in cgs; find that M g. Definition of angular size: R a E sin θ. Stars are gases Plug in known values in cgs; find that R cm. a Provide a quantitative relation between temperature and density of a star that indicates when we can treat it as a gas, and show that it holds at the center of the Sun. To check if the center of the Sun can be treated as a gas, we can compare Coulomb energy to thermal energy. The ideal gas law is reasonable when the thermal energy T is larger than the Coulomb energy Ze /r. This occurs when kt Ze /r 16 r Ze /kt 17 where r is the interparticle distance and Z is the charge of the ion Z 1 for a gas composed only of ionized hydrogen. By thinking of the number density n as r 3, we can write r in terms of mass density: r n 1/3 ρ/ 1/3. Then we can rewrite equation 17: ρ k/e 3 T 3 18

3 For atomic hydrogen, µ.5. Plugging this in, our condition for treating a star as an ideal gas is in cgs units: ρ T 3 19 The sun s central temperature is T c 1 7 K. By equation 19, we require ρ c g cm 3. Since the sun s central density ρ c is only 15 g cm 3, we may treat the sun as an ideal gas throughout, and need not consider Coulomb interactions between particles. b Use a scaling argument to determine the stellar mass at which the ideal gas assumption breaks down. Now we want to know how ρ scales with stellar mass M. In the following derivation we only care about approximate scaling arguments, so don t worry about prefactors. The internal energy can be approximated as U T M at the central temperature T. Now recall that by the virial theorem, the internal energy is approximately the gravitational energy Ω GM /R. Solve for T and find that T M/R. We can then assume that all stars have roughly the same central temperature which is actually a good approximation for main-sequence stars, so the central density M R. Then ρ M/R 3 M, so M ρ 1/. 1/ Plugging in values for the Sun, we find M lim ρlim M. This yields a limiting mass of M.16M, which is about the mass of brown dwarfs or giant planets not stars!. Therefore, we will never have to consider Coulomb interactions for main sequence stars. 3. A toy star Assume that a star obeys the density model ρr ρ c 1 r. R a Find an expression for the central density in terms of R and M. Solve for total mass M by integrating over the density profile: M π π dφ π πρ c R 3 ρ sin θdθ ρ c r r3 R 3 R3 r ρrdr 1 dr 3 π 3 ρ cr 3 Then solve for central density: ρ c 3M πr 3. b Find the pressure as a function of radius. Doing the same integral as in the previous problem, we know that mr π 3 ρ cr 3 1 3r R 5 Now use the equation of hydrostatic equilibrium: dp dr Gm ρr 6 r G π 3 ρ cr 1 3r ρ c 1 r 7 R R π 3 Gρ cr 1 7r R + 3r R 8 3

4 Integrate equation 8 to get the total pressure: P r π 3 Gρ c π [ r 3 Gρ c r 7r R + 3r3 R dr 9 7r3 1R + 3r 16R + C Use the zero boundary condition P R to solve for the integration constant C: P R π [ 1 3 Gρ cr C ] R 31 C R R 33 Now let s solve for the central pressure P c P r : ] 3 P c π 3 Gρ cc 3 5π 36 Gρ cr 35 Okay, finally we can substitute stuff into equation 3 to write the full equation for pressure: P r π [ r 3 Gρ c 7r3 1R + 3r 16R 5 ] 8 R 36 [ P c 1 r 8 r 3 9 r ] R 5 R 5 R So we find that P r P c f r R as expected. Now plug in the answer for part a to get P c as a function of M and R: We can simplify this to P c 5 GM π R. P c 5π 3M 36 G πr 3 R 38 c What is the central temperature T c, assuming an ideal gas equation of state? How does it scale with mean particle mass? Ideal gas: P ρt Solve this for central temperature, plugging in answer from b for P c and answer from a for ρ c : T c P c ρ c 39 5π 36 Gρ cr 5π 36 G 3M πr R µm 3 p 1 This simplifies to T c 5GM 1R which scales as T c. is the mean particle mass.

5 d Find the ratio of radiation pressure to gas pressure at the center of the star as a function of total stellar mass in M. At what mass does radiation pressure become comparable to the ideal gas pressure? Radiation pressure is given by 1 3 a ot. The ratio at the center of the star is therefore Then plug in T c from part c: 1 P gas 3 a Tc o 5 GM π R P gas 1 3 a o 5GM 1R 5 GM π 15π 1555 a og 3 M 3 R µmp Assuming solar composition µ.6, we can rewrite this in terms of solar masses as: P gas 7. 1 M M When P gas 1, the mass of the star is M 37M. Note that this is not an exact result, since our formula for T c assumes that radiation pressure is negligible. e Write the total gravitational potential energy of this toy star and verify the virial theorem. The total gravitational potential is Ω Gmr r dm. For simplicity s sake, let s convert this to an integral over radius so we can plug in the definition of mr from part b and the definition of ρr: Ω G 16π 3 ρ c G 16π 3 ρ c Gmrπrρdr 5 G π 3 ρ cr 3 1 3r R [ R 5 r 7r5 R + 3r6 R ] 5 7R6 R + 3R7 8R πrρ c 1 r dr 6 R dr 7 The gravitational potential is G 6π 315 ρ cr 5 Now let s do math with the other side of the virial theorem, plugging in P r from equation 3: 3 P dv 3 1π π dφ 8π 3 Gρ c 8π 3 Gρ c π r π 3 Gρ c r [ R 5 sin θdθ r 8 r P rdr 9 7r3 1R + 3r 16R 5 8 R 7r5 1R + 3r6 16R 5r 1 7R R5 11 5R5 1 ] 8 R dr 5 dr π 315 Gρ cr 5 53 So 3 P dv Ω, and the virial theorem is exactly satisfied. 5

6 . Stellar coronae a Solve hydrostatic equilibrium for the density profile as a function of radius given a density ρ b at the base radius r b R. Start with hydrostatic equilibrium and assume an ideal gas equation of state with constant temperature T. Also assume that the mass of the corona is negligible compared to the mass of the star M, so mr M: dp dr T dρ dr dρ dr Gmr r ρr 5 GM ρr 55 r GM r ρr 56 T Equation 56 is a separable differential equation: dρ GM dr ρ T r 57 GMµmp ρr Aexp 58 r T Now use the given boundary condition to solve for integration constant A: GMµmp ρ b Aexp r b T A ρ b exp GMµm p r b T 59 6 So the final equation is ρr ρ b exp GMµm p GMµmp exp r b T r T 61 b What is the pressure in the corona as r? Comment on the implications of a low-density, low-pressure ISM surrounding the star. As r, we find the limit ρr ρ b exp GMµmp r b T. Since we ve assumed an ideal gas P ρt, this yields a pressure P ρ b T exp GMµmp r b T. This is finite! Since the stellar corona is surrounded by an ISM with much lower pressure, we expect the corona to be able to produce a stellar wind. 5. Using the MESA stellar evolution code a Run the default stellar model located in mesa/star/work/. At time step 1, what is the core temperature and surface temperature of the model? The core temperature is log T c 5.8 [K], or T c K. The surface temperature is T eff 35 K. b Evolve a 1M model up the red giant branch. Make an HR diagram and plot log L as a function of time. Recall that an H-R diagram should have temperature increasing from right to left. Also note that in astronomy, log means log 1 and not ln. Finally, make sure to include units! 6

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

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

Observed Properties of Stars - 2 ASTR 2110 Sarazin

Observed Properties of Stars - 2 ASTR 2110 Sarazin Observed Properties of Stars - 2 ASTR 2110 Sarazin Properties Location Distance Speed Radial velocity Proper motion Luminosity, Flux Magnitudes Magnitudes Stellar Colors Stellar Colors Stellar Colors Stars

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

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

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

Astronomy 7A Final Exam December 14, 2012

Astronomy 7A Final Exam December 14, 2012 Astronomy 7A Final Exam December 14, 2012 Name: Section: There are 6 problems of varying length that are required, and 1 bonus problem that is optional. Write your answers on these sheets showing all of

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

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

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

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

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

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department. Final Exam

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department. Final Exam MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department Physics 8.282J EAPS 12.402J May 20, 2005 Final Exam Name Last First (please print) 1. Do any

More information

Stars + Galaxies: Back of the Envelope Properties. David Spergel

Stars + Galaxies: Back of the Envelope Properties. David Spergel Stars + Galaxies: Back of the Envelope Properties David Spergel Free-fall time (1) r = GM r 2 (2) r t = GM 2 r 2 (3) t free fall r3 GM 1 Gρ Free-fall time for neutron star is milliseconds (characteristic

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

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

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department. Quiz 2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department. Quiz 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department Astronomy 8.282J 12.402J April 13, 2005 Quiz 2 Name Last First (please print) 1. Work any

More information

Static Stellar Structure

Static Stellar Structure 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

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

Ay123 Set 3 solutions

Ay123 Set 3 solutions Ay123 Set 3 solutions Mia de los Reyes October 23 218 1. Equation of state and the Chandrasekhar mass (a) Using the Fermi-Dirac distribution for non-relativistic electrons, derive the relationship between

More information

Ay 20: Basic Astronomy and the Galaxy Fall Term Solution Set 4 Kunal Mooley (based on solutions by Swarnima Manohar, TA 2009)

Ay 20: Basic Astronomy and the Galaxy Fall Term Solution Set 4 Kunal Mooley (based on solutions by Swarnima Manohar, TA 2009) Ay 20: Basic Astronomy and the Galaxy Fall Term 2010 Solution Set 4 Kunal Mooley (based on solutions by Swarnima Manohar, TA 2009) Reporting an answer to unnecesary number of decimal places should be avoided.

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

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

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

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

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

Revision: Sun, Stars (and Planets) See web slides of Dr Clements for Planets revision. Juliet Pickering Office: Huxley 706

Revision: Sun, Stars (and Planets) See web slides of Dr Clements for Planets revision. Juliet Pickering Office: Huxley 706 Revision: Sun, Stars (and Planets) See web slides of Dr Clements for Planets revision Juliet Pickering Office: Huxley 706 Office hour (Pickering): Thursday 22nd May 12-11 pm Outline overview of first part

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: Robert Hibbins Phone: 94 82 08 34 Examination date: 04-06-2013 Examination time: 09:00 13:00 Permitted

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: 31-05-2014 Examination time: 09:00 13:00 Permitted examination

More information

Preliminary Examination: Astronomy

Preliminary Examination: Astronomy Preliminary Examination: Astronomy Department of Physics and Astronomy University of New Mexico Spring 2017 Instructions: Answer 8 of the 10 questions (10 points each) Total time for the test is three

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

Ay 20 Basic Astronomy and the Galaxy Problem Set 2

Ay 20 Basic Astronomy and the Galaxy Problem Set 2 Ay 20 Basic Astronomy and the Galaxy Problem Set 2 October 19, 2008 1 Angular resolutions of radio and other telescopes Angular resolution for a circular aperture is given by the formula, θ min = 1.22λ

More information

B ν (T) = 2hν3 c 3 1. e hν/kt 1. (4) For the solar radiation λ = 20µm photons are in the Rayleigh-Jean region, e hν/kt 1+hν/kT.

B ν (T) = 2hν3 c 3 1. e hν/kt 1. (4) For the solar radiation λ = 20µm photons are in the Rayleigh-Jean region, e hν/kt 1+hν/kT. Name: Astronomy 18 - Problem Set 8 1. Fundamental Planetary Science problem 14.4 a) Calculate the ratio of the light reflected by Earth at 0.5 µm to that emitted by the Sun at the same wavelength. 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

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

Observed Properties of Stars - 2 ASTR 2120 Sarazin

Observed Properties of Stars - 2 ASTR 2120 Sarazin Observed Properties of Stars - 2 ASTR 2120 Sarazin Properties Location Distance Speed Radial velocity Proper motion Luminosity, Flux Magnitudes Magnitudes Hipparchus 1) Classified stars by brightness,

More information

Today in Astronomy 328

Today in Astronomy 328 The Sun as a typical star: Central density, temperature, pressure The spectrum of the surface (atmosphere) of the Sun The structure of the sun s outer layers: convection, rotation, magnetism and sunspots

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

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

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

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

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

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

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

Energy transport: convection

Energy transport: convection Outline Introduction: Modern astronomy and the power of quantitative spectroscopy Basic assumptions for classic stellar atmospheres: geometry, hydrostatic equilibrium, conservation of momentum-mass-energy,

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

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

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

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

AST Homework V - Solutions

AST Homework V - Solutions AST 341 - Homework V - Solutions TA: Marina von Steinkirch, steinkirch@gmail.com State University of New York at Stony Brook November, 010 1 (1 point) Derive the homologous form of the luminosity equation

More information

Astro Week 1. (a) Show that the transit duration for a non-central transit (see Figures) is: R R. b = a cos i

Astro Week 1. (a) Show that the transit duration for a non-central transit (see Figures) is: R R. b = a cos i Astro-286 - Week 1 1. Radial Velocity (10 pt) What is the expected amplitude of velocity oscillations of 1 M star that is orbited by a Jupiter mass planet (m J = 0.001 M ) at 1 AU separation? What is the

More information

Molecular Weight & Energy Transport

Molecular Weight & Energy Transport Molecular Weight & Energy Transport 7 September 20 Goals Review mean molecular weight Practice working with diffusion Mean Molecular Weight. We will frequently use µ,, and (the mean molecular weight per

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 Spectra ASTR 2120 Sarazin. Solar Spectrum

Stellar Spectra ASTR 2120 Sarazin. Solar Spectrum Stellar Spectra ASTR 2120 Sarazin Solar Spectrum Solar Prominence Sep. 14, 1999 Solar Activity Due to rotation, convection, and magnetic field (Section 7.2 review) Charged Particles in Magnetic Fields

More information

Review from last class:

Review from last class: Review from last class: Properties of photons Flux and luminosity, apparent magnitude and absolute magnitude, colors Spectroscopic observations. Doppler s effect and applications Distance measurements

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

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

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

! p. 1. Observations. 1.1 Parameters

! p. 1. Observations. 1.1 Parameters 1 Observations 11 Parameters - Distance d : measured by triangulation (parallax method), or the amount that the star has dimmed (if it s the same type of star as the Sun ) - Brightness or flux f : energy

More information

AST1100 Lecture Notes

AST1100 Lecture Notes AST1100 Lecture Notes 4 Stellar orbits and dark matter 1 Using Kepler s laws for stars orbiting the center of a galaxy We will now use Kepler s laws of gravitation on much larger scales. We will study

More information

The Hertzprung-Russell Diagram. The Hertzprung-Russell Diagram. Question

The Hertzprung-Russell Diagram. The Hertzprung-Russell Diagram. Question Key Concepts: Lecture 21: Measuring the properties of stars (cont.) The Hertzsprung-Russell (HR) Diagram (L versus T) The Hertzprung-Russell Diagram The Stefan-Boltzmann Law: flux emitted by a black body

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

From Last Time: We can more generally write the number densities of H, He and metals.

From Last Time: We can more generally write the number densities of H, He and metals. From Last Time: We can more generally write the number densities of H, He and metals. n H = Xρ m H,n He = Y ρ 4m H, n A = Z Aρ Am H, How many particles results from the complete ionization of hydrogen?

More information

Based on the reduction of the intensity of the light from a star with distance. It drops off with the inverse square of the distance.

Based on the reduction of the intensity of the light from a star with distance. It drops off with the inverse square of the distance. 6/28 Based on the reduction of the intensity of the light from a star with distance. It drops off with the inverse square of the distance. Intensity is power per unit area of electromagnetic radiation.

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

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

Astronomy 111 Midterm #1

Astronomy 111 Midterm #1 Astronomy 111 Midterm #1 Prof. Douglass 11 October 2018 Name: You may consult only one page of formulas and constants and a calculator while taking this test. You may not consult any books, digital resources,

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

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

Electromagnetic Spectra. AST443, Lecture 13 Stanimir Metchev

Electromagnetic Spectra. AST443, Lecture 13 Stanimir Metchev Electromagnetic Spectra AST443, Lecture 13 Stanimir Metchev Administrative Homework 2: problem 5.4 extension: until Mon, Nov 2 Reading: Bradt, chapter 11 Howell, chapter 6 Tenagra data: see bottom of Assignments

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Friday, January 17, 2014 1:00PM to 3:00PM General Physics (Part I) Section 5. Two hours are permitted for the completion of this section

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

Midterm Study Guide Astronomy 122

Midterm Study Guide Astronomy 122 Midterm Study Guide Astronomy 122 Introduction: 1. How is modern Astronomy different from Astrology? 2. What is the speed of light? Is it constant or changing? 3. What is an AU? Light-year? Parsec? Which

More information

Atmospheric Thermodynamics

Atmospheric Thermodynamics Atmospheric Thermodynamics R. Wordsworth February 12, 2015 1 Objectives Derive hydrostatic equation Derive dry and moist adiabats Understand how the theory relates to observed properties of real atmospheres

More information

Binary Stars (continued) ASTR 2120 Sarazin. γ Caeli - Binary Star System

Binary Stars (continued) ASTR 2120 Sarazin. γ Caeli - Binary Star System Binary Stars (continued) ASTR 2120 Sarazin γ Caeli - Binary Star System Visual Binaries: Types of Binary Stars Spectroscopic Binaries: Eclipsing Binaries: Periodic changes in brightness, stars block one

More information

Astronomy 111 Review Problems Solutions

Astronomy 111 Review Problems Solutions Astronomy 111 Review Problems Solutions Problem 1: Venus has an equatorial radius of 6052 km. Its semi-major axis is 0.72 AU. The Sun has a radius of cm. a) During a Venus transit (such as occurred June

More information

Bremsstrahlung. Rybicki & Lightman Chapter 5. Free-free Emission Braking Radiation

Bremsstrahlung. Rybicki & Lightman Chapter 5. Free-free Emission Braking Radiation Bremsstrahlung Rybicki & Lightman Chapter 5 Bremsstrahlung Free-free Emission Braking Radiation Radiation due to acceleration of charged particle by the Coulomb field of another charge. Relevant for (i)

More information

Stellar Scaling Relations and basic observations

Stellar Scaling Relations and basic observations Stellar Scaling Relations and basic observations (roughly sections 3.2-3.5 in Choudhuri s book) Astrophysics-I, HS2017 week 3, Oct. 17 & 18, 2017 Benny Trakhtenbrot Stellar structure - recap density mass

More information

Spectroscopy, the Doppler Shift and Masses of Binary Stars

Spectroscopy, the Doppler Shift and Masses of Binary Stars Doppler Shift At each point the emitter is at the center of a circular wavefront extending out from its present location. Spectroscopy, the Doppler Shift and Masses of Binary Stars http://apod.nasa.gov/apod/astropix.html

More information

Ay Fall 2004 Lecture 6 (given by Tony Travouillon)

Ay Fall 2004 Lecture 6 (given by Tony Travouillon) Ay 122 - Fall 2004 Lecture 6 (given by Tony Travouillon) Stellar atmospheres, classification of stellar spectra (Many slides c/o Phil Armitage) Formation of spectral lines: 1.excitation Two key questions:

More information

Ay 1 Midterm. Due by 5pm on Wednesday, May 9 to your head TA s mailbox (249 Cahill), or hand it directly to any section TA

Ay 1 Midterm. Due by 5pm on Wednesday, May 9 to your head TA s mailbox (249 Cahill), or hand it directly to any section TA Ay 1 Midterm Due by 5pm on Wednesday, May 9 to your head TA s mailbox (249 Cahill), or hand it directly to any section TA You have THREE HOURS to complete the exam, but it is about two hours long. The

More information

Components of Galaxies Stars What Properties of Stars are Important for Understanding Galaxies?

Components of Galaxies Stars What Properties of Stars are Important for Understanding Galaxies? Components of Galaxies Stars What Properties of Stars are Important for Understanding Galaxies? Temperature Determines the λ range over which the radiation is emitted Chemical Composition metallicities

More information

Problem set: solar irradiance and solar wind

Problem set: solar irradiance and solar wind Problem set: solar irradiance and solar wind Karel Schrijver July 3, 203 Stratification of a static atmosphere within a force-free magnetic field Problem: Write down the general MHD force-balance equation

More information

Sources of radiation

Sources of radiation Sources of radiation Most important type of radiation is blackbody radiation. This is radiation that is in thermal equilibrium with matter at some temperature T. Lab source of blackbody radiation: hot

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

Linear Theory of Stellar Pulsation

Linear Theory of Stellar Pulsation Linear Theory of Stellar Pulsation Mir Emad Aghili Department of Physics and Astronomy, University of Mississippi, University, MS, 38677 Introduction The first pulsating star to be discovered was omicron

More information

Stellar Winds. Star. v w

Stellar Winds. Star. v w Stellar Winds Star v w Stellar Winds Geoffrey V. Bicknell 1 Characteristics of stellar winds Solar wind Velocity at earth s orbit: Density: Temperature: Speed of sound: v 400 km/s n 10 7 m 3 c s T 10 5

More information

Our Galaxy. We are located in the disk of our galaxy and this is why the disk appears as a band of stars across the sky.

Our Galaxy. We are located in the disk of our galaxy and this is why the disk appears as a band of stars across the sky. Our Galaxy Our Galaxy We are located in the disk of our galaxy and this is why the disk appears as a band of stars across the sky. Early attempts to locate our solar system produced erroneous results.

More information

Astronomy 112: The Physics of Stars. Class 14 Notes: The Main Sequence

Astronomy 112: The Physics of Stars. Class 14 Notes: The Main Sequence Astronomy 112: The Physics of Stars Class 14 Notes: The Main Sequence In the last class we drew a diagram that summarized the basic evolutionary path of stars, as seen from their centers. In this class

More information

Giant planet formation. Core accretion = core nucleation = core instability = bottom-up. M total M rocky core M gas envelope

Giant planet formation. Core accretion = core nucleation = core instability = bottom-up. M total M rocky core M gas envelope Giant planet formation M total M rocky core M gas envelope Core accretion = core nucleation = core instability = bottom-up Runaway gas accretion when M envelope ~ M core Miguel & Brunini 08 Critical core

More information

Astro 162 Planetary Astrophysics Solution to Problem Set 3

Astro 162 Planetary Astrophysics Solution to Problem Set 3 Astro 162 Planetary Astrophysics Solution to Problem Set 3 Problem 1. Disk Heaven Consider once again the minimum-mass solar nebula, a circumstellar disk of gas and dust of solar composition orbiting the

More information

ASTRONOMY QUALIFYING EXAM August 2014

ASTRONOMY QUALIFYING EXAM August 2014 ASTRONOMY QUALIFYING EXAM August 2014 L = 3.9 10 33 erg s 1 M = 2 10 33 g M bol = 4.74 R = 7 10 10 cm 1 AU = 1.5 10 13 cm 1 pc = 3.26 Ly. = 3.1 10 18 cm a = 7.56 10 15 erg cm 3 K 4 c = 3 10 10 cm s 1 σ

More information

Astronomy 100 Spring 2006 Lecture Questions Twelve Weeks Review

Astronomy 100 Spring 2006 Lecture Questions Twelve Weeks Review Astronomy 100 Spring 2006 Lecture Questions Twelve Weeks Review 16-1 Fusion in the Sun The solar corona has temperatures roughly the same as temperatures in the Sun's core, where nuclear fusion takes place.

More information

Chapter 15. Thermodynamics of Radiation Introduction

Chapter 15. Thermodynamics of Radiation Introduction A new scientific truth does not triumph by convincing its opponents and maing them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it

More information

Astronomy 242: Review Questions #3 Distributed: April 29, 2016

Astronomy 242: Review Questions #3 Distributed: April 29, 2016 Astronomy 242: Review Questions #3 Distributed: April 29, 2016 Review the questions below, and be prepared to discuss them in class next week. Modified versions of some of these questions will be used

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

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

Structure & Evolution of Stars 1

Structure & Evolution of Stars 1 Structure and Evolution of Stars Lecture 2: Observational Properties Distance measurement Space velocities Apparent magnitudes and colours Absolute magnitudes and luminosities Blackbodies and temperatures

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

UNIVERSITY of NORTH CAROLINA at CHAPEL HILL. Doctoral Written Examination in Physics, Part I: Classical mechanics and Statistical mechanics

UNIVERSITY of NORTH CAROLINA at CHAPEL HILL. Doctoral Written Examination in Physics, Part I: Classical mechanics and Statistical mechanics UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Doctoral Written Examination in Physics, 2012 Part I: Classical mechanics and Statistical mechanics Saturday, May 12, 2012 Instructions: Please work in the assigned

More information