Galaxies: Structure, Dynamics, and Evolution Galaxies: Structure, Dynamics, and Evolution (from website)
“Galaxies are the basic building blocks of the universe, and we use them to trace the evolution of the universe. Fundamental processes such as star formation, recycling and enrichment of gas, formation of planets etc. all take place in galaxies. The course describes the structure of the galaxies, including dark matter, stars, and gas as well as the large scale structure in which galaxies are embedded. It discusses ongoing surveys of the nearby and distant universe. A special focus will be on the evolution of galaxies. The course builds on the bachelor lecture course “Galaxies and Cosmology” (Sterrenstelsels en Kosmologie), and assumes that the material in this course is known to the student. A very brief recapitulation will be given of the most important material.” 6 EC course Galaxies: Structure, Dynamics, and Evolution (from website)
The approach we take in this course will be very empirical / observational.
We find these interesting physical objects called galaxies nearby with observations. In this course, we will outline their properties and try to explain them using various physical ideas.
This contrasts with the approach taken in the galaxy formation course originally taught by Joop Schaye -- where one tries to form galaxies ab initio just from theory. Lectures
Rychard Bouwens Oort 459 [email protected]
Lecture Hours: Huygens 414 Friday 11:15-13:00
Course Website: http://www.strw.leidenuniv.nl/~bouwens/galstrdyn/ Textbook?
Useful Textbooks for the course will be “Galaxy Formation and Evolution” by Houjun Mo, Frank van den Bosch, and Simon White
“Galaxy Dynamics” by James Binney & Scott Tremaine
“Galactic Astronomy” by James Binney & Michael Merrifield
The textbook includes a useful discussion of the material, but the course will not be organized to follow the presentation in the book.
However, I will advise you as to where you can find the relevant material in the textbook. Layout of the Course
Sep 9: Review: Galaxies and Cosmology Sep 16: No Class — Science Day Sep 23: Review: Disk Galaxies and Galaxy Formation Basics Sep 30: Disk Galaxies (II) Oct 7: Disk Galaxies (III) Oct 14: No Class Oct 21: Review: Elliptical Galaxies, Vlasov/Jeans Equations Oct 28: Elliptical Galaxies (I) Nov 4: Elliptical Galaxies (II) Nov 11: Dark Matter Halos Nov 18: Large Scale Structure Nov 25: Analysis of Galaxy Stellar Populations Dec 2: Lessons from Large Galaxy Samples at z<0.2 Dec 9: Evolution of Galaxies with Redshift Dec 16: Galaxy Evolution at z>1.5 / Review for Final Exam Jan 13: Final Exam How will you be evaluated? Final Exam (Written) -- 65% Homework -- 35% How many homework sets? You will have ~6 homework sets. Who am I?
My name is Rychard Bouwens (studied in the United States: Berkeley & Santa Cruz)
I study the most distant galaxies in the universe
A good understanding of galaxies is very important for my research Discovery of Plausible Galaxy just ~400-450 Myr after Big Bang
Candidate for the most distant galaxy ever discovered...
Bouwens et al. 2011 (Nature, January 27, 2011) 11.09 414 Myr GN-z11 25.9 140 Oesch16 CANDELS
9.8 480 Myr Abell2744-JD 27.0 70 Zitrin14 Abell2744
11.09 414 Myr GN-z11 25.9 Oesch16 8.68 580 Myr EGS-zs8-1 25.2 Zitrin15 7.73 680 Myr EGS-zs7-1 25.0 Oesch15 Credit: Dan Coe How will this class be taught?
40% Powerpoint....
60% Derivations on the blackboard... Teaching Assistants
Daniel Lam Oort 570 [email protected] Gabriela Calistro Rivera Oort 551 [email protected]
Both will also be available by appointment to answer your questions, and he also may hold office hours Who are you?
Why don’t we go around the class and introduce ourselves briefly?
Name Program -- Physics or Astronomy? Master’s Student? First or Second Year? Why interested in course? What’s your background?
How many of you have taken Huub’s or Marijn’s course “Galaxies and Cosmology” when you were a bachelor student?
How manyPass of you around are from thesheet physics program? How many of you have taken a Jarle’s course on galaxiesto collect or Koen’s info course on cosmology? Please ask questions
This is *your* course. It is your opportunity to learn.
By asking questions, you allow me to clarify issues It is assumed that you all are familiar with the following material from the Bachelor course here.
7-2-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c01-5 7-2-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c01-6
Background (assumed known) Clusters of galaxies, the universe Candidate dark matter particles Brief content of the course Galaxies and Cos- moly (Bachelors) 8) Galaxy formation Universe expansion 1) Introduction Growth of galaxies by gravity Galaxy scaling relations What is a galaxy ? Classifications 9) Galaxy formation - forming the stars Photometry, exponentials, r1/4 profiles, luminosity Gas cooling and star formation function formation of disks dynamical friction and mergers 2) Keeping a galaxy together: Gravity tidal tails in mergers Potentials for spherical systems 10) Observing galaxy formation 3) Galactic Dynamics High redshift galaxies from HST Equilibrium Fair samples of galaxies at high redshift collisions, Virial Theorem
4) Galactic Dynamics continued Timescales Orbits
5) Collisionless Boltzmann Equation equilibrium, phase mixing derivation of distribution function
6) Velocity Moments Jeans equations comparison to observations
7) Mass distribution and dark matter Evidence for dark matter from rotation curves Solar neighborhood, Oort limit Elliptical galaxies and hot gas But I know that many of you have not done your education here... so we will review the Bachelor material before moving onto the new material from this course... Key concepts from bachelor course 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlung from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlung from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans Keyequations. concepts from bachelor course
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
8. Stars dont collide calculate from typical density, velocity, and cross- section of stars. To understand equilibrium, one can ignore stellar collisions. The main challenge is to understand a system with > 1011 particles moving in their common gravitational potential. 1 9. Individual interactions between stars can be ignored. Hubble sequence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers The relaxation time is the time needed to a↵ect the motion of the star by indi- vidual star-star interactions. This timescale is very high (1016 years) - too high to be relevant for galaxies. As a re- sult, we can describe the potential as a smooth, 3D, potential ⇢(~x)), instead of the potential produced by 1011 particles. Similarly, we can derive the phase-space density (a 6D function f(~x,~v)), by calculating the average density in space and velocity of the 1011 or more particles.
10. Galaxy formation is driven by the formation of the halo At very high redshift, the density of the universe is close to critical. Any perturbation will drive the density to above critical. Regions with density above critical density will collapse and become self-gravitating.
11. Galaxies form inside the collapsed halos, as the gas can cool radia- tively, become self-gravitating, and form stars.
12. The equations of motion for a self-gravitating system are simple. They imply that any system can be scaled in mass, size, and velocity if the scaling factors satisfy one single relaiton This implies that any equilibrium model can be scaled in mass and size !
13. In practice, the halos grow in mass due to merging with smaller halos. Merging is very e↵ective in the universe, due to the fact that the halos are extended and not point-sources.
14. The time-evolution of the distribution function is defined by the dis- tribution function at that time, spatial derivatives, and the gradients in the potential (Vlasov-Equation). This follows directly from the conservation of stars.
2 8. Stars dont collide calculate from typical density, velocity, and cross- section of stars. To understand equilibrium, one can ignore stellar collisions. The main challenge is to understand a system with > 1011 particles moving in their common gravitational potential.
9. Individual interactions between stars can be ignored. Hubble sequence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers The relaxation time is the time needed to a↵ect the motion of the star by indi- vidual star-star interactions. This timescale is very high (1016 years) - too high to be relevant for galaxies. As a re- sult, we can describe the potential as a smooth, 3D, potential ⇢(~x)), instead of the potential produced by 1011 particles. Similarly, we can derive the phase-space density (a 6D function f(~x,~v)), by calculating the average density in space and velocity of the 1011 Keyor concepts more particles. from bachelor course
10. Galaxy formation is driven by the formation of the halo At very high redshift, the density of the universe is close to critical. Any perturbation will drive the density to above critical. Regions with density above critical density will collapse and become self-gravitating.
11. Galaxies form inside the collapsed halos, as the gas can cool radia- tively, become self-gravitating, and form stars.
12. The equations of motion for a self-gravitating system are simple. They imply that any system can be scaled in mass, size, and velocity if the scaling factors satisfy one single relaiton This implies that any equilibrium model can be scaled in mass and size !
13. In practice, the halos grow in mass due to merging with smaller halos. Merging is very e↵ective in the universe, due to the fact that the halos are extended and not point-sources.
14. The time-evolution of the distribution function is defined by the dis- tribution function at that time, spatial derivatives, and the gradients in the potential (Vlasov-Equation). This follows directly from the conservation of stars. 15. Integrals of motion are functions of x and vvwhichareconstantalong an orbit. They are not explicit functions of time. Examples: Energy; angular momentum in a spherical potential. Most 3D densities allow for 3 integrals of motion, 2 of which are non-classical.2
16. Equilibrium models can easily made by requiring that the distribution function is a function of integrals of motion. By definition, the distribution function is an integral of motion itself.
17. Another way to construct equilibrium models is to make a library of orbits, calculate their spatial densities, and calculate the weight function which reproduces the density distribution for which the orbits were calcu- lated. This is Schwarzschilds method.
18. Using modern telescopes, we can trace galaxies to z =6andbeyond, and study galaxy formation and evolution directly.
3 15. Integrals of motion are functions of x and vvwhichareconstantalong an orbit. They are not explicit functions of time. Examples: Energy; angular momentum in a spherical potential. Most 3D densities allow for 3 integrals Key ofconcepts motion, 2 of from which are bachelor non-classical. course
16. Equilibrium models can easily made by requiring that the distribution function is a function of integrals of motion. By definition, the distribution function is an integral of motion itself.
17. Another way to construct equilibrium models is to make a library of orbits, calculate their spatial densities, and calculate the weight function which reproduces the density distribution for which the orbits were calcu- lated. This is Schwarzschilds method.
18. Using modern telescopes, we can trace galaxies to z =6andbeyond, and study galaxy formation and evolution directly.
3 I will begin this course by summarizing key points from the Leiden Bachelor course on galaxies “Galaxies and Cosmology” taught by Huub Rottgering or Marijn Franx REVIEW
1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlung from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 Think for a moment about what complex entities galaxies are: Elliptical Galaxy Whirlpool Galaxy ESO 325-G004 Messier51
Sombrero Galaxy Messier104
Note: (1) the remarkable spiral structure. (2) prominent nucleus Think for a moment about what complex entities galaxies are: Elliptical Galaxy Whirlpool Galaxy ESO 325-G004 Messier51
Sombrero Galaxy Messier104
Note: (1) how homogeneous this galaxy looks Think for a moment about what complex entities galaxies are: Elliptical Galaxy Whirlpool Galaxy ESO 325-G004 Messier51
Sombrero Galaxy Messier104
Note: (1) the faint disk for this mostly spherical galaxy (2) notice the dark band (dust lane) Contemplate how different galaxies look on different scales:
Different components in galaxies
Are these part of the same galaxy? They are different parts of the same galaxy! Diverse View of Galaxies at Different Wavelengths
A wide variety of different physical processes cause the emission of light from galaxies.
Almost of all of these processes emit light over some characteristic range of wavelengths.
By viewing galaxies over different wavelength ranges, we can obtain a better physical understanding of the many physical processes important for the formation and evolution of galaxies. Different Multi-wavelength views of our own galaxy M31 Xray (center) Accretion Disks
x-ray binaries M31 IRAS 100 micron
Thermal emission from Dust M31 Radio Synchrotron Emission from Supernovae M81 optical Light from Normal Stars Light from M81 UV Massive OB Stars Light from x-ray M81 Xray binary star systems Light from M81 H alpha + continuumMassive O Stars Synchrotron M81 radio continuum 20 cmEmission from Supernovae 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlungREVIEW from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 Elliptical Galaxies
Figures from Binney & Merrifield 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlungREVIEW from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 Dwarf Elliptical Galaxies
Figures from Binney & Merrifield 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlungREVIEW from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 Lenticular Galaxies (Transition-Type Galaxies)
Figures from Binney & Merrifield transition Elliptical and spiral: S0 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlungREVIEW from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 Early-Type Spiral Galaxies
Figures from Binney & Merrifield earlytype spiral galaxy Spiral Galaxies Late type spiral Late type: bulge decreases, arm more open, knots along arm 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlungREVIEW from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 Figures from Binney & Merrifield Barred Spiral Figures from Binney & Merrifield Figures from Binney & Merrifield Irregulars: hardly any structure
1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electronsREVIEW (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlung from hot gas to ...
2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers.
3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the factMorphological that the potential classifications inside a spherical only shell “work” is constant (hence no force!). Outsideapproximately, of the shell, the given potential how is the complex same as that the of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined onlymorphologies by the mass inside ofr ,galaxies as if the massare. were at the center of the spherical distribution.
4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.
5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.
6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 What is some of the essential physics that I will assume you understand? 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-1 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-2
2 Gravitational force and potential (BT 2 - Sphere ρ = ρ(r) I’m going2.1) to assume that you are familiar with • 2 2 x2 Classical ellipsoid ρ = ρ(m ) where m = a2 + • y2 z2 how toThe calculate matter in galaxies the gravitational (whether stars, gas, force dark mat- between b2 + c2 ter, etc) is kept from escaping by gravity. Before we Thin disk twostudy bodies... the motions and of individual also how particles, to we construct show how a • we can calculate the gravitation force and potential The density follows from the potential by Poisson’s from an extendedgravitational density distribution. potential equation: 4πGρ(⃗x)=⃗ 2Φ(⃗x) The gravitational force caused by a point mass M at ∇ ⃗x0 on a unit mass at position ⃗x is The mass in some volume can easily be derived from the force field: Integrate both sides of Poisson’s equa- ⃗ ⃗x0 ⃗x tion over the volume enclosing a total mass M. F (⃗x)=GM − 3 ⃗x0 ⃗x For the left hand side we obtain: | − | 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-1 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-2 In general, the gravitational force is related to the po- 4πG ρd⃗x =4πGM tential Φ by !V 2 Gravitational force and potential (BT 2 - Sphere ρ = ρ(r) • F⃗ (⃗x)= ⃗ Φ(⃗x) 2 Using the divergence theorem, we obtain for the right 2.1) −∇ 2 2 x Classical ellipsoid ρ = ρ(m ) where m = a2 + hand side: so that• y2 z2 The matter in galaxies (whether stars, gas, dark mat- b2 + c2 GM Φ(⃗x)= 2 ⃗ 2 ter, etc) is kept from escaping by gravity. Before we Thin disk − ⃗x0 ⃗x Φd⃗x = Φ d S study the motions of individual particles, we show how • | − | !V ∇ !S ∇ · we can calculate the gravitation force and potential TheThe gravitational density follows potential from for theextended potentialdensity by Poisson’s dis- Combining left and right side gives Gauss’s theorem: from an extended density distribution. tributionequation:ρ(⃗x) can be obtained by integrating over the density distribution 2 ⃗ 2 The gravitational force caused by a point mass M at 4πGρ(⃗x)= Φ(⃗x) 4πGM = ⃗ Φ d S ∇ 3 ! ∇ · ⃗x0 on a unit mass at position ⃗x is ρ(⃗x0)d ⃗x0 The mass inΦ some(⃗x)= volumeG can easily be derived from the force field: Integrate− !!! both⃗x sides0 ⃗x of Poisson’s equa- the integral of the normal component of ⃗ Φ over | − | ⃗ ⃗x0 ⃗x tion over the volume enclosing a total mass M. any→ closed surface equals 4πG times the mass∇ con- F (⃗x)=GM − 3 ⃗x0 ⃗x Note:For the left hand side we obtain: tained within that surface | − | The triple integration is often expensive Easier for special geometries, mass stratifications The potential energy can be shown to be: In general, the gravitational force is related to the po- 4πG ρd⃗x =4πGM tential Φ by !V F⃗ (⃗x)= ⃗ Φ(⃗x) Using the divergence theorem, we obtain for the right −∇ hand side: so that GM Φ(⃗x)= 2 ⃗ 2 − ⃗x0 ⃗x Φd⃗x = Φ d S | − | !V ∇ !S ∇ · The gravitational potential for extended density dis- Combining left and right side gives Gauss’s theorem: tribution ρ(⃗x) can be obtained by integrating over the density distribution 4πGM = ⃗ Φ d2S 3 ! ∇ · ρ(⃗x0)d ⃗x0 Φ(⃗x)= G − !!! ⃗x0 ⃗x the integral of the normal component of ⃗ Φ over | − | any→ closed surface equals 4πG times the mass∇ con- Note: tained within that surface The triple integration is often expensive Easier for special geometries, mass stratifications The potential energy can be shown to be: 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-1 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-2
2 Gravitational force and potential (BT 2 - Sphere ρ = ρ(r) 2.1) • 2 2 x2 Classical ellipsoid ρ = ρ(m ) where m = a2 + • y2 z2 The matter in galaxies (whether stars, gas, dark mat- b2 + c2 ter, etc) is kept from escaping by gravity. Before we Thin disk study the motions of individual particles, we show how • 1.we Galaxies can calculate omit radiation the gravitation in just about force every wavelength and potential you can imag- The density follows from the potential by Poisson’s ine difrom↵erent an processes extended at didensity↵erent wavelengths, distribution. from (optical) stellar light equation: of normal stars to relativistic electrons (radio, high energy) to hot accretion 4πGρ(⃗x)=⃗ 2Φ(⃗x) disksThe (UV, gravitational Xray) to bremsstrahlung force caused from by hot a gas point to ... mass M at ∇ ⃗x0 on a unit mass at position ⃗x is 2. Galaxies are usually classified based on optical imaging. Hubble se- The mass in some volume can easily be derived from quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, the force field: Integrate both sides of Poisson’s equa- REVIEW⃗x0 ⃗x mergers. ⃗ tion over the volume enclosing a total mass M. F (⃗x)=GM − 3 ⃗x0 ⃗x For the left hand side we obtain: 3. The potential of a smooth density| distribution− | can be calculated by an integralIn general, in 3 dimensions. the gravitational The potential force of is a relatedspherical to system the ispo- simpler. Use the fact that the potential inside a spherical shell is constant (hence no 4πG ρd⃗x =4πGM tential Φ by !V force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell withF⃗ ( the⃗x)= same⃗ mass.Φ(⃗x Hence) the force at a radius r Using the divergence theorem, we obtain for the right is determined only by the mass inside−r,∇ as if the mass were at the center of hand side: the sphericalso that distribution. GM Φ(⃗x)= 2 ⃗ 2 4. For a self-gravitating system, we− can⃗x0 derive⃗x the virial theorem, 2K + Φd⃗x = Φ d S W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.| − | !V ∇ !S ∇ · The gravitational potential for extended density dis- Combining left and right side gives Gauss’s theorem: 5.tribution Galaxy massρ(⃗x) distributionscan be obtained can be byderived integrating from observed over stellar the ve- locities,density gas rotation distribution curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 4πGM = ⃗ Φ d2S 14), the relation between enclosed mass and circular3 velocity, and the equa- ! ∇ · ρ(⃗x0)d ⃗x0 tions for hydrostaticΦ equilibrium(⃗x)= G of gas (which are very similar to the Jeans equations. − !!! ⃗x0 ⃗x the integral of the normal component of ⃗ Φ over | − | any→ closed surface equals 4πG times the mass∇ con- 6.Note: The mass distributions of galaxies show strong evidence for dark mat- tained within that surface ter,The i.e., the triple mass integration distribution is continues often expensive much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals,Easier large for masses special of clusters. geometries, mass stratifications The potential energy can be shown to be:
7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe.
1 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-3 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-4
3.1 Potential for spherical systems (BT 2.1, W =1/2 ρ(⃗x)Φ(⃗x)d⃗x 2.2) 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-3 13-9-07 see http://www.strw.leidenuniv.nl/˜For a !spherically symmetric franx/college/ distribution, mf-sts-07-c2-4 the potential can be calculated much more simply.Newton’s Theorems: We derive this as follows. Assume that we “build” up the galaxy3.1 Potential slowly. We have for a spherical galaxy with systems a density f (BTρ, 2.1, 2.2) First Theorem: W =1/2 ρ(⃗x)Φ(⃗x)d⃗x with 0 13-9-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c2-5 13-9-07 see http://www.strw.leidenuniv.nl/˜2 2 franx/college/ mf-sts-07-c2-6 proportional to δm1/r1 and δm2/r2,andthereforeequal, with the same solid angle δΩ is: but of opposite sign. Hence the matter in the cone does not 2 2 proportional to δm1/r1 and δm2/r2,andthereforeequal, contributewith any the net same force solid at the angle locationδΩ ⃗is:r. If we sum over all GM δΩ but of opposite sign. Hence the matter in the cone does not cones, we find no net force ! δΦ2 = − p⃗′ ⃗q 4π contribute any net force at the location ⃗r. If we sum over all GM δΩ | − | cones, we find no net force ! Newton’s Second Theorem:δΦ2 = • The gravitational force on a− bodyp⃗′ outside⃗q 4π a closed Since p⃗ ⃗q′ = p⃗′ ⃗q , δΦ1 = δΦ2. Sum over all solid | − | angles| to− obtain| | − | Newton’s Second Theorem: spherical shell of matter is the same as it would be 13-9-07 seeif http://www.strw.leidenuniv.nl/˜ all the shell’s matter13-9-07 were franx/college/ see concentrated http://www.strw.leidenuniv.nl/˜ mf-sts-07-c2-5 into a franx/college/13-9-07 mf-st sees-07-c2-5 http://www.strw.leidenuniv.nl/˜Φ13-9-071 = Φ franx/college/2 see http://www.strw.leidenuniv.nl/˜ mf-sts-07-c2-6 franx/college/ mf-sts-07-c2-6 • The gravitational force on a body outside a closed Since p⃗ ⃗q′ = p⃗′ ⃗q , δΦ1 = δΦ2. Sum over all solid pointangles at| to its− obtain center.| | − | spherical shell of matter is the same as it would be 2 2 2 2 Since Φ2 is the potential inside a sphere with mass M and proportional to δm1/r1 and δmproportional2/r2,andthereforeequal,Φ1 = to δΦm21/r1 and δm2/r2,andthereforeequal,withradius the samer,itisequalto solid angle δΩΦ2is:=withGM/r the same,andthisisequalto solid angle δΩ is: if all the shell’s matter were concentrated into a but of opposite sign. Hence the matterbut of inopposite the cone sign. does Hence not the matter in the cone does not − point at its center. Φ1. This is the same as the potential at r if all the mass is contribute anySince netΦ force2 is at the the potential locationcontribute⃗r inside. If any we a net sum sphere force over with at all the mass locationM ⃗rand. If we sumconcentrated over all at the center. GM δΩ GM δΩ δΦ2 = δΦ2 = cones, we findradius no netr,itisequalto force ! cones,Φ2 we= findGM/r no net force,andthisisequalto ! − p⃗′ ⃗q 4π − p⃗′ ⃗q 4π If we have a spherical shell with mass M with Φ1. This is the same as the potential− at r if all the mass is | − | | − | Newton’sconcentrated Second Theorem: at the center.Newton’sBT2.1:page Second 35 Theorem: We can now calculate potential of spherical system radius a and we lie at distance r (point p) from • • Since p⃗ ⃗q′ = p⃗′ ⃗q , δΦ1Since= δΦp⃗2. Sum⃗q′ = overp⃗′ all solid⃗q , δΦ1 = δΦ2. Sum over all solid The gravitational force on a bodyThe gravitational outside a closed force on a body outsidewith| a− closed density| |ρ(r−).| Divide system| − up| into| shells,− | and the center of this sphere (outside of the spherical shell of matter is thespherical same as shell it would of matter be is the sameangles asadd it to would obtain contribution be from eachangles shell. to obtain Distinguish between shells with radius rΦ,1r= 2 2 proportional to δm1/r1 and δm2/r2,andthereforeequal, with the same solid angle δΩ is: but of opposite sign. Hence the matter in the cone does not contribute any net force at the location ⃗r. If we sum over all GM δΩ cones, we find no net force ! δΦ2 = − p⃗ ⃗q 4π | ′ − | Newton’s Second Theorem: • Since p⃗ ⃗q′ = p⃗′ ⃗q , δΦ1 = δΦ2. Sum over all solid The gravitational force on a body outside a closed Given Newton’s| − | theorems,| − it| is very easy to calculate the potential spherical shell of matter is the same as it would be angles to obtainfor a mass with spherical symmetry. if all the shell’s matter were concentrated into a Φ1 = Φ2 point at its center. In calculatingSince the Φpotential2 is the at potential a given point inside in a space sphere from with a spherical mass M mass,and one must distinguishradius ther,itisequalto mass in shells atΦ radii2 = greaterGM/r than,andthisisequalto the given point in space. Φ1. This is the same as the potential− at r if all the mass is 1) For massconcentrated in shells at atsmaller the center. radius r (than radius r’ where the potential is being evaluated), treat the mass as if it were concentrated at the center of the sphere We can now calculate potential of spherical system 2) For mass inwith shells density at largerρ (radiusr). Divide (than radius system r’ where up into the potential shells, and is being evaluated), treat the massadd as contribution if it were concentrated from each at the shell. center Distinguish of the sphere between but viewed from a shells with radiusdistancer′ r’, rthe′ 2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers. 3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution. REVIEW 4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations. 5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These Why is the virial theorem important? methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tionsThere for are hydrostatic many reasons, equilibrium but here of gas are (which two simple are very applications similar to the of Jeans equations. this thereom of which you should be aware: 6.A. The One mass simple distributions application of galaxiesof the virial show theorem strong evidence is to calculate for dark mat- ter, i.e.,the themass mass of a distribution system given continues its observed much velocity further out dispersion than the light. Typical examples are the rotationand physical curves size of galaxies, hot X-ray halos of el- lipticals, large masses of clusters. B. Another application of virial theorem is to compute the 7. Equilibrium:velocity dispersion Galaxies given are in the (rough) mass equilibrium: and size of thean object dynamical time is typically 108 years - much shorter than the age of the universe. 1 What do I assume you understand about the structure of galaxies? Picture of a Galaxy Billions of Stars Do galaxies have (visible matter) other components than luminous matter (stars)? To look for other components: We want to quantify the mass of galaxies inside some radius. How can we? 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlung from hot gas to ... 2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers. 3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution. 4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations.REVIEW 5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations. 6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters. 7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe. 1 Methods Relevant for Galaxies Method 1: Rotation Curve rotating rotating (approaching) (receding) Rotation Velocity Line of Sight Velocity Radius on Sky Method 1: Rotation Curve rotating rotating (approaching) (receding) What data are used for this? Hα (optical spectroscopy) -- star-forming/ionized regions CO (mm arrays) -- molecular hydrogen HI (radio 21 cm) -- atomic hydrogen Method 1: Rotation Curve 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-11 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-12 rotating rotating 7.2 Mass distribution and dark matter in spi- Historically, optical rotation curves like these on the (approaching) ral galaxies (receding) following figure indicated for the first time the presence of dark matter. Dark Matter Halos in Spiral Galaxies BT 10.1- 6 measure rotation curve of cold gas H alpha (optical) How do •weCO (mmderive arrays) masses? • H I (vla, westerbork) • assume circular orbits: v2(r) GM( 2 M( Isothermal sphere: v = constant, ρ r−2 c ∝ Point Mass: v 1/√r c ∝ Measurement in practice: Take edge-on system, measure maximum veloc- • ity of gas at each point possible problems: extinction, confusion Take inclined galaxies, measure velocities every- • where Very modern data are based on 21cm HI emission lines. Model by fitting a circular velocity field with The HI disks can often be followed to very large radii, unknown• inclination and rotation curve and hence the rotation curve can be followed to well 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-11 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-12 7.2 Mass distribution and dark matter in spi- Historically, optical rotation curves like these on the ral galaxies following figure indicated for the first time the presence 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-11 of29-10-07see dark matter. http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-12 Dark Matter Halos in Spiral Galaxies BT 10.1- 67.2 Mass distributionMethod and dark 1: matter in spi- Historically, optical rotation curves like these on the ral galaxies following figure indicated for the first time the presence Rotation Curve of dark matter. rotatingmeasureDark Matter rotation Halos curve in of cold Spiral gas Galaxies BT 10.1- rotating 6 H alpha (optical) (approaching)• CO (mm arrays) (receding) • H I (vla, westerbork) measure• rotation curve of cold gas assumeH circular alpha (optical) orbits: • CO (mm arrays) • H I (vla, westerbork)v2(r) GM( Take edge-oninclined galaxies, system, measure measure maximum velocities veloc-every- • itywhere of gas at each point Very modern data are based on 21cm HI emission lines. possibleModel problems: by fitting a extinction, circular velocity confusion field with The HI disks can often be followed to very large radii, unknown• inclination and rotation curve and hence the rotation curve can be followed to well Take inclined galaxies, measure velocities every- • where Very modern data are based on 21cm HI emission lines. Model by fitting a circular velocity field with The HI disks can often be followed to very large radii, unknown• inclination and rotation curve and hence the rotation curve can be followed to well 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-15 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-16 model looks like: Obviously, an additional mass component is necessary to explain the rotation curve. Fit rotation curves: Constant M/L for starlight 2 If the only matter inadd spiral halo, galaxies with ρwas= ρfrom0/(1 the + ( r/aobserved) ) stars, then the rotation curve would look as follows: Example for NGC 3198 Rotation Velocity We can now model the galaxy. Take the surface bright- (rotation curve from stars alone) ness profile, and calculate the rotation curve if the mass-to-light ratio were constant: 2 Vc (implied from stars) = GMstars(R)/R In such a case, the rotation curve would have an approximately Keplerian shape 1/2 (i.e., vc ∝1/r ) at large radii Problems: The fit is never unique. Different M/L’s for the disk, and different values of a, ρ0 for the 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-15 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-16 model looks like: Obviously, an additional mass component is necessary to explain the rotation curve. Fit rotation curves: Constant M/L for starlight 2 However, theadd observed halo, with rotationρ = ρ 0curve/(1 + looks (r/a) flat,) even at large radii, and isExample very different for NGC from 3198 expectations if only stars contribute (observed rotation curve) Rotation Velocity We can now model the galaxy. Take the surface bright- (rotation curve from stars alone) ness profile, and calculate the rotation curve if the mass-to-light ratio were constant: 2 2 Vc (observed) > Vc (implied from stars) = GMstars(R)/R This implies there must be another component... Problems: The fit is never unique. Different M/L’s for the disk, and different values of a, ρ0 for the 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-15 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-16 model looks like: Obviously, an additional mass component is necessary to explain the rotation curve. Fit rotation curves: Constant M/L for starlight 2 If the only matter inadd spiral halo, galaxies with ρwas= ρfrom0/(1 the + ( r/aobserved) ) stars, then the rotation curve would look as follows: Example for NGC 3198 (observed rotation curve) Rotation Velocity We can now model the galaxy. Take the surface bright- (rotation curve from stars alone) ness profile, and calculate the rotation curve if the mass-to-light ratio were constant: 2 Vc (observed) = GMall(R)/R 2 Vc (observed) = G(Mstars(R)+Mhalo(R))/R = GMstars(R)/R + GMhalo(R)/R in principle, this allows us to measure the dark matter in galaxies very precisely. The problem is thatProblems: there is Thesome fit uncertainty is never unique. in measuring Different the M/L’s mass in stars. for the disk, and different values of a, ρ0 for the 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-17 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-18 halo will give fits which are all good. large bright galaxies - slightly falling, compact Hence: it is very hard to determine how much of galaxies, falling slightly more the mass is due to the halo, within the outer most radius Never Keplerian - always a lot of dark matter ! Solution: determine the “minimum halo mass”, • by calculating “maximum disk”. But this is only the minimum ! Homework Assignment: Various authors have assembled large samples of galax- 1) Calculate the total mass, the mass in the disk, and ies with rotation curves. An example: Casertano and the mass in the halo for NGC 3198 at a radius of 30kpc. van Gorkom (1991, AJ 101, 1231) Use the decomposition into components shown in the figure above. Observations of the rotation curve always rapidly rise from center Rotation Velocity Velocity of the rotation Curve is nearly always flat at the largest radii Resulting rotation curves: Rapid rise near center • nearly flat out to most distant point, with some • variations: dwarf galaxies: still rising intermediate galaxies: flat 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-17 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-18 halo will give fits which are all good. large bright galaxies - slightly falling, compact Hence: it is very hard to determine how much of galaxies, falling slightly more the mass is due to the halo, within the outer most radius Never Keplerian - always a lot of dark matter ! Solution: determine the “minimum halo mass”, • by calculating “maximum disk”. But this is only the minimum ! Homework Assignment: Various authors have assembled large samples of galax- 1) Calculate the total mass, the mass in the disk, and ies with rotation curves. An example: Casertano and the mass in the halo for NGC 3198 at a radius of 30kpc. van Gorkom (1991, AJ 101, 1231) Use the decomposition into components shown in the figure above. Observations of the rotation curve always rapidly rise from center Rotation Velocity For large luminous galaxies, the rotation curve falls slightly at largest radii (vc independent of radius) Resulting rotation curves: Rapid rise near center • nearly flat out to most distant point, with some • variations: dwarf galaxies: still rising intermediate galaxies: flat 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-17 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-18 halo will give fits which are all good. large bright galaxies - slightly falling, compact Hence: it is very hard to determine how much of galaxies, falling slightly more the mass is due to the halo, within the outer most radius Never Keplerian - always a lot of dark matter ! Solution: determine the “minimum halo mass”, • by calculating “maximum disk”. But this is only the minimum ! Homework Assignment: Various authors have assembled large samples of galax- 1) Calculate the total mass, the mass in the disk, and ies with rotation curves. An example: Casertano and the mass in the halo for NGC 3198 at a radius of 30kpc. Observationsvan of the rotation Gorkom curve always (1991, rapidly rise AJfrom center 101, 1231) Use the decomposition into components shown in the figure above. Rotation Velocity Nevertheless, for dwarf galaxies, the rotation curve is still slightly rising at largest radii. Different dependencies of the circular velocities on radius for different galaxies tell us about mass profiles of dark matter and stars in these sources. Resulting rotation curves: Rapid rise near center • nearly flat out to most distant point, with some • variations: dwarf galaxies: still rising intermediate galaxies: flat 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-25 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-26 7.4 X-ray gas in halosMethod of Ellipticals 2: BT 10.1-7(c) x-ray profile What is equilibrium of gas ? Balance pressure with gravity Many luminous ellipticals have huge X-ray halos For spherical systems, gravity balances the pressure Many galaxies have luminous x-ray halos: gradient if Example: NGC 720 dp GM(r) = ρ NGC720 NGC720 dr − r2 Use the ideal gas law ρk T p = B m k T Since p = ρ v2 , this implies v2 = 1 v2 = B x x 3 m Now we find dρk T/m GM(r)ρ x-ray optical B = dr − r2 k r2 dρT M(r)=− B mG ρ dr k TrrdρT k Tr d ln ρ d ln T = b = b − mG ρT dr mG !− d ln r − d ln r " Note how comparable this equation is to the isotropic spherical formula for stellar systems. Observations: example NGC 720 from analysis of the spectrum (continuum and lines) for NGC 720: T =7 106K × 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-25 29-10-07see http://www.strw.leidenuniv.nl/˜Method franx/college/2: mf-sts-07-c7-26 7.4 X-ray gas in halos of Ellipticals BT x-ray profile 10.1-7(c) What is equilibrium of gas ? We can measureBalance the pressure enclosed with mass gravity out to larger radii using the x-ray light profile Many luminous ellipticals have huge X-ray halos assumingFor hydrostatic spherical equilibrium. systems, gravity For a system balances in hydrostatic the pressure equilibrium, gas pressuregradient counteracts if the gravitational force provided by the enclosed mass. Example: NGC 720 dp GM(r) = ρ dr − r2 Use the ideal gasPressure law Gravitational Gradient Force ρk T p = B Can determine the density of the gas and mpressure gradient from the x-ray light profile and the measured gas temperature. k T Since p = ρ v2 , this implies v2 = 1 v2 = B x x 3 m Now we find dρk T/m GM(r)ρ B = dr − r2 k r2 dρT M(r)=− B mG ρ dr k TrrdρT k Tr d ln ρ d ln T = b = b − mG ρT dr mG !− d ln r − d ln r " Note how comparable this equation is to the isotropic spherical formula for stellar systems. Observations: example NGC 720 from analysis of the spectrum (continuum and lines) for NGC 720: T =7 106K × Method 3: From Stellar Velocity Dispersion (using Jeans Equations) We will review this method in Lecture 5-6 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlung from hot gas to ... 2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers. 3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution. 4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations. 5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations. REVIEW 6. The mass distributions of galaxies show strong evidence for dark mat- ter, i.e., the mass distribution continues much further out than the light. Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters. 7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe. 1 Methods Relevant for Galaxy Clusters Method 4: From Velocity Dispersion In clusters of galaxies, galaxies are observed to show a huge dispersion in their observed velocities along the line of sight: Galaxy Cluster 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-29 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-30 7.5 Clusters of galaxies BT 10.2-4 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-29 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-30 measure redshifts of cluster7.5 galaxies Clusters of galaxies BT 10.2-4 velocity dispersions of 1000 km/s or higher are mea- sured ! measure redshifts of cluster galaxies velocity dispersions of 1000 km/s or higher are mea- suredComa ! cluster 15 Careful model shows: total mass is 2 10 M⊙ (H0 = 70 km/sec/Mpc). × Obtained by measuring 15 redshifts / Doppler shifts Careful model shows: total mass is 2 10 M⊙ (H0 = Total amount70 km/sec/Mpc). of light: 279 galaxies brighter× than L∗, of all galaxies in cluster 10 Line of sight velocity sight Line of L∗ = 0.7 10 L⊙. Total amount of light is 4.33 L∗ Total amount of light: 279 galaxies brighter than L∗, 13 10 × × N= 0.8 10L∗ =L 0.7⊙ 10 L⊙. Total amount of light is 4.33 L∗ Center 13 OuterParts × × N= 0.8 10 L⊙ Hence mass to light ratio = M/L = 21015/0.81013 = Hence mass to light ratio = M/L = 21015/0.81013 = 250 M⊙/L250⊙M. ⊙/L⊙. Similar studySimilar (Kent study (Kent and and Sargent) Sargent) Perseus: Perseus: M/LM/LV =V = 420 M/L⊙ 420 M/L⊙ 14 15 Typical masses of clusters: 10 10 M⊙ 14 15 Typical masses of clusters: 10 −10 M⊙ Normal galaxy: MLV < 10M/L−⊙, hence a lot of dark Kent and Gunn 1982 matter. Normal galaxy: MLV < 10M/L⊙, hence a lot of dark Kent and Gunn 1982 matter. Assuming virial equilibrium, we can derive the mass for a galaxy cluster from the observed spread in velocities: GM / R ~ v2 M ~ v2 R / G Careful model shows: total mass is 2 × 1015M⊙ (H0 = 70 km/sec/Mpc). Total amount of light: 279 galaxies brighter than L∗, L∗ = 0.7 x 1010L⊙. Total amount of light is 4.33×L∗ × N= 0.8 x1013L⊙ Hence mass to light ratio = M/L = 21015 / 0.8 x1013 = 250 M⊙/L⊙. 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-31 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-32 7.6 Dark matter in universe as a whole BT 10.3-1 The emissivity of light in the universe in the V band is 8 −3 j0 =1.7 0.610 hL⊙Mpc ± The critical density for the universe is: 3H2 ρ = 0 Method 5: c 8πG From x-ray light itself This density is enough to stop the expansion at t = 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-31 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-32 infinity 7.6 Dark matter in universe as a whole BT 10.3-1 Define the density parameter Ω0 = ρ0/ρc,whereρ0 is the actual density of the universe 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-25 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-26 The emissivity of light in the universe in the V band is 8 −3 Express Ω0 in terms of M/L ratio of galaxies: j0 =1.7 0.610 hL⊙Mpc ± 7.4 X-ray gas in halos of Ellipticals BT −4 −1 10.1-7(c) WhatSimilar is equilibrium mass determinationsThe of critical gas density ? for from the universe gas is: . Temperatures Ω0 =6.110 h M/L/(M/L)⊙ can now be measured easily with X-ray2 satellites. Re- 3H0 Balancesults: pressure with gravity ρc = 8πG The critical mass-to-light ratio for Ω =1is given by Many luminous ellipticals have huge X-ray halos For spherical systems, gravity balances the pressure gradientDerive ifConfirms total massThis high densityassuming masses is enough hydrostatic to stop the equilibrium expansion at t = • infinity M/Lc = 1600h(M/L)⊙ Example: NGC 720 Mass budget: Definedp the densityGM parameter(r) Ω0 = ρ0/ρc,whereρ0 is the actual= densityρ of the universe Clusters imply M/Lc = 300 600h(M/L)⊙,hence Gas mass fractiondr − 20-30r2 % of total mass more realistic values are − • Express Ω0 in terms of M/L ratio of galaxies: Star massPressure fraction Gravitational< 10 % of mass • −4 −1 Use the idealRemaining: gas law darkΩ matter0 =6.110 60-70h M/L/ %(M/L ! )⊙ Similar mass determinations from gas . Temperatures Gradient Force Ω0 =0.2 0.4 can now be measured easily with X-ray satellites. Re-• − sults: The critical mass-to-lightρkBT ratio for Ω =1is given by Confirms high masses (same method asp =just explained for galaxies) • M/Lm c = 1600h(M/L)⊙ from clusters. Mass budget: Clusters imply M/Lc = 300 600h(M/L)⊙,hence Gas mass fraction 20-30 % of total mass 2 more realistic values2 are 1 −2 kBT • Star mass fraction < 10 % of mass Since p = ρ vx , this implies vx = v = • 3 m Remaining: dark matter 60-70 % ! Ω0 =0.2 0.4 • Now we find − dρkfromT/m clusters. GM(r)ρ B = dr − r2 k r2 dρT M(r)=− B mG ρ dr k TrrdρT k Tr d ln ρ d ln T = b = b − mG ρT dr mG !− d ln r − d ln r " Note how comparable this equation is to the isotropic spherical formula for stellar systems. Observations: example NGC 720 from analysis of the spectrum (continuum and lines) for NGC 720: T =7 106K × Method 6: From Gravitational Lensing Measure the mass of a galaxy cluster from the impact it has on light passing by the cluster Diagram here shows such a measurement using multiple images of same sources, but often measurement made using distortion of galaxy shapes From analyses of the x-ray light and the gravitational lensing of background galaxies around galaxy clusters, we find very similar masses for galaxy clusters. Bottom Line: What is composition of galaxy clusters by mass? Mass budget: • Star mass fraction < 10 % of mass • Gas mass fraction 20-30 % of total mass • Remaining: dark matter 60-70 % ! 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-31 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-32 7.6 Dark matter in universe as a whole BT 10.3-1 What does this imply about the matter density of the universe? The emissivity of light in the universe in the V band is 8 −3 In assessing the amountj0 =1 of matter.7 0in. the610 universe,hL⊙ usuallyMpc this is done relative to the density of matter in the universe± necessary to stop the expansion of the universe. The critical density for the universe is: The critical density of the universe (needed to stop expansion at time infinity) is 3H2 ρ = 0 c 8πG We then compare the observed density of matter in the universe ρ0 with the This densitycritical density is enough ρc using the to parameter stop the Ω0. expansionWe define it as atfollows:t = infinity Ω0 = ρ0 /ρc Define the density parameter Ω0 = ρ0/ρc,whereρ0 is the actual density of the universe Express Ω0 in terms of M/L ratio of galaxies: −4 −1 Similar mass determinations from gas . Temperatures Ω0 =6.110 h M/L/(M/L)⊙ can now be measured easily with X-ray satellites. Re- sults: The critical mass-to-light ratio for Ω =1is given by Confirms high masses • M/Lc = 1600h(M/L)⊙ Mass budget: Clusters imply M/Lc = 300 600h(M/L)⊙,hence Gas mass fraction 20-30 % of total mass more realistic values are − • Star mass fraction < 10 % of mass • Remaining: dark matter 60-70 % ! Ω0 =0.2 0.4 • − from clusters. 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-31 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-32 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-31 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-32 7.6 Dark matter in universe as a whole BT 10.3-1 7.6 Dark matter in universe as a whole BT 10.3-1 The emissivity of light in the universe in the V band is The emissivity of light in the universe8 in the− V3 band is j0 =1.7 0.610 hL⊙Mpc ± 8 −3 j0 =1.7 0.610 hL⊙Mpc ± The critical density for the universe is: The critical density for the universe is: 2 32H0 ρc =3H ρ = 80πG c 8πG ThisWhat density does is enough this toimply stop theabout expansion the atmattert = Thisinfinity density is enough to stop the expansion at t = infinity density of the universe? Define the density parameter Ω = ρ /ρ ,whereρ is Define the density parameter Ω0 =0 ρ0/ρ0c,wherec ρ0 is 0 thethe actual actual density density of of the the universe universe We can express Ω0 in terms of the mass to light ratio of galaxies (the basic ExpressExpressΩΩ00inin terms termsbuilding of of M/LblocksM/L ofratio theratio universe): of galaxies: of galaxies: −4−4−1−1 SimilarSimilar mass mass determinations determinations from from gas gas . Temperatures . Temperatures ΩΩ0 0=6=6.110.110 h hM/L/M/L/(M/L(M/L)⊙ )⊙ can nowcan now be measured be measured easily easily with with X-ray X-ray satellites. satellites. Re- Re- sults:sults: WhatTheThe value critical critical of the mass-to-lightmass mass-to-light to light ratio ratio of ratio galaxies for forΩ do=1Ω we=1is need givenis for given the by universe by to Confirms high masses be at the critical density? Confirms• high masses M/Lc = 1600h(M/L)⊙ • M/Lc = 1600h(M/L)⊙ Mass budget: Mass budget: Clusters imply M/Lc = 300 600h(M/L)⊙,hence Gas mass fraction 20-30 % of total mass SincemoreClusters the observed realistic imply mass-to-light valuesM/L arec ratios= 300 of− galaxies600 ish 200-400(M/L )(M/L)⊙,hence at large radii, Gas• massStar mass fraction fraction 20-30< 10 %% of of total mass mass more realistic values are − ⦿ • this implies Ω0 = 0.2-0.4. Star• Remaining: mass fraction dark< matter10 % of 60-70 mass % ! • • Ω0 =0.2 0.4 Remaining: dark matter 60-70 % ! Ω0 =0−.2 0.4 • − from clusters. from clusters. 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-33 29-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c7-34 What does this imply about the matter other indicators give: Measurements on the largest scales concern those density of the universe? caused by inflows around clusters (BT 10.3-2) matter falls into clusters of galaxies. Assume that the density contrast of the cluster is the same as that of the light ρ j cluster = cluster = δ ρuniverse juniverse where j is emissivity. The total mass of the cluster is proportional to Ω0 3 Mass R ρ δρ δΩ0 ∝ ∝ universe ∝ Hence, the acceleration of galaxies outside the cluster will depend on Ω0. As we probe larger physical scales outside galaxies, we find a lot of matter! Example: determine Ω0 from “Virgo centric infall” All analyses suggest there is a lot of mass outside the centers of galaxies... ⇒ Dark Matter Performing these analyses at very large scales, we can try to measure the mass density of the universe Picture of a Galaxy Huge Amount (dark matter) Billions of Stars (visible matter) What distinguishes a galaxy from a globular cluster? What distinguishes a galaxy from a globular cluster? A galaxy is a collection of stars, baryonic material (gas, etc), and dark matter that are generally held together by the gravitational forces from the dark matter. The concept of being gravitationally bound to a dark matter halo is an important one for galaxies. What about the stars in a galaxy? How do they behave? 1. Galaxies omit radiation in just about every wavelength you can imag- ine di↵erent processes at di↵erent wavelengths, from (optical) stellar light of normal stars to relativistic electrons (radio, high energy) to hot accretion disks (UV, Xray) to bremsstrahlung from hot gas to ... 2. Galaxies are usually classified based on optical imaging. Hubble se- quence: ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers. 3. The potential of a smooth density distribution can be calculated by an integral in 3 dimensions. The potential of a spherical system is simpler. Use the fact that the potential inside a spherical shell is constant (hence no force!). Outside of the shell, the potential is the same as that of a point-mass at the center of the shell with the same mass. Hence the force at a radius r is determined only by the mass inside r, as if the mass were at the center of the spherical distribution. 4. For a self-gravitating system, we can derive the virial theorem, 2K + W =0.Thisrelationcanbeusedtoestimatemasses,andscalingrelations. 5. Galaxy mass distributions can be derived from observed stellar ve- locities, gas rotation curves, hot X-ray gas, and many other methods These methods use the Jeans equations (moments of the Vlasov equation: see point 14), the relation between enclosed mass and circular velocity, and the equa- tions for hydrostatic equilibrium of gas (which are very similar to the Jeans equations.How might we characterize the dynamical properties of a galaxy? 6. The mass distributions of galaxies show strong evidence for dark mat- ter,How i.e., the might mass distributionwe expect continues such much objects further to out behave than the light.or Typical examples are the rotation curves of galaxies, hot X-ray halos of el- lipticals, large masses of clusters.evolve with time? 7. Equilibrium: Galaxies are in (rough) equilibrium: the dynamical time is typically 108 years - much shorter than the age of the universe. Assume: radius ~ 5 kpc / mass ~ 3 x 1010 solar masses velocities ~ (ɸ(R))1/2 (from virial relation) velocities ~ (GM/R)1/21 (from virial relation) 1 pc ~ 3.09 x 1018 cm => velocities ~ 164 km/s, 1 solar mass ~ 2 x 1033 g G ~ 6.67 x 10-8 cm3 g-1 s-2 tdyn = 2 (radius) / velocity tdyn = 2 (5 kpc) / (164 km/s) ~ 6e7 years REVIEW 8. Stars dont collide calculate from typical density, velocity, and cross- section of stars. To understand equilibrium, one can ignore stellar collisions. The main challenge is to understand a system with > 1011 particles moving in their common gravitational potential. 9. Individual interactions between stars can be ignored. Hubble sequence: ellipticals,Let’s S0s,quantify spirals this (barred by estimating and unbarred), the time Sa-Sd,scale for irregulars, collisions: mergers The relaxation time is the timeradius needed (galaxy) to ~ a ↵5ect kpc the motion of the star by indi- vidualLet’s star-starassume: interactions.# of stars This (galaxy) timescale ~ 1 isx 10 very10 stars high (1016 years) - too high to be relevant forstar galaxies. diameter As a~ re- 1.4 sult, x 10 we6 km can describe the potential as a smooth, 3D, potentialall stars⇢ (have~x)), a instead mass equal of the to potential the sun produced by 1011 particles. Similarly, we can derive the phase-space density (a 6D function In 6e7 years, a typical star crosses pathes with N stars: f(~x,~v)), by calculating the average density in space and velocity of the 1011 or more particles. N = (Distance Covered)(# stars / size3) Collisions per crossing time = N πr2 10. Galaxy formation is driven by the formation of the halo At very high = (10 kpc)(1e10 stars/(5 kpc)3)π(1.4e6 km)2 = 5e-12 per crossing time redshift, the density of the universe is close to critical. Any perturbation will drive the density= to 5e-12 above / 6e7 critical. ~ 8e-20 Regions / year with density above critical density will collapseHence stars and collide become with self-gravitating. each other very rarely! 11. Galaxies form inside the collapsed halos, as the gas can cool radia- tively, become self-gravitating, and form stars. 12. The equations of motion for a self-gravitating system are simple. They imply that any system can be scaled in mass, size, and velocity if the scaling factors satisfy one single relaiton This implies that any equilibrium model can be scaled in mass and size ! 13. In practice, the halos grow in mass due to merging with smaller halos. Merging is very e↵ective in the universe, due to the fact that the halos are extended and not point-sources. 14. The time-evolution of the distribution function is defined by the dis- tribution function at that time, spatial derivatives, and the gradients in the potential (Vlasov-Equation). This follows directly from the conservation of stars. 2 2-10-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c3b-1 2-10-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c3b-2 Galactic Dynamics - Continued Realistic densities decrease after some radius, so that the force will be determined by the density distribu- 3.6 Time scales (BT 4 to start 4.1) tion on a galactic scale (characterized by the half mass radius). dynamical timescale, particle interaction timescale 3.7 Relaxation time Short range encounters do not dominate Is gravitational force dominated by short or long range → encounters? (N.B. in a gas, only short range forces are Approximate force field with a smooth density ρ(x) relevant). instead of point masses. REVIEWIn a galaxy, the situation is different. Contrary of situation in gas: only consider long 8b. In contrast to situations with fluids, the • range encounters (long range scale of the galaxy) force on individualConsider stars does force not withcome which stars in cone attract star in ∼ apex of cone. primarily from its immediate neighbors, but from stars at all distances in the galaxy Assume all stars have mass m. Analyze perturbations Force from region of galaxy on a star due to the fact that density is not smooth, but consists of individual stars. Simplify, and look first at single = Gm(ρr2drdΩ)/r2 star-star encounter. = GmρdrdΩ Force is independent of r! This means that a given star in a galaxy What is effect of a single encounter with point mass on feels essentially the same force from motion of star? stars at 1 kpc and stars at 4 kpc. Exact: BT 7.1: hyperbolic Keplerian encounter • § This is very different from Estimate: straight line trajectory past stationary hydrodynamics where short range 2 • perturber pressure forces dominate! Force 1/r , with r the distance from apex. If ρ is ∼ almost constant,BT4: pages then 187 the mass in a shell with width dr increases as r2dr. Hence differential force is constant at each r,andwe have to integrate all the way out to obtain the total force.