# Lesson 3: Isothermal Hydrostatic Spheres. B68: a self-gravitating stable cloud. Hydrostatic self-gravitating spheres. P = "kt 2.

Save this PDF as:

Size: px
Start display at page:

Download "Lesson 3: Isothermal Hydrostatic Spheres. B68: a self-gravitating stable cloud. Hydrostatic self-gravitating spheres. P = "kt 2."

## Transcription

1 Lesson 3: Isothermal Hydrostatic Spheres B68: a self-gravitating stable cloud Bok Globule Relatively isolated, hence not many external disturbances Though not main mode of star formation, their isolation makes them good test-laboratories for theories Hydrostatic equilibrium: Assumptions: isothermal cloud spherical symmetry molecular material (µ=.3) dp(r) GM (r)#(r) = " dr r Equation of state (ideal gas): Gravitational potential: P = "kt = "c s µm p dm (r) = 4"r #(r) dr 1

2 The boundary condition problem These equations can be directly integrated from the center, choosing ρ(0). But if we want to construct a cloud with given mass and given external pressure, we must try many ρ(0) to get a match. Example solutions are shown on the next slide. Numerical solutions: Numerical solutions: Plotted logarithmically (which we will usually do from now on) Bonnor-Ebert Sphere

3 Numerical solutions: Different starting ρ o : a family of solutions Numerical solutions: Singular isothermal sphere (limiting solution) Numerical solutions: Boundary condition: Pressure at outer edge = pressure of GMC 3

4 Numerical solutions: Another boundary condition: Mass of clump is given Must replace One boundary ρ c inner condition BC with one too of many outer BCs Summary of BC problem: For inside-out integration the paramters are ρ c and r o. However, the physical parameters are M and P o We need to reformulate the equations: Write everything dimensionless Consider the scaling symmetry of the solutions All solutions are scaled versions of each other 4

5 Scale-free formulation dp(r) GM (r)#(r) = " dr r d"(r) "(r)dr = d ln" GM (r) = # dr r c s " = " c e #\$ 1/ % " = 4#G\$ ( c ' * r & c s ) " = # /c s % 1 d ) M (r) = #(r) ' 4"r dr ' & *, \$ 1 - M (r) = \$ r dp(r) 4"r -r (' #(r)g dr + ' r #(r)g -P(r) = #(r) -r Scale-free formulation substituting ξ and Ψ yields the Lane-Emden equation 1 d [ " d" " ] d# d" = e\$# "(# = 0) = 0; d" d# #=0 = 0 For M(r) we can write: r % 1 ( M (r) = \$ 4"s #(s)ds = ' * & 4"# 0 ) 0 1/ 3/ cs % ( ' * & G ) + d, d+ = M (+) Scale-free formulation Now we consider models with mass of the cloud M(R) where R is at the edge of the cloud, corresponding to ξ=ξ 1. We get: \$ 1 ' M (R) = & ) % 4"# 0 ( 1/ 3/ cs \$ ' d+ \$ & ) * 1 = c ' s d+ & ) R* 1 % G ( d* *1 % G ( d* *1 and therefore: GM (R) R = c s 1 " 1 d# d" "1 5

6 Scale-free formulation Similarly: 8 P(R) = "c c s = s 4#M (R)G \$ & 4 d% ) 3 1 ( + ' d\$ * " c 4#" c R 3 " 0 3M (R) = \$ 1 3( d% /d\$) \$1 e,% 1 \$ 1 Numerical calculations can be used to solve the Lane-Emden equation and give R, P(R) and ρ c /ρ 0 Solutions for Ψ and ρ/ρ c Scale-free formulation It turns out that P(R) is maximal for ξ 1 =6.5 # R m = 0.76 c & s % ( \$ G" ' 1/ Compare with the expression for the Jeans length % # kt ( " J = ' * & µm H G\$ 0 ) 1/ 6

7 Interpretation of maximum pressure When P(R) increases, the central density ρ 0 will increase and R will decrease (fixed M). Cloud mass becomes more and more concentrated toward the center At P(R) =P max gravity is about to overtake If we increase the pressure, cloud starts to collapse and a smaller P(R) would be required for stability Start of an inside-out collapse. Dimensionless mass Dimensionless mass: 7

8 Stability of BE spheres Many modes of instability One is if dp o /dr o > 0 Run-away collapse, or Run-away growth, followed by collapse Dimensionless equivalent: dm/d(ρ c /ρ o ) < 0 unstable unstable Stability of BE spheres Maximum density ratio =1 / 14.1 Bonnor-Ebert mass m 1 = 1.18 Ways to cause BE sphere to collapse: Increase external pressure until M BE <M Load matter onto BE sphere until M>M BE 8

9 A simple numerical model Temperature: 30 K Outer radius: 5000 AU Initial condition: BE sphere with ρ c = 1.x10-17 g/cm 3 ρ(r) A simple numerical model A more `realistic non-static model: Make perturbation, but keep mass the same. ρ(r) A simple numerical model ρ(r) Strong wobbles, but it remains stable 9

10 A simple numerical model Now add a little bit of mass (10%) to nudge it over the BE limit: ρ(r) Cloud collapses in a global way (not really inside-out) BE Sphere : Observations of B68 Alves, Lada, Lada 001 Magnetic field support / ambipolar diff. As mentioned in previous chapter, magnetic fields can partly support cloud and prevent collapse. Slow ambipolar diffusion moves fields out of cloud, which could trigger collapse. Models by Lizano & Shu (1989) show this elegantly: Magnetic support only in x-y plane, so cloud is flattened. Dashed vertical line is field in beginning, solid: after some time. Field moves inward geometrically, but outward w.r.t. the matter. 10

11 Magnetic field support In presence of B-field, the stability analysis changes. Magnetic fields can provide support against gravity. Replace Jeans mass with critical mass, defined as: M cr = 0.1 " \$ B ' \$ M R ' G 1/ #103 M sun & )& ) % 30µG( % pc( Magnetic field support Consider an initially stable cloud. We now compress it. The density thereby increases, but the mass of the cloud stays constant. Jeans mass decreases: M J " 1 # If no magnetic fields: there will come a time when M>M J and the cloud will collapse. But M cr stays constant (magnetic flux freezing) So if B-field is strong enough to support a cloud, no compression will cause it to collapse. Ambipolar diffusion Forces acting: f L = " 1 8# \$B f drift = n n m n n i v% in v drift 11

12 Drag force: Collisions between ions and neutrals. The rate of collisions between ions and neutrals per neutral atom is: n i v" in σ in elastic scattering coefficient v relative velocity of ions as seen from neutral rest frame n I number density of ions <..> average over distribution function ions Momentum transfer: m i ( v r i " v r # m n ) n & % ( \$ m n + m i ' Momentum transfer ions -> neutrals per unit volume " m f drag = n n m i % n \$ '( v r i ( v r n )n i v) in # m n + m i & with v i -v n =v drift and m n +m i m i f drag = n n m n n i v" in v drift Example: infinite cylinder of uniform density f grav = "G#r# % & f drift = n i m n n n v\$ in v drift ' v = drift "G# r n i n n m n v\$ in drift timescale t drift = r v drift assume low ionization rate, neutrals are dominated by H, He \$ " = ( n i m i + n n m n ) # n H m H 1+ 4 n ' He & ) % ( n H 1

13 Example: infinite cylinder of uniform density t drift = v" \$ in & #Gm H % for <vσ in > cm 3 s -1 n i n H ' 1 ) ( 1+ 4n He /n H ( ) \$ t drift " 5 #10 13 & % n i n H HI cloud: n i /n H Ambipolar diffusion not important Dense cloud: n i /n H t drift ~ 5 x 10 6 yr t drift >>t ff, so ambipolar diffusion not important during collapse ' ) years ( Ambipolar diffusion Magnetic pressure builds up during cloud contraction. Ambipolar diffusion acts to reduce magnetic pressure Contraction continues Once collapse set in, magnetic field remains frozen into the matter. 13

### Heating & Cooling in Molecular Clouds

Lecture 8: Cloud Stability Heating & Cooling in Molecular Clouds Balance of heating and cooling processes helps to set the temperature in the gas. This then sets the minimum internal pressure in a core

### Dinamica del Gas nelle Galassie II. Star formation

Dinamica del Gas nelle Galassie II. Star formation Overview on ISM Molecular clouds: composition and properties. Plasmas Charge neutrality, infinite conductivity; Field freezing; Euler equation with magnetic

### Three-dimensional Simulation of Magnetized Cloud Fragmentation Induced by Nonlinear Flows and Ambipolar Diffusion

accepted by Astrophysical Journal Letters Three-dimensional Simulation of Magnetized Cloud Fragmentation Induced by Nonlinear Flows and Ambipolar Diffusion Takahiro Kudoh 1 and Shantanu Basu 2 ABSTRACT

### IV. Molecular Clouds. 1. Molecular Cloud Spectra

IV. Molecular Clouds Dark structures in the ISM emit molecular lines. Dense gas cools, Metals combine to form molecules, Molecular clouds form. 1. Molecular Cloud Spectra 1 Molecular Lines emerge in absorption:

### Sound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8

References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that

### Planetesimal Dynamics Formation of Terrestrial Planets from Planetesimals

Planetesimal Dynamics Formation of Terrestrial Planets from Planetesimals Protoplanetary disk Gas/Dust Planetesimals...... 10 6 yr 10 5-6 yr Protoplanets 10 7-8 yr Terrestrial planets Eiichiro Kokubo National

### Lecture 19: Planet Formation I. Clues from the Solar System

Lecture 19: Planet Formation I. Clues from the Solar System 1 Outline The Solar System:! Terrestrial planets! Jovian planets! Asteroid belt, Kuiper belt, Oort cloud Condensation and growth of solid bodies

### DYNAMICS OF GALAXIES

DYNAMICS OF GALAXIES 2. and stellar orbits Piet van der Kruit Kapteyn Astronomical Institute University of Groningen the Netherlands Winter 2008/9 and stellar orbits Contents Range of timescales Two-body

### Cloud Formation, Evolution and Destruction

Chapter 4 Cloud Formation, Evolution and Destruction We now begin to trace the journey towards a star. How long does this take? The answer is surprisingly short: a good many clouds already contain new

### A i A i. µ(ion) = Z i X i

Lecture 2 Review: calculation of mean atomic weight of an ionized gas (µ) Given a mass fraction X i (or abundance) for an ionic (or atomic) species with atomic weight A i, we can can calculate µ by: For

### 8 Radiative Cooling and Heating

8 Radiative Cooling and Heating Reading: Katz et al. 1996, ApJ Supp, 105, 19, section 3 Thoul & Weinberg, 1995, ApJ, 442, 480 Optional reading: Thoul & Weinberg, 1996, ApJ, 465, 608 Weinberg et al., 1997,

### Our Galaxy, the Milky Way

Our Galaxy, the Milky Way In the night sky, the Milky Way appears as a faint band of light. Dusty gas clouds obscure our view because they absorb visible light. This is the interstellar medium that makes

### Lecture 3 Properties and Evolution of Molecular Clouds. Spitzer space telescope image of Snake molecular cloud (IRDC G11.11-0.11

Lecture 3 Properties and Evolution of Molecular Clouds Spitzer space telescope image of Snake molecular cloud (IRDC G11.11-0.11 From slide from Annie Hughes Review CO t in clouds HI: Atomic Hydrogen http://www.atnf.csiro.au/research/lvmeeting/magsys_pres/

### Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

### Chapter 7 Neutron Stars

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

### School of Biotechnology

Physics reference slides Donatello Dolce Università di Camerino a.y. 2014/2015 mail: donatello.dolce@unicam.it School of Biotechnology Program and Aim Introduction to Physics Kinematics and Dynamics; Position

### On a Flat Expanding Universe

Adv. Studies Theor. Phys., Vol. 7, 2013, no. 4, 191-197 HIKARI Ltd, www.m-hikari.com On a Flat Expanding Universe Bo Lehnert Alfvén Laboratory Royal Institute of Technology, SE-10044 Stockholm, Sweden

### Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

### Domain walls may form in the early universe during a phase transition involving the spontaneous breaking of a discrete symmetry. If the mass density (

UTPT-93-18 hep-ph/9307359 July 1993 Cosmic Balloons B. Holdom 1 Department of Physics University of Toronto Toronto, Ontario Canada M5S 1A7 ABSTRACT Cosmic balloons, consisting of relativistic particles

### Chapter 4. Electrostatic Fields in Matter

Chapter 4. Electrostatic Fields in Matter 4.1. Polarization A neutral atom, placed in an external electric field, will experience no net force. However, even though the atom as a whole is neutral, the

### 4.1 Momentum equation of the neutral atmosphere

Chapter 4 Dynamics of the neutral atmosphere 4.1 Momentum equation of the neutral atmosphere Since we are going to discuss the motion of the atmosphere of a rotating planet, it is convenient to express

### L3: The formation of the Solar System

credit: NASA L3: The formation of the Solar System UCL Certificate of astronomy Dr. Ingo Waldmann A stable home The presence of life forms elsewhere in the Universe requires a stable environment where

### ENERGY TRANSPORT WITHIN A STAR

M. Pettini: Structure and Evolution of Stars Lecture 8 ENERGY TRANSPORT WITHIN A STAR 8.1 Introduction Up to now, we have considered how energy is generated within the interior of stars by the processes

### AS CHALLENGE PAPER 2014

AS CHALLENGE PAPER 2014 Name School Total Mark/50 Friday 14 th March Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Marks allocated for each question

### The Origin of the Solar System and Other Planetary Systems

The Origin of the Solar System and Other Planetary Systems Modeling Planet Formation Boundary Conditions Nebular Hypothesis Fixing Problems Role of Catastrophes Planets of Other Stars Modeling Planet Formation

### = = GM. v 1 = Ωa 1 sin i.

1 Binary Stars Consider a binary composed of two stars of masses M 1 and We define M = M 1 + and µ = M 1 /M If a 1 and a 2 are the mean distances of the stars from the center of mass, then M 1 a 1 = a

### Chapter 8 Steady Incompressible Flow in Pressure Conduits

Chapter 8 Steady Incompressible Flow in Pressure Conduits Outline 8.1 Laminar Flow and turbulent flow Reynolds Experiment 8.2 Reynolds number 8.3 Hydraulic Radius 8.4 Friction Head Loss in Conduits of

### PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

### Gauss s Law for Gravity

Gauss s Law for Gravity D.G. impson, Ph.D. Department of Physical ciences and Engineering Prince George s Community College December 6, 2006 Newton s Law of Gravity Newton s law of gravity gives the force

### MAE 20 Winter 2011 Assignment 3 solutions

MAE 20 Winter 2011 Assignment 3 solutions 4.3 Calculate the activation energy for vacancy formation in aluminum, given that the equilibrium number of vacancies at 500 C (773 K) is 7.57 10 23 m -3. The

### Ay 20 - Lecture 9 Post-Main Sequence Stellar Evolution. This file has many figures missing, in order to keep it a reasonable size.

Ay 20 - Lecture 9 Post-Main Sequence Stellar Evolution This file has many figures missing, in order to keep it a reasonable size. Main Sequence and the Range of Stellar Masses MS is defined as the locus

### Galaxy Formation. Leading questions for today How do visible galaxies form inside halos? Why do galaxies/halos merge so easily?

8-5-2015see http://www.strw.leidenuniv.nl/ franx/college/ mf-sts-2015-c9-1 8-5-2015see http://www.strw.leidenuniv.nl/ franx/college/ mf-sts-2015-c9-2 Galaxy Formation Leading questions for today How do

Steady Heat Conduction In thermodynamics, we considered the amount of heat transfer as a system undergoes a process from one equilibrium state to another. hermodynamics gives no indication of how long

### Lecture 7 Formation of the Solar System. Nebular Theory. Origin of the Solar System. Origin of the Solar System. The Solar Nebula

Origin of the Solar System Lecture 7 Formation of the Solar System Reading: Chapter 9 Quiz#2 Today: Lecture 60 minutes, then quiz 20 minutes. Homework#1 will be returned on Thursday. Our theory must explain

### How did the Solar System form?

How did the Solar System form? Is our solar system unique? Are there other Earth-like planets, or are we a fluke? Under what conditions can Earth-like planets form? Is life common or rare? Ways to Find

### Lecture 10 Formation of the Solar System January 6c, 2014

1 Lecture 10 Formation of the Solar System January 6c, 2014 2 Orbits of the Planets 3 Clues for the Formation of the SS All planets orbit in roughly the same plane about the Sun. All planets orbit in the

### The Layout of the Solar System

The Layout of the Solar System Planets fall into two main categories Terrestrial (i.e. Earth-like) Jovian (i.e. Jupiter-like or gaseous) [~5000 kg/m 3 ] [~1300 kg/m 3 ] What is density? Average density

### Giant Molecular Clouds

Giant Molecular Clouds http://www.astro.ncu.edu.tw/irlab/projects/project.htm Galactic Open Clusters Galactic Structure GMCs The Solar System and its Place in the Galaxy In Encyclopedia of the Solar System

### KINETIC MOLECULAR THEORY OF MATTER

KINETIC MOLECULAR THEORY OF MATTER The kinetic-molecular theory is based on the idea that particles of matter are always in motion. The theory can be used to explain the properties of solids, liquids,

### Proceedings of the NATIONAL ACADEMY OF SCIENCES

Proceedings of the NATIONAL ACADEMY OF SCIENCES Volume 55 * Number 1 * January 15, 1966 DYNAMICS OF SPHERICAL GALAXIES, II* BY PHILIP M. CAMPBELL LAWRENCE RADIATION LABORATORY, LIVERMORE, CALIFORNIA Communicated

### Natural Convection. Buoyancy force

Natural Convection In natural convection, the fluid motion occurs by natural means such as buoyancy. Since the fluid velocity associated with natural convection is relatively low, the heat transfer coefficient

### Elliptical Galaxies. Houjun Mo. April 19, 2004. Basic properties of elliptical galaxies. Formation of elliptical galaxies

Elliptical Galaxies Houjun Mo April 19, 2004 Basic properties of elliptical galaxies Formation of elliptical galaxies Photometric Properties Isophotes of elliptical galaxies are usually fitted by ellipses:

### Protobinaries. von Cornelia Weber, Bakk.rer.nat. Donnerstag, 23. Mai 13

Protobinaries von Cornelia Weber, Bakk.rer.nat. Overview Motivation Molecular Clouds Young Stellar Objects Multiplicity of YSO Orion Molecular Cloud Aims of my thesis Motivation Binary and Multiple system

### Degeneracy of Electrons

Properties of Degenerated Fermi-Gas in Astrophysics Hsin-Yu Chen Introduction The degenerated fermi-gas is a dominated component in a highly dense region in Astronomy, such as the center of a white dwarf.

Chapter 6 Atmospheric Aerosol and Cloud Processes Spring 2015 Cloud Physics Initiation and development of cloud droplets Special interest: Explain how droplet formation results in rain in approximately

### Chapter 15.3 Galaxy Evolution

Chapter 15.3 Galaxy Evolution Elliptical Galaxies Spiral Galaxies Irregular Galaxies Are there any connections between the three types of galaxies? How do galaxies form? How do galaxies evolve? P.S. You

### 12.307. 1 Convection in water (an almost-incompressible fluid)

12.307 Convection in water (an almost-incompressible fluid) John Marshall, Lodovica Illari and Alan Plumb March, 2004 1 Convection in water (an almost-incompressible fluid) 1.1 Buoyancy Objects that are

### Patterns in the Solar System. Patterns in the Solar System. ASTR 105 The Solar System

ASTR 105 The Solar System 1. Orderly motions 2.Two kinds of planets 3.Two kinds of small bodies 4.Exceptions to the rules Today: Group Lab at the end of class Next THURSDAY 03/10: First Group Project Orderly

### Imaging a sphere of mass M, constant density ρ, radius R, and absorption by mass of κ λ. The the optical depth is. 3M τ = κ λ ρr = κ λ 4πR 3R = κ 3M

1 Lectures 18 and 19 Optical Depth vs. Density Imaging a sphere of mass M, constant density ρ, radius R, and absorption by mass of κ λ. The the optical depth is 3M τ = κ λ ρr = κ λ 4πR 3R = κ 3M λ (1)

### Nuclear fusion in stars. Collapse of primordial density fluctuations into galaxies and stars, nucleosynthesis in stars

Nuclear fusion in stars Collapse of primordial density fluctuations into galaxies and stars, nucleosynthesis in stars The origin of structure in the Universe Until the time of formation of protogalaxies,

### Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

### Physics 2112 Unit 6: Electric Potential

Physics 2112 Unit 6: Electric Potential Today s Concept: Electric Potential (Defined in terms of Path Integral of Electric Field) Unit 6, Slide 1 Big Idea Last time we defined the electric potential energy

### Exemplar Problems Physics

Chapter Eight GRAVITATION MCQ I 8.1 The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration

### UNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.

UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Constituent exam in: AST4320 Cosmology and Extragalactic Astronomy Day of examination: Thursday 8. October 2015 Examination hours: 10.00 13.00

### Numerical Analysis of the Jeans Instability

June 15, 2010 Background Goal Refine our understanding of Jeans Length and its relation to astrophysical simulations. Currently, it is widely accepted that one needs four cells per Jeans Length to get

### 1. Gravitational forces and potentials (BT 2-2.1) Intermezzo: divergence and divergence theorem (BT: B.3) 2. Potential for spherical systems (BT 2.

Overview 1. Gravitational forces and potentials (BT 2-2.1) Intermezzo: divergence and divergence theorem (BT: B.3) Poisson equation Gauss s theorem Potential energy 2. Potential for spherical systems (BT

### The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C. = 2(sphere volume) = 2 = V C = 4R

3.5 Show that the atomic packing factor for BCC is 0.68. The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C Since there are two spheres associated

### Welcome to the first lesson of third module which is on thin-walled pressure vessels part one which is on the application of stress and strain.

Strength of Materials Prof S. K. Bhattacharya Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture -15 Application of Stress by Strain Thin-walled Pressure Vessels - I Welcome

### Solar Ast ro p h y s ics

Peter V. Foukal Solar Ast ro p h y s ics Second, Revised Edition WI LEY- VCH WILEY-VCH Verlag Co. KCaA Contents Preface 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.2.3 2.3

### Summary: Four Major Features of our Solar System

Summary: Four Major Features of our Solar System How did the solar system form? According to the nebular theory, our solar system formed from the gravitational collapse of a giant cloud of interstellar

### Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

### Forces on the Rocket. Rocket Dynamics. Equation of Motion: F = Ma

Rocket Dynamics orces on the Rockets - Drag Rocket Stability Rocket Equation Specific Impulse Rocket otors Thrust orces on the Rocket Equation of otion: = a orces at through the Center of ass Center of

### Lecture 14. Introduction to the Sun

Lecture 14 Introduction to the Sun ALMA discovers planets forming in a protoplanetary disc. Open Q: what physics do we learn about the Sun? 1. Energy - nuclear energy - magnetic energy 2. Radiation - continuum

### Modern Atomic Theory

Reading: Ch. 9, sections 1-4 Ch. 7, sections 5-6 (lec) Ch. 7, sections 1-3 (lab) Modern Atomic Theory Homework: Chapter 9: 37*, 39*, 41 Chapter 7: 59, 61*, 63, 65 (lec.) Chapter 7: 39, 41, 43, 47 (lab)

### FREESTUDY HEAT TRANSFER TUTORIAL 3 ADVANCED STUDIES

FREESTUDY HEAT TRANSFER TUTORIAL ADVANCED STUDIES This is the third tutorial in the series on heat transfer and covers some of the advanced theory of convection. The tutorials are designed to bring the

### Nonlinear evolution of unstable fluid interface

Nonlinear evolution of unstable fluid interface S.I. Abarzhi Department of Applied Mathematics and Statistics State University of New-York at Stony Brook LIGHT FLUID ACCELERATES HEAVY FLUID misalignment

### 2.5 Physically-based Animation

2.5 Physically-based Animation 320491: Advanced Graphics - Chapter 2 74 Physically-based animation Morphing allowed us to animate between two known states. Typically, only one state of an object is known.

### EQUILIBRIUM AND ELASTICITY

Chapter 12: EQUILIBRIUM AND ELASTICITY 1 A net torque applied to a rigid object always tends to produce: A linear acceleration B rotational equilibrium C angular acceleration D rotational inertia E none

### The Evolution of GMCs in Global Galaxy Simulations

The Evolution of GMCs in Global Galaxy Simulations image from Britton Smith Elizabeth Tasker (CITA NF @ McMaster) Jonathan Tan (U. Florida) Simulation properties We use the AMR code, Enzo, to model a 3D

### Chapter 22: Electric Flux and Gauss s Law

22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we

### In order to solve this problem it is first necessary to use Equation 5.5: x 2 Dt. = 1 erf. = 1.30, and x = 2 mm = 2 10-3 m. Thus,

5.3 (a) Compare interstitial and vacancy atomic mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion. Solution (a) With vacancy diffusion,

### Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows

Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical

### In studying the Milky Way, we have a classic problem of not being able to see the forest for the trees.

In studying the Milky Way, we have a classic problem of not being able to see the forest for the trees. A panoramic painting of the Milky Way as seen from Earth, done by Knut Lundmark in the 1940 s. The

### 1 Wetting your feet. 2 Scaling. 8.298 Lies / Check your understanding: Solutions

1 Wetting your feet 1.1 Estimate how many liters are in a barrel of oil and how many barrels of oil the United States imports every year. A: A barrel may be a few feet high, so h 1m, and have a diameter

### Topic 3. Evidence for the Big Bang

Topic 3 Primordial nucleosynthesis Evidence for the Big Bang! Back in the 1920s it was generally thought that the Universe was infinite! However a number of experimental observations started to question

### Gravitational instabilities in protostellar discs and the formation of planetesimals

Gravitational instabilities in protostellar discs and the formation of planetesimals Giuseppe Lodato - Università degli Studi di Milano 17 February 2011 - Bologna Gravitational instabilities in protostellar

### Name Date Per Teacher

Reading Guide: Chapter 28.1 (read text pages 571-575) STRUCTURE OF THE SUN 1e Students know the Sun is a typical star and is powered by nuclear reactions, primarily the fusion of hydrogen to form helium.

### Motion of a Leaky Tank Car

1 Problem Motion of a Leaky Tank Car Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 8544 (December 4, 1989; updated October 1, 214) Describe the motion of a tank car initially

### A. Kinetic Molecular Theory (KMT) = the idea that particles of matter are always in motion and that this motion has consequences.

I. MOLECULES IN MOTION: A. Kinetic Molecular Theory (KMT) = the idea that particles of matter are always in motion and that this motion has consequences. 1) theory developed in the late 19 th century to

### Statistical Mechanics, Kinetic Theory Ideal Gas. 8.01t Nov 22, 2004

Statistical Mechanics, Kinetic Theory Ideal Gas 8.01t Nov 22, 2004 Statistical Mechanics and Thermodynamics Thermodynamics Old & Fundamental Understanding of Heat (I.e. Steam) Engines Part of Physics Einstein

### Be Stars. By Carla Morton

Be Stars By Carla Morton Index 1. Stars 2. Spectral types 3. B Stars 4. Be stars 5. Bibliography How stars are formed Stars are composed of gas Hydrogen is the main component of stars. Stars are formed

### Solar cycle. Auringonpilkkusykli. 1844 Heinrich Schwabe: 11 year solar cycle. ~11 years

Sun Solar cycle Auringonpilkkusykli 1844 Heinrich Schwabe: 11 year solar cycle ~11 years Auringonpilkkusykli Solar cycle Butterfly diagram: Edward Maunder 1904 New cycle Spots appear at mid-latitudes Migration

### PHYSICAL QUANTITIES AND UNITS

1 PHYSICAL QUANTITIES AND UNITS Introduction Physics is the study of matter, its motion and the interaction between matter. Physics involves analysis of physical quantities, the interaction between them

### A Guide to Calculate Convection Coefficients for Thermal Problems Application Note

A Guide to Calculate Convection Coefficients for Thermal Problems Application Note Keywords: Thermal analysis, convection coefficients, computational fluid dynamics, free convection, forced convection.

### 39th International Physics Olympiad - Hanoi - Vietnam - 2008. Theoretical Problem No. 3

CHANGE OF AIR TEMPERATURE WITH ALTITUDE, ATMOSPHERIC STABILITY AND AIR POLLUTION Vertical motion of air governs many atmospheric processes, such as the formation of clouds and precipitation and the dispersal

### AMBIPOLAR DIFFUSION REVISITED

RevMexAA (Serie de Conferencias), 36, 73 8 (29) AMBIPOLAR DIFFUSION REVISITED F. C. Adams,2 29: Instituto de Astronomía, UNAM - Magnetic Fields in the Universe II: From Laboratory and Stars to the Primordial

### Introduction to Nuclear Radiation 9/04. Purpose of the Experiment

Modern Physics Lab Introduction to Nuclear Radiation 9/04 Purpose of the Experiment - become familiar with detectors for radioactive decay products - apply statistical analysis techniques to data - understand

### Units and Dimensions in Physical Chemistry

Units and Dimensions in Physical Chemistry Units and dimensions tend to cause untold amounts of grief to many chemists throughout the course of their degree. My hope is that by having a dedicated tutorial

### ............... [2] At the time of purchase of a Strontium-90 source, the activity is 3.7 10 6 Bq.

1 Strontium-90 decays with the emission of a β-particle to form Yttrium-90. The reaction is represented by the equation 90 38 The decay constant is 0.025 year 1. 90 39 0 1 Sr Y + e + 0.55 MeV. (a) Suggest,

### Gauss's Law. Gauss's Law in 3, 2, and 1 Dimension

[ Assignment View ] [ Eðlisfræði 2, vor 2007 22. Gauss' Law Assignment is due at 2:00am on Wednesday, January 31, 2007 Credit for problems submitted late will decrease to 0% after the deadline has passed.

### Basic Principles in Microfluidics

Basic Principles in Microfluidics 1 Newton s Second Law for Fluidics Newton s 2 nd Law (F= ma) : Time rate of change of momentum of a system equal to net force acting on system!f = dp dt Sum of forces

### (1) The size of a gas particle is negligible as compared to the volume of the container in which the gas is placed.

Gas Laws and Kinetic Molecular Theory The Gas Laws are based on experiments, and they describe how a gas behaves under certain conditions. However, Gas Laws do not attempt to explain the behavior of gases.

### Problem #1 [Sound Waves and Jeans Length]

Roger Griffith Astro 161 hw. # 8 Proffesor Chung-Pei Ma Problem #1 [Sound Waves and Jeans Length] At typical sea-level conditions, the density of air is 1.23 1 3 gcm 3 and the speed of sound is 3.4 1 4

### Unit 8 Lesson 2 Gravity and the Solar System

Unit 8 Lesson 2 Gravity and the Solar System Gravity What is gravity? Gravity is a force of attraction between objects that is due to their masses and the distances between them. Every object in the universe

### Differential Balance Equations (DBE)

Differential Balance Equations (DBE) Differential Balance Equations Differential balances, although more complex to solve, can yield a tremendous wealth of information about ChE processes. General balance

### KINETIC THEORY. 1.Name any one scientist who explained the behavior of gases considering it to be made up of tiny particles.

KINETIC THEORY ONE MARK OUESTION: 1.Name any one scientist who explained the behavior of gases considering it to be made up of tiny particles. 2.Based on which idea kinetic theory of gases explain the

### Numerical Model for the Study of the Velocity Dependence Of the Ionisation Growth in Gas Discharge Plasma

Journal of Basrah Researches ((Sciences)) Volume 37.Number 5.A ((2011)) Available online at: www.basra-science -journal.org ISSN 1817 2695 Numerical Model for the Study of the Velocity Dependence Of the

### Chapter Biography of J. C. Maxwell Derivation of the Maxwell Speed Distribution Function

Chapter 10 10.1 Biography of J. C. Maxwell 10.2 Derivation of the Maxwell Speed Distribution Function The distribution of molecular speeds was first worked out by Maxwell before the development of statistical