ASTR 1020 1st Edition Lecture 10Units and Dimensional AnalysisUnits are more important that numbers!Tracking units can tell you the equation you want in most cases1. Write down units you have and the units you want2. Build an equation: How can I get the units I want with the units I have?3. Fill in the numbers and do the math4. Check to make sure your answer makes sense Ex: Lifetime of the Sun- The Sun has a mass of 2x10^30 kg. 70% of this mass is hydrogen but only about 13% of that hydrogen is found in the core and available for fusion. What is the total mass of fuel for the Sun?HAVE:Mass kg / percents [no unit]WANT: mass kg[kg] x % = [kg]2 x 10^30[kg] x .7 x .13 = 1.8 x 10^29 [kg]- Fusing one kg of hydrogen into helium releases 6.3x10^14 joules of energy. How many joules does you answer in part a represent? HAVE: mass [kg] / energy per mass [J/kg]WANT: energy [J][kg]x[J/kg] = [J]1.8 x 10^29 [kg] x 6.3 x 10^14 [J/kg] = 1.1 x 10^44 [J]- We know that the luminosity of the Sun (the energy emitted at the surface) must be equal to the amount of energy being generated in the core. The luminosity of the sun is measured to be 3.8 x 10^26 watts. 1 watt = 1 joule/second. How many kilograms of fuel per second does this represent? HAVE: energy per mass [J/kg] / energy per time [J/sec]WANT: mass per time [kg/sec]Ex: Parallax- Assume that stars are distributed are distributed uniformly in space with a density of .1 stars per cubic parsec. How many stars would you expect to have accurate parallax measurements from the ground? How many from Gaia? - VOLUME of a sphere: 4/3 pi d^3HAVE: distance [pc] /density [stars/pc^3]WANT: number of stars[pc]^3 x [stars/pc^3] = stars4/3 pi (100[pc])^3 x .1 [stars/pc^3] = 41,866.67 starsBirth of Stars Part 2If we see massive stars, we are seeing star-forming “stellar nurseries” Gravitational/Kinetic/ThermalTotal energy remains constant, but changes formsPressure of Gases- Pressure of a gas comes from the gas particles bouncing off a surfaceIn a balloon: Pressure of gas inside=pressure of gas outsideP = n K T “Ideal Gas Law:” (thermal pressure)- Radiation –light pressure- Magnetic –magnetic exert outward pressure- Degeneracy –electrons crowding together GravityUniversal Law of Gravitation tells us the force of gravity between any two massesFg = G (M1M2/d^2)GRAVITY VS PRESSURE- Gas atoms pull together via gravitational force- Gas atoms push apart due to thermal pressure- Gravity contracts clouds  density rises, pressure rises- Equilibrium –gravity and pressure in balance (gravity then starts to win)“Jeans Mass” determines if the cloud will collapse or not (will gravity or pressure win?)- If cloud mass is greater than the Jeans Mass, Gravity overcomes pressure- Molecular cloud (T~30 K, n=300 atoms/cm^3) must have > 171 Msun to collapse- Gas cools via radiation (IR, radio photons). Keeps temperature constant as densityincreases (gravity keeps winning) What starts the collapse?- Something needs to change T, n, or M to cause clouds to collapse- Supernova shockwave? Colliding clouds? Turbulence in galaxy? Spontaneous cooling?Molecular Cloud Cores- Molecular Clouds (100s to 1000s Msun)- Cloud=“lumpy” –some regions collapse faster (each will form a few stars)Stages of collapse- Cloud starts to collapse due to its own gravity- Spins faster, may fragment (conservation of angular momentum)- Radiates energy as it collapses (energy)- Flattens into a disk (momentum)Accretion Disk- Accrete –to grow by accumulation- Common phenomenon: solar systems, black holes, galaxies all form disks- Angular momentum is lost (magnetic fields?)- Potential energy kineticthermal- Proto-planetary diskCloud CoreProtostar- Meanwhile, center of disk is getting very hot and very dense- Densest parts of cloud become opaque. Light can’t escape. Heat trappedProtostar ContractionOnce a protostar becomes opaque, it can continue to radiate from its surface.Gravitational energythermal energy as protostar contracts. Some energy radiated, somestays in protostarA gas sphere held together by its own gravity contracts as it radiates, growing even hotter1. Cloud collapses to form protostars, heating up as it shrinks2. Collapse continues, temperature stabilizes as convection circulates energy outwards (moves downwards)3. As core temperatures reach million K, fusion beings but our protostar is still not stable, still collapsing (moves left on HR diagram)4. HHe fusion begins. Main Sequence, sheds cocoon of gas. Stellar thermostat keeps luminosity, temperature stable How long does it take?- OB stars can live their entire lives before M-dwarfs even reach the main sequence- Protostars of different masses follow different life tracks towards the main sequenceInitial mass FunctionHow much of the mass of a star forming cloud goes into forming what kind/mass of star?For every massive O-star that is born, there are ~200 low-mass M-stars also born!-Most stars in the galaxy are low-mass, main sequence starsWhy so few massive stars?Massive stars require massive Molecular Cloud coresProbably has to do with scale of turbulent motion within clouds***200 times more M stars, but each is 10,000,000 times fainter than a single O star*****Massive blue stars dominate the