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# Astrophysics 1 ASTR 3730

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This 10 page Class Notes was uploaded by Stephan Kuvalis on Thursday October 29, 2015. The Class Notes belongs to ASTR 3730 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/231962/astr-3730-university-of-colorado-at-boulder in Astronomy at University of Colorado at Boulder.

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Date Created: 10/29/15

Stellar structure For an isolated static spherically symmetric star four basic laws equations needed to describe structure 39 Conservation of mass 39 Conservation of energy at each radius the change in the energy flux equals the local rate of energy release o Equation of hydrostatic equilibrium at each radius forces due to pressure differences balance gravity o Equation of energy transport relation between the energy flux and the local gradient of temperature Basic equations are supplemented by o Equation of state pressure of a gas as a function of its density and temperature Opacity how transparent it is to radiation 39 Nuclear energy generation rate as fpT ASTR 3730 Fall 2003 Conservation of mass Let r be the distance from the center Density as function of radius is pr Let m be the mass inferiorto r then conservation of mass implies that dm 4m2pdr M Write this as a differential equation 1St stellar structure equa on ASTR 3730 Fall 2003 Equation of hydrostatic equilibrium PUd0 ConSIder small cylindrical element between radius r and radius r dr in the star rdr Surface area 2 d8 Mass Am Pr Mass of gas in the star at smaller gravity radii m mr Radial forces acting on the element GmAm 2 Iquot Gravity inward Fg gravitational constant G 667 x 10398 dyne cm2 g392 ASTR 3730 Fall 2003 Pressure net force due to difference in pressure between upper and lower faces Fp PrdS Pr drdS l d l PrdS Pr x dr dS dr d P drdS dr Mass of element Am pdrdS Applying Newton s second law Fma to the cylinder AmfFg Fp M d Pdrds T r2 dr acceleration O everywhere if star static ASTR 3730 Fall 2003 Setting acceleration to zero and substituting for Am GmpdrdS dP 2 r dr Equation of hydrostatic equilibrium 0 drdS 2nol stellar structure equation If we use enclosed mass as the dependent variable can combine these two equations into one dP dP dr Gm 1 X x dm dr dm r2 p 4m2p dP Gm alternate form of hydrostatic dm 4W4 equilibrium equation ASTR 3730 Fall 2003 dP Gm Properties of hydrostatic equilibrium equation d 2p 739 r 1 Pressure always decreases outward 2 Pressure gradient vanishes at r O 3 Condition at surface of star P O to a good first approximation 2 and 3 are boundary conditions for the hydrostatic equilibrium equation but why two ASTR 3730 Fall 2003 Virial theorem Can use the hydrostatic equation to derive the virial theorem which is very useful for estimating properties of stars Start with d P Gm dm 47174 Multiply both sides by volume V 4 3nr3 Vd if x G dm lG mdm 3 4m 3 r Now integrate over the whole star LHS gives by parts deP PV deV l ButPOatrRandVOatrO so this term vanishes ASTR 3730 Fall 2003 If we have a small mass dm at radius r gravitational potential energy is dQ 9Emn r Hence integrating RHS of previous equation over the star f l dmlfd9lg 3 r 3 3 where Q is the gravitational potential energy of the star ie the energy required to assemble the star by bringing gas from infinity very large radius Putting the pieces together f PdV 192 3 VltrRgt version of the 0 9 3 deV virial theorem 0 ASTR 3730 Fall 2003 With some assumptions about the pressure can progress further Often can write the pressure in the form P y 1pu o p is the density o u is the internal energy per unit mass per gram of gas o y is a constant example for an ideal monatomic gas y 5 3 where this is the ratio of the specific heat at constant pressure to that at constant volume Substitute this equation of state into the virial theorem we 0 23 fy 1p1udV pu has units of g cm393 x erg g erg cm393 it is the internal energy per unit volume ASTR 3730 Fall 2003 Integral of internal energy per unit volume over all volume in the star is just the total internal energy of the star U 0 Q 3y 1U gravitational potential total internal energy energy of the star of the star If in some application we can estimate eitherQ or U can use this relation to find the other ASTR 3730 Fall 2003

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