1B11 Foundations of Astronomy Sun (and stellar) Models

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1B11 Foundations of AstronomySun (and stellar)
Models

• Silvia Zane, Liz Puchnarewicz
• emp_at_mssl.ucl.ac.uk
• www.ucl.ac.uk/webct
• www.mssl.ucl.ac.uk/

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1B11 Physical State of the stellar (Sun) interior
Fundamental assumptions

• Although stars evolve, their properties change
  so slowly that at any time it is a good
  approximation to neglect the rate of change of
  these properties.
• Stars are spherical and symmetrical about their
  centre all physical quantities depend just on r,
  the distance from the centre..

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1B11 1) Equation of hydrostatic equilibrium.
Concept 1) Stars are self-gravitating bodies in
dynamical equilibrium ? balance of gravity and
internal pressure forces.

• Consider a small volume element at distance r
  from the centre, cross section S2?r, thickness dr

(1)
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1B11 2) Equation of distribution of mass.

• Consider the same small volume element at
  distance r from the centre, cross section S2?r,
  thickness dr

(2)
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1B11 First consequence upper limit on central P

• From (1) and (2)

At all points within the star rltRs hence
1/r4gt1/RS4
For the Sun Pcgt4.5 1013 Nm-24.5 108 atm
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1B11 Toward the E-balance equation The virial
theorem

• Thermal energy/unit volume unfkT/2(?/?mH)fkT/2
• Ratio of specific heats ?cP/cV(f2)/f
  (f3?5/3)

• U total thermal Energy ? total gravitational
  energy

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1B11 Toward the E-balance equation The virial
theorem
For a fully ionized gas ?5/3 and 2U?0 Total
Energy of the star EU ?

• E is negative and equal to ?/2 or U
• A decrease in E leads to a decrease in ? but an
  increase in U and hence T.
• A star, with no hidden energy sources, c.omposed
  of a perfect gas, contracts and heat up as it
  radiates energy

Stars have a negative heat capacity they heat
up when their total energy decreases.
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1B11 Toward the E-balance equation
Sources of stellar energy since stars lose
energy by radiation, stars supported by thermal
pressure require an energy source to avoid
collapse.

• Energy loss at stellar surface as measured by
  stellar luminosity is compensated by energy
  release from nuclear reactions through the
  stellar interior.

?rnuclear energy released per unit mass per s.
Depends on T, ? and chemical composition

• During rapid evolutionary phases
  (contraction/expansion)

TdS/dt accounts for the gravitational energy term
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1B11 The equations of Stellar structure
Summary

• P,k,?r are functions of ?,T, chemical
  composition (basic physics provides these
  expressions)
• In total 4, coupled, non-linear partial
  differential equations ( 3 constitutive
  relations) for 7 unknowns P, ?,T, M, L, k, ?r
  as a function of r.
• These completely determine the structure of a
  star of given composition, subject to suitable
  boundary conditions.
• in general, only numerical solutions can be
  obtained (computer).

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1B11 The equations of Stellar structure
Using mass as independent variable (better from a
theoretical point of view)

• Boundary conditions M0, L0 and r0 MMs
  L4RS 2?Teff4 and P2/3g/k
• These equations must be solved for specified Ms
  and composition.

Uniqueness of solution the Vogt Russel theorem
For a given chemical composition, only a single
equilibrium configuration exists for each mass
thus the internal structure is fixed.
This theorem has not been proven and is not
rigorously true there are unknown exceptions
(for very special cases)
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1B11 Last ingredient Equation of State
Perfect gas

• Nnumber density of particles ?mean particle
  mass in units of mH.
• Define

• X mass fraction of H (Sun0.70)
• Y mass fraction of He (Sun 0.28)
• Z mass fraction of heavy elements (metals)
  (Sun0.02)

• XYZ1

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1B11 Last ingredient Equation of State
If the material is assumed to be fully ionized

• Aaverage atomic weight of heavier elements
  each metal atom contributes A/2 electrons
• Total number of particles

• N(2X3Y/4Z/2)? /mH
• (1/?) 2X3Y/4Z/2)

• Very good approximation is standard
  conditions!

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1B11 Deviations from a perfect gas
The most important situations in which a perfect
gas approximation breaks down are
1) When radiation pressure is important (very
massive stars)
2) In stellar interiors where electrons becomes
degenerate (very compact stars, with extremely
high density) here the number density of
electrons is limited by the Pauli exclusion
principle)