Download R - AMUSE code

stellar evolution
Simon Portegies Zwart
Sterrewacht Leiden
> from amuse.lab import *
> bodies = Salpeter(N, Mmin, Mma
> stellar = MESA()
> stellar.add_particles(bodies)
> stellar.evolve_model(t_end)
>write_to_file(stellar, “stars.hdf5”)
Initialize
Evolve
Store data
The Sun – best studied example
Stellar interiors not directly observable. Solar neutrinos emitted at core and
detectable. Helioseismology - vibrations of solar surface can be used to probe
density structure
Must construct models of stellar interiors – predictions of these models are tested by
comparison with observed properties of individual stars
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Observable properties of stars
Basic parameters to compare theory and observations:
• Mass (M)
• Luminosity (L)
– The total energy radiated per second i.e. power (in W)
∞
L=∫0 L λ dλ
• Radius (R)
• Effective temperature (Te)
– The temperature of a black body of the same radius as the star that would
radiate the same amount of energy. Thus
L= 4πR2 σ Te4
where σ is the Stefan-Boltzmann constant (5.67× 10-8 Wm-2K-4)
⇒ 3 independent quantities
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What are the main physical processes which determine
the structure of stars ?
• Stars are held together by gravitation – attraction
exerted on each part of the star by all other parts
• Collapse is resisted by internal thermal pressure.
• These two forces play the principal role in determining
stellar structure – they must be (at least almost) in
balance
• Thermal properties of stars – continually radiating into
space. If thermal properties are constant, continual
energy source must exist
• Theory must describe - origin of energy and transport
to surface
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Equation of hydrostatic support
Balance between gravity and internal pressure is known as hydrostatic equilibrium
Mass of element
dm = ρ  r  dsdr
where ρ(r)=density at r
Consider forces acting in radial direction
1. Outward force: pressure exerted by stellar material
on the lower face:
P r  ds
2. Inward force: pressure exerted by stellar material
on the upper face, and gravitational attraction of all
stellar material lying within r
GM(r)
P(r + dr)ds +
dm
2
r
GM(r)
= P(r + dr)ds +
r (r)dsdr
2
r
ﾠ
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In hydrostatic equilibrium:
GM  r 
P r  ds=P r dr  ds 2
ρ  r dsdr
r
GM  r 
⇒ P  r dr −P  r  =- 2
ρ  r dr
r
If we consider an infinitesimal element, we write
P(r + dr) - P(r) dP(r)
=
dr
dr
for δr→0 Hence rearranging above we get
dP(r)
GM(r)r (r)
=dr
r2
The equation of hydrostatic support
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Equation of mass conservation
Mass M(r) contained within a star of radius r is determined by the density of the gas ρ( r).
Consider a thin shell inside the star with radius
r and outer radius r+δr
2
dV =4 r dr
2
⇒ dM=dVρr =4 r drρ r 
dM  r 
2
⇒
=4  r ρ  r 
dr
In the limit where δr → 0
This the equation of mass conservation
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Stellar energy production
Cooling
● Contraction
● Chemical Reactions
● Nuclear Reactions
●
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Solving the equations of stellar structure
Hence we now have four differential equations, which govern the structure of stars
(note – in the absence of convection).
dM  r 
=4 r 2 ρ r 
dr
dP(r)
GM(r)r (r)
=dr
r2
dL r 
=4 r 2 ρ r  e r 
dr
3ρ  r k R  r 
dT  r 
=L r 
2
3
dr
64  r sT  r 
We will consider the quantities:
P = P (ρ, T, chemical composition)
κkR = κR(ρ, T, chemical composition)
ε
= ε (ρ, T, chemical composition)
Where
r = radius
P = pressure at r
M = mass of material within r
ρ = density at r
L = luminosity at r (rate of energy flow across
sphere of radius r)
T = temperature at r
kR = Rosseland mean opacity at r
ε
= energy release per unit mass per unit time
The equation of state
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Boundary conditions
Two of the boundary conditions are fairly obvious, at the centre of the star
M=0, L=0 at r=0
At the surface of the star its not so clear, but we use approximations to allow solution. There
is no sharp edge to the star, but for the the Sun
ρ(surface)~10-4 kg m-3. Much smaller than mean density ρ(mean)~1.4×103 kg m-3
(which we derived). We know the surface temperature (Teff=5780K) is much smaller than its
minimum mean temperature (2×106 K).
Thus we make two approximations for the surface boundary conditions:
ρM=M, ρ = 0 kg/m3 and T = 0K at r=rs
i.e. that the star does have a sharp boundary with the surrounding vacuum
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Use of mass as the independent variable
The above formulae would (in principle) allow theoretical models of stars with a given radius.
However from a theoretical point of view it is the mass of the star which is chosen, the stellar
structure equations solved, then the radius (and other parameters) are determined. We
observe stellar radii to change by orders of magnitude during stellar evolution, whereas mass
appears to remain constant. Hence it is much more useful to rewrite the equations in terms
of M rather than r.
If we divide the other three equations by the equation of mass conservation, and invert the
latter:
dr
1
=
dM 4pr 2 r
dP
GM
=dM
4pr 4
ﾠ
dL
=e
dM
With boundary conditions:
r=0, L=0 at M=0
ρ=0, T=0 at M=Ms
dT
3k R L
=dM
64p 2 r 4 acT 3
We specify Ms and the chemical composition and now have a well defined set of
relations
ﾠ to solve. It is possible to do this analytically if simplifying assumptions are
made, but in general these need to be solved numerically on a computer.
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Example set of models
Eggleton code
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Geneva code
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Theoretical isochrones from Geneva models
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Examples of young and old clusters
NGC6231 young cluster
Age~ 6Myrs
Pleiades young open cluster
Age~ 100Myrs
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47 Tuc : globular cluster. Age= 810Gyrs
NGC188: old open cluster . Age=
7Gyrs
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