8. TIME DEPENDENCE 132 this scenario on the same radiative star as before are shown in figure 8.3. The trend is just the same as before, with the temperature falling by the same amount it initially rose, and the luminosity falling everywhere back down to the internally generated value.
These results confirm that radiative stars exponentially damp temperature differ- ences as a function of depth. Additionally, the quick response of radiative stars to changes in external illumination mean that they track the present-day properties of the pulsar wind. This, combined with possible anisotropies in their thermal profiles, means that they may still be useful for exploring the environments pulsars produce.
8. TIME DEPENDENCE 133
−1 0 1 2 3 4 5 6 7 8
−0.16
−0.14
−0.12
−0.10
−0.08
−0.06
−0.04
−0.02 0.00 0.02
∆T/T
Lin=100Lsun,∆t =1e8s
−1 0 1 2 3 4 5 6 7 8
Log Σ 100
120 140 160 180 200 220
L/Lsun
t=0e7s t=1e7s t=2e7s t=3e7s t=4e7s t=5e7s t=6e7s t=7e7s t=8e7s t=9e7s t=10e7s
Figure 8.3: ∆T /T0 (top) and L/L (bottom) versus log Σ (in g/cm2) for a star of mass M, radius R, and luminosity 100L. The external heat was put in at Σ = 103g/cm2 and linearly decreased from 100L to zero over the course of 108s.
Color represents time, with the simulation beginning at violet and ending with red.
8. TIME DEPENDENCE 134
1 2 3 4 5 6 7
−0.012
−0.010
−0.008
−0.006
−0.004
−0.002 0.000
∆T/T
Lin=0.1Lsun,∆t =1e8s
1 2 3 4 5 6 7
Log Σ 0.10
0.12 0.14 0.16 0.18 0.20
L/Lsun
t=0e7s t=1e7s t=2e7s t=3e7s t=4e7s t=5e7s t=6e7s t=7e7s t=8e7s t=9e7s t=10e7s
Figure 8.4: ∆T /T0 (top) and L/L (bottom) versus log Σ (in g/cm2) for a star of mass 0.3M, radius 2.65R, and luminosity 0.1L. The external heat was put in at Σ = 103g/cm2 and linearly decreased from 0.1L to zero over the course of 108s.
Color represents time, with the simulation beginning at violet and ending with red.
8. TIME DEPENDENCE 135
1 2 3 4 5 6 7
−0.012
−0.010
−0.008
−0.006
−0.004
−0.002 0.000
∆T/T
Lin=0.1Lsun,∆t =2e8s
1 2 3 4 5 6 7
Log Σ 0.10
0.12 0.14 0.16 0.18 0.20
L/Lsun
t=0e7s t=2e7s t=4e7s t=6e7s t=8e7s t=10e7s t=12e7s t=14e7s t=16e7s t=18e7s t=20e7s
Figure 8.5: ∆T /T0 (top) and L/L (bottom) versus log Σ (in g/cm2) for a star of mass 0.3M, radius 2.65R, and luminosity 0.1L. The external heat was put in at Σ = 103g/cm2 and linearly decreased from 0.1L to zero over the course of 108s.
The simulation was then run for an additional 108s with no external heating. Color represents time, with the simulation beginning at violet and ending with red.
8. TIME DEPENDENCE 136
1 2 3 4 5 6 7
−0.05
−0.04
−0.03
−0.02
−0.01 0.00 0.01
∆T/T
Lin=0.1Lsun,∆t =1e8s
1 2 3 4 5 6 7
Log Σ 0.0
0.2 0.4 0.6 0.8 1.0 1.2
L/Lsun
t=0e7s t=1e7s t=2e7s t=3e7s t=4e7s t=5e7s t=6e7s t=7e7s t=8e7s t=9e7s t=10e7s
Figure 8.6: ∆T /T0 (top) and L/L (bottom) versus log Σ (in g/cm2) for a star of mass 0.3M, radius 2.65R, and luminosity 0.1L. The external heat was put in at Σ = 103g/cm2 and linearly decreased from L to zero over the course of 108s.
The simulation was then run for an additional 108s with no external heating. Color represents time, with the simulation beginning at violet and ending with red.
8. TIME DEPENDENCE 137 of adjustment forL in a convection cell is set by the convective turnover time l/vc5, and so what occurs is that the temperature adjusts due to the nonzero slope of L in m for a time l/vc, after which the flux has uniformly risen to match the outer boundary condition. At this point and in this region, the flux ceases to vary, and hence∂tT falls to zero. The expected change inT is then expected to be roughly
δT ≈∂tT δt≈ ∆L cpδm
δz vc
= ∆L
4πr2ρcpvc
, (8.22)
where δm andδz refer to the mass and thickness of a spherical shell of material, and
∆L is the change in luminosity, which should be equal to the external luminosity.
Now the convective flux may be written as Fc ≈ρvc3 ≈ Lin
4πr2, (8.23)
so
δT ≈ Le 4πr2ρcp
Lin 4πr2ρ
!−1/3
= 1 cp
Le 4πr2ρ
!2/3Le Lin
1/3
. (8.24)
This may also be written as
δlnT ≈ δT T ≈ vc2
vs2
Le Lin
. (8.25)
Near the surface of a fully convective star, we usually havevc ≈vs/10, so forLe= Lin we expect δT /T ≈ 10−2. Furthermore, at higher pressures the sound speed rises relative to the convection speed, and so the difference drops off. This may be understood as following from the above result thatδT ∝ρ−2/3. The endpoint of this process occurs when the moving "wavefront" of the flux change reaches the nuclear burning regime. At this stage the temperature will drop significantly more, for there is nowhere else for the wavefront to go.
To understand what happens next, we first remark that the convection zone will adjust to maintain an adiabatic gradient on a timescale set by the convective turnover rate. As a result, the timescale for the entire star to adjust to maintain this gradient is
τadj =
Z dr vc =
Z 1 ρvcdΣ≈
Z
ρ−1 F ρ
!−1/3
dΣ =F−1/3
Z
ρ−2/3dΣ, (8.26)
5Recall that this is why Acorn only takes time-steps which are at least max(l/vc) in size.
8. TIME DEPENDENCE 138 where the integral is taken over the entire star. In a convective atmosphere, P ∝ργ, so ρ ∝ P1/γ. If we make the thin atmosphere approximation throughout the star, just to gain an order-of-magnitude estimate, thenP =gΣ, and so
τadj ≈g−1F−1/3
Z
ρ−2/3dP =g−1F−1/3ρ−2/30
Z P P0
−2/3γ
dP (8.27)
= 1
1− 3γ2 g−1F−1/3ρ−2/30 P02/3γPf1−2/3γ−P01−2/3γ. (8.28) Now γ is typically 5/3 outside of the ionization zone, so 2/3γ=2/5, and hence
τadj ≈ 5P02/5 3gF1/3ρ2/30
Pf3/5−P03/5≈ 5P02/5Pf3/5
3gF1/3ρ2/30 = 5v2s,0 3γgvc,0
Pf P0
3/5
= v2s,0 gvc,0
Pf P0
3/5
, (8.29) where we have made the approximation that the core pressure vastly exceeds the pressure at the top of the convection zone. Nowvc,0 is typically aroundvs,0/10, and vs,0 is typically around 106cm/s≈100sg, so the prefactor is around 103s. Typically P0 ∼ 105erg/cm3, and Pf ∼ gM/(4πR2) ∼ 1015erg/cm3 for the sun, so the overall timescale is around 103+6 = 109s, scaling roughly as M6/5R−12/5. For the fully convective star considered in simulation, M = 0.3M and R = 2.65R, so this timescale is smaller by a factor of 25, giving around 4×107s, or roughly a year.
On the other hand, the core adjusts its temperature in time τcore = mcorecpTcore
Le =fMLin
Le τK =fMGM2
2RLe, (8.30)
where fM ≈1/10 is the fraction of the star’s mass in the core andτK is the Kelvin timescale for the star. This is typically of order ten million years, and so ifLe =Lin the core’s adjustment timescale is of order a million years. As this is much shorter than the timescale required to maintain adiabaticity, the star may be approximated as being adiabatic at all times after the cooling wavefront reaches the core.