2. ONE-DIMENSIONAL MODEL 18 in the absence of shear turbulence the primary criterion determining if convection occurs is
∇rad >∇ad. (2.13)
If this condition is satisfied then convection occurs. Loosely speaking this criterion may be thought of as indicating that the temperature gradient needed to carry the thermal flux through radiation is too high relative to the buoyancy experienced by an adiabatically expanding packet of gas. The result is a convective instability.
The only remaining piece of physics we need to compute stellar structures with the above equations is κ. This we take from tables such that those of OPAL8 and Ferguson9, as discussed in Appendix B.1. A plot of the opacities produced by these tables at X = 0.7, Y = 0.27, Z = 0.03 is shown in figure 2.1.
2. ONE-DIMENSIONAL MODEL 19
3 4 5 6 7 8 9
log T
−10
−8
−6
−4
−2 0 2 4 6
log ρ
−4
−3
−2
−1 0 1 2 3 4 5
log κ
Figure 2.1: The vertical axis is logρ (with ρ measured in g/cm3), the horizontal is logT (with T measured in K), and the color represents logκ (with κ measured in cm2/g. White regions are those without data.
2. ONE-DIMENSIONAL MODEL 20 addition of the thin envelope assumption, such that M−m andr are taken not to change, except where they appear explicitly as parameters for differentiation.
Though Acorn only computes stellar envelopes, this is more than enough to examine the vicinity of Σh. The heat input was modeled by changing the luminosity of the star as a function of column density, according to
L(Σ) =Lin+Lee−Σ/Σh. (2.14) A value of Σh = 103g/cm2 was used here, as per the discussion in Chapter 1.
To begin with, we consider models where the external illumination is imposed whilst holding the star’s radius and intrinsic luminosity fixed. The following three representative models for companion stars were chosen for the simulations:
• The Sun: M =M,Lin=L,R =R
• Low-mass nuclear-burning: M = 0.3M,Lin = 10−2L,R = 0.43R
• Brown dwarf: M = 0.02M,Lin = 10−4L,R= 0.14R
The full output from Acorn for each of these cases for a variety of external luminosities may be found in Appendix E.
The first aspect of these models worth investigating is the region of validity of the thin-envelope approximation, which is the assumption that r≈R everywhere in the envelope. To see where this holds, we have plotted the radius as a function of Σ in figure 2.2. For the 1M star, the thin-envelope approximation is good down to Σ = 106g/cm2 or so, where deviations reach roughly 10%. For the 0.3M star, the approximation is valid everywhere with no external heating, down to Σ = 107g/cm2for Le= Li, and to Σ = 106g/cm2 forLe= 10Li. For both of these stars, deviations grow rapidly past the regime of validity. Finally, for the 0.02M star, the approximation is typically only valid within 10% down to Σh. Past this, however, deviations grow much more slowly than for the other two stars, and so the approximation may be safely used down to around 105g/cm2, where deviations reach 15%. Fortunately there are no phenomena which are both sensitive to the high-Σ failure of this approximation and are of significant quantitative interest, so this approximation is a safe one to make. Subsequent plots will be truncated in their range of Σ to that in which the approximation is valid to within 50%.
We now turn to the thermal structure of the star. Figure 2.3 shows the log of temperature versus the log of column density for nine scenarios. The three stars of interest are represented by the columns, while three different external luminosities are represented by the rows. The top row has no external illumination, the middle
2. ONE-DIMENSIONAL MODEL 21
−2 0 2 4 6 8 10
3 4 5 6 7
R, Le=0(R/R⊙)2 L⊙
1e10 M=1.0
−2 0 2 4 6 8 10
1 2 3 4 5 6 7
R, Le=1(R/R⊙)2 L⊙
1e10
−2 0 2 4 6 8 10
Log Σ 1
2 3 4 5 6 7
R, Le=10(R/R⊙)2 L⊙
1e10
−2 0 2 4 6 8 10
2.70 2.75 2.80 2.85 2.90 2.95 3.00
1e10 M=0.3
−2 0 2 4 6 8 10
1.0 1.5 2.0 2.5 3.0
1e10
−2 0 2 4 6 8 10
Log Σ 0.5
1.0 1.5 2.0 2.5 3.0
1e10
−2 0 2 4 6
9.60 9.62 9.64 9.66 9.68 9.70 9.72
9.74 1e9 M=0.02
−2 0 2 4 6
0.65 0.70 0.75 0.80 0.85 0.90 0.95
1e10
−2 0 2 4 6
Log Σ 0.0
0.2 0.4 0.6 0.8 1.0
1e10
Figure 2.2: Radial coordinate (in cm) versus log of Σ (in g/cm2) for nine different scenarios. Σ here is computed as the mass above the point of interest divided by 4πR2. The columns are the three different stars under consideration. Moving left to right, they areM =M,Lin =L,R =R, M = 0.3M,Lin= 10−2L,R = 0.43R, andM = 0.02M,Lin = 10−4L,R= 0.14R. The rows represent different quantities of external luminosity. From top to bottom, these are Le = 0, Le = LRR2
2, Le = 10LR2
R
2. The vertical grey bar goes from the edge of the photosphere (whereτ = 2/3) to the heating depth (Σ = 103g/cm2). Blue regions are dominated by convective heat transport, red by radiative transport.
2. ONE-DIMENSIONAL MODEL 22 row has the illumination equal toLR2/R2, and the bottom row has it equal to ten times that. Note that for each mass, the radius was held constant. As a result, the top row represents a nearly unmodified system, while the bottom row represents a system dominated by the external heating.
Looking first at the sun, we see that adding external heating begins by shutting down convection at the base of the envelope, and eventually leads to almost completely radiative transport at high external luminosity. The only regions which remain convective are those in the vicinity of the ionization zone, where the adiabatic gradient is very low to begin with. This may be understood as a result of the external heat decreasing the temperature gradient between the core and the heating depth, while increasing it between this depth and the surface. In the former region this suffices to switch the transport from convective to radiative, while the latter is very stable against convection and so remains radiative. That the effect of the heating is so much deeper than the heating depth may be viewed as due to the imposition of a different boundary condition at this depth. In particular, the fact that we maintain a fixed radius as we vary the flux means that the surface temperature scales as L1/4net. In the 0.3M star we see the same thing, though with convection holding on in a larger region in the middle plot. In the 0.02M star, the same process is evidently occurring, though the transition to radiative transport is not apparent until the final plot. This is as we expect: at the lower temperatures which dominate in these stars, radiative transport is less efficient and so the need for convective heat transport is greater.
One interesting feature of note is the change in thermal gradient betweenT = 104K and T = 104.5K. This occurs when the ionization zone is convective, which it almost always is, and results from a decrease in the adiabatic gradient within the zone. The reason this feature is not visible in each of the nine scenarios plotted in figure 2.3 is that in not all scenarios does the ionization zone fall within the envelope.
The next aspect of these models worth examining is the pressure scale height, hs. This sets the characteristic length scale for turbulence, wind shearing, and convective motion, and so will be of interest at every stage of our analysis. The log of this height is shown in figure 2.4. In each of the models, hs increases monotonically into the star past the photosphere, starting around 106.5cm near the surface and reaching values only a few orders of magnitude smaller thanRat the base of the envelope. In general, we expect hs to follow a power-law as a function of Σ, and indeed this is what we see.
Deviations from this are typically due to changes in the mode of heat transport, or to the ionization of material at various points.
Now we may also compute the efficiency of convection, Γ, defined as the ratio of the heat carried by a convecting gas packet to the heat lost radiatively along the
2. ONE-DIMENSIONAL MODEL 23
−2 0 2 4 6 8 10
3.5 4.0 4.5 5.0 5.5 6.0 6.5
Log T, Le=0(R/R⊙)2L⊙
M=1.0
−2 0 2 4 6 8
4.0 4.5 5.0 5.5 6.0
Log T, Le=1(R/R⊙)2L⊙
−2 0 2 4 6 8
Log Σ 4.0
4.5 5.0 5.5 6.0
Log T, Le=10(R/R⊙)2L⊙
−2 0 2 4 6 8 10
3.5 4.0 4.5 5.0 5.5
M=0.3
−2 0 2 4 6 8 10
3.5 4.0 4.5 5.0 5.5 6.0
−2 0 2 4 6 8
Log Σ 4.0
4.5 5.0 5.5 6.0
−2 0 2 4 6
3.2 3.4 3.6 3.8 4.0
M=0.02
−2 0 2 4 6
4.0 4.5 5.0 5.5
−2 −1 0 1 2 3 4 5 Log Σ
4.0 4.2 4.4 4.6 4.8 5.0 5.2
Figure 2.3: Log ofT (in K) versus log of Σ (in g/cm2) for the same nine scenarios defined in figure 2.2. Σ here is computed as the mass above the point of interest divided by 4πR2. The columns are the three different stars under consideration.
The rows represent different quantities of external luminosity. The vertical grey bar goes from the edge of the photosphere (where τ = 2/3) to the heating depth (Σ = 103g/cm2). Blue regions are dominated by convective heat transport, red by
radiative transport.
2. ONE-DIMENSIONAL MODEL 24
−2 0 2 4 6 8 10
6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
Log hs, Le=0(R/R⊙)2L⊙
M=1.0
−2 0 2 4 6 8
7.0 7.5 8.0 8.5 9.0 9.5 10.0
Log hs, Le=1(R/R⊙)2L⊙
−2 0 2 4 6 8
Log Σ 7.5
8.0 8.5 9.0 9.5 10.0
Log hs, Le=10(R/R⊙)2L⊙
−2 0 2 4 6 8 10
6.0 6.5 7.0 7.5 8.0 8.5 9.0
M=0.3
−2 0 2 4 6 8 10
6.5 7.0 7.5 8.0 8.5 9.0 9.5
−2 0 2 4 6 8
Log Σ 7.0
7.5 8.0 8.5 9.0 9.5
−2 0 2 4 6
6.5 7.0 7.5
M=0.02
−2 0 2 4 6
7.0 7.5 8.0 8.5 9.0
−2 −1 0 1 2 3 4 5 Log Σ
7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8
Figure 2.4: Log of pressure scale height versus log of Σ (in g/cm2) for the same nine scenarios defined in figure 2.2. The columns are the three different stars under consideration. The rows represent different quantities of external luminosity. The vertical grey bar goes from the edge of the photosphere (whereτ = 2/3) to the heating depth (Σ = 103g/cm2). Blue regions are dominated by convective heat transport, red by radiative transport.
2. ONE-DIMENSIONAL MODEL 25 way. While a variety of expressions exist for this, we will make use of the one used in the Gob stellar integration code13. The results of doing so are shown in figure 2.5.
This quantity is of interest because it is a good indicator of the extent to which the balance between convection and radiation has been altered by the external heating, as well as because it indicates the extent to which the convective gradient deviates from the adiabatic one. In each of the unperturbed stars, convection is either highly efficient at the heating depth or becomes very efficient close to the heating depth. In shallower regions the efficiency decreases until convection ceases, with a sharp drop in efficiency at the boundary. Importantly, the region over which the efficiency is low is very small, as the slope of Γ with respect to Σ is large near the radiative-convective transition. In the perturbed stars, convection does not always occur in the same region, as the additional heat may turn it off in the vicinity of the surface, but where it does occur all of the same statements regarding its efficiency hold.
Finally, it is also useful to examine how κvaries through each of the stellar models of interest, and so this is shown in figure 2.6. Referencing figure 2.1, we see a few points worthy of discussion. First, many of the stellar tracks go outside of the known opacity data. In most of these cases the stars are convective, with highly efficient convection, and so the opacity is irrelevant. In every combination of the two low-mass stars with the two lowest-heating values, however, we get a radiative region outside of the known opacity data. In each case the issue arises because ρ is too large. The opacity tables are internally stored using ρ/T3 and T as the independent variables14. Below 106K the tables form a rectangular grid in these variables. As a result, these tracks have exceeded the maximum value of ρ/T3 for which we have data, while remaining in an acceptable temperature range. The simulation code in these cases simply returns the opacity at the correct temperature and maximum value of ρ/T3 for which data exists. Fortunately, however, examination of the corresponding regions in figure 2.3 indicates that these regions are actually quite small in pressure-space, and only appear stretched in this plot out becauseρ changes more rapidly here.
The second feature worth noting is that the convergence of the various tracks corresponding to the radiative atmospheres supports our conclusions regarding the decay of heating into radiative zones. Likewise, the lack of convergence between the analogous convective envelopes supports our conclusions regarding the continuation of heating into convection zones. Additionally, the vast majority of each track, whether measured by pressure-space or arc-length in logρ,logT space, is spent in regions
13Ibid.
14The first of these,ρ/T3, is often called R, and usually defined withρ measured in units of 1g/cm3and T measured in units of 106K.
2. ONE-DIMENSIONAL MODEL 26
−2 0 2 4 6 8 10
−4
−2 0 2 4 6 8
Log Γ, Le=0(R/R⊙)2 L⊙
M=1.0
−2 0 2 4 6 8
−4
−3
−2
−1 0 1 2 3
Log Γ, Le=1(R/R⊙)2 L⊙
−2 0 2 4 6 8
Log Σ
−8
−7
−6
−5
−4
−3
−2
−1
Log Γ, Le=10(R/R⊙)2 L⊙
−2 0 2 4 6 8 10
0 2 4 6 8 10 12
M=0.3
−2 0 2 4 6 8 10
−4
−2 0 2 4 6 8
−2 0 2 4 6 8
Log Σ
−8
−7
−6
−5
−4
−3
−2
−1
−2 0 2 4 6
0 2 4 6 8 10
M=0.02
−2 0 2 4 6
−4
−2 0 2 4 6 8
−2 −1 0 1 2 3 4 5 Log Σ
−8
−7
−6
−5
−4
−3
−2
−1
Figure 2.5: Log of convective efficiency (Γ, see text) versus log of Σ (in g/cm2) for the same nine scenarios defined in figure 2.2. The columns are the three different stars under consideration. The rows represent different quantities of external luminosity.
The vertical grey bar goes from the edge of the photosphere (where τ = 2/3) to the heating depth (Σ = 103g/cm2). Blue regions are dominated by convective heat transport, and all other regions have been omitted due to Γ only being defined in convective zones.
2. ONE-DIMENSIONAL MODEL 27
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log T
−10
−8
−6
−4
−2 0
log ρ
−3
−2
−1 0 1 2 3 4 5
log κ
Figure 2.6: The vertical axis is logρ (with ρ measured in g/cm3), the horizontal is logT (with T measured in K), and the color represents logκ (with κ measured in cm2/g. White regions are those without data. The nine stellar models defined in figure 2.2 are plotted as tracks on top of the opacity. The terminus marker indicates which track is which: the three sizes of markers correspond in increasing order to the three stellar masses under consideration, and the three kinds of markers correspond in order of increasing number of sides to increasing external illumination. Blue regions are dominated by convective heat transport, red by radiative transport.
2. ONE-DIMENSIONAL MODEL 28 where
∂κ
∂T
<0. (2.15)
This fact will become relevant later in the next section. Finally, the fact that the minimum in κ lies at temperatures comparable to those in the ionization zone means that∇rad tends to peak where∇ad is at a minimum, which encourages the formation of a convection region around the ionization zone. This is seen even in the case of heavy external illumination, which generally pushes stars towards radiative transport even at depths much below where the additional heat is deposited.