THE EFFECT OF DENSITY ON THE THERMAL STRUCTURE OF GRAVITATIONALLY DARKENED Be STAR DISKS

Figures 1–4 show the changes produced by increasing rotation on the temperature of disks for a variety of densities using the same four temperature diagnostics as McGill et al. (2011): the density-weighted average temperature, the volume-weighted average temperature, the maximum temperature, and the minimum temperature. The density-weighted average temperature is defined as

$$\begin{equation} \overline{T}_{\rho } = \frac{1}{M_{\rm {disk}}}\,\int T(R,z)\,\rho (R,z)\,dV \; \end{equation} \tag{ 3 }$$

and the volume-weighted average temperature is defined as

$$\begin{equation} \overline{T}_{\rm {V}} = \frac{1}{V_{\rm {disk}}}\,\int T(R,z)\,dV \;. \end{equation} \tag{ 4 }$$

In order to avoid numerical effects, the median of the 20 hottest and 20 coolest disk locations are taken to represent the maximum and minimum disk temperatures.

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The most important trend seen in Figures 1–4 is in the density-weighted average temperatures. There is a decrease in this temperature with both increasing rotation rate and increasing density. This trend is seen in all spectral types. In addition, for all spectral types, the curves for different ρ_o are well separated and decrease smoothly. For spectral types B2V, B3V, and B5V, the slope increases with increasing rotation but flattens out near critical rotation. For spectral types B2V and B3V, the four curves for each ρ_o are essentially parallel and for B5V are nearly parallel. Parallel curves are not seen in the B0V model, and the decrease in temperature with rotation is greatest for the largest density model.

The temperature minimums also decrease with both increasing rotation rate and increasing density, analogous to the density-weighted average temperatures. The size of the decrease in the minimum temperature is also larger for the least dense models. For spectral types B2V, B3V, and B5V, the curve for the most dense model, ρ_o = 5.0 × 10⁻¹¹ g cm⁻³, is essentially flat. This causes the temperature minimums to converge at v_crit to values between 5000 and 6000 K for all spectral types. The curves for different ρ_o are distinct, but there is significant overlap. The temperature minimums in the non-rotating models are larger for the earlier spectral types, and the earlier models experience a larger decrease in temperature with both rotation and density. The drop in the value of the minimum temperatures with rotation and density is barely noticeable in the B5V models as all curves are nearly flat and cluster at ≈5000 K.

One of the most interesting effects produced by rotation on Be star disk temperatures is seen in the volume-weighted average temperatures of the B2V, B3V, and B5V models. These values initially decrease with moderate rotation but begin to increase approaching critical rotation. The strength of both the initial decease and the increase near critical rotation is larger for early spectral types, excluding B0V. For the two highest densities of the critically rotating B2V models, the volume-weighted average temperatures are essentially the same as the non-rotating models. Generally, the volume-weighted average temperatures are hotter for the least dense models, but the separations between the averages are not as large as for the density-weighted average temperatures or the temperature minimums, and there is a great deal of overlap. The effect of density is strongest in the B2V model and very small by B5V. For moderate rotation, lower density models generally have smaller volume-weighted temperature averages than denser models, but the lowest density B2V models (ρ = 5.0 × 10⁻¹²), and the non-rotating and slowest rotating models of the second lowest density B2V models (ρ = 1.0 × 10⁻¹¹) do not follow this trend. For rapid rotation, lower density models generally have larger volume-weighted temperature averages than denser models, but this does not occur for the B5V models.

The volume-weighted temperature averages of the B0V model do not behave like the other spectral types. There is only a very small change in temperature with rotation, which is somewhat larger for larger densities. The least dense B0V model is of nearly constant temperature. The densest, non-rotating models are hotter than the least dense models, but by critical rotation, this trend has reversed. All of the values are tightly clustered between 13,900 and 14,500 K.

The maximum disk temperatures change very little with rotation or density. This is because these temperatures are found in parts of the disk that are not shielded by the dense, mid-plane and are directly illuminated by the pole of the star. The maximum temperatures are determined essentially by spectral type. There is a small but noticeable decrease in temperature with rotation in all models, which flattens out near critical rotation. Like the volume-weighted average temperatures, the temperature maximums begin to increase for extreme rotation, significantly for the B3V models but very weakly for the other spectral types.

One of the most complex behaviors seen in Figures 1–4 is how the relationship between the volume-weighted and the density-weighted average temperatures is affected by rotation and disk density. These effects can be separated into three categories based on density, the high- and low-density extremes, and the case of intermediate density. (1) For the highest density B0V model, the two highest density B2V and B3V models, and the highest density B5V model, the volume-weighted average temperatures are always higher than the density-weighted average temperatures. This is due to the presence of a large, cool region in the equatorial plane at high density. (2) For the lowest density B0V models and the two lowest density B5V models, the volume-weighted average temperatures are always lower than the density-weighted average temperatures. This is due to the absence of a significant cool region in the equatorial plane at low density. (3) All models of intermediate density experience a transition: for low rotation rates there is either not a cool region in the equatorial plane or it is too small to cause the density-weighted average temperatures to be lower than the volume-weighted average temperatures. At higher rotation rates, gravitational darkening causes the development of a larger cool region and the density-weighted average temperatures become less than the volume-weighted average temperatures.

Figures 5–8 show radial temperature profiles of these disks obtained by averaging the temperatures within each vertical column perpendicular to the mid-plane of the disk. The average temperature at each radial distance is density weighted and is found using

$$\begin{equation} \bar{T}(R)=\frac{\int ^{z_{\rm {max}}}_{0} \, T(R,z) \rho (R,z) dz}{\int ^{z_{\rm {max}}}_{0} \rho (R,z) dz}. \end{equation} \tag{ 5 }$$

As seen in the previous figures, lower density models have higher temperatures. The profiles follow two patterns. (1) For low-density models, the temperature at the inner edge of the disk begins fairly hot, increases to a maximum, and then the temperature decreases as the radius increases, sometimes reaching a constant value. (2) For sufficiently large densities, the temperature at the front of the disk still begins fairly hot, but the temperature drops to a minimum and then increases again at larger radii, sometimes reaching a maximum value and then dropping again. For all spectral types, the temperature minimum becomes cooler and the extent of the cool zone increases in size with rotation. The slope of the temperature increase at large radii becomes smaller with increasing rotation. If a temperature maximum occurs, its size is reduced with rotation and the slope of the temperature drop-off at large radii becomes shallower.

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Figures 9–12 show vertical temperature profiles of these disks obtained by averaging the temperatures at different radii but at the same scale height, u = z/H(R). The average temperature is density weighted and is found using

$$\begin{equation} \bar{T}(u)=\frac{\int ^{R_{\rm { max}}}_{0} \, T(R,u) \rho (R,u) A(R,u) dR}{\int ^{R_{\rm { max}}}_{0} \rho (R,u) A(R,u) dR}, \end{equation} \tag{ 6 }$$

where A(R, u) is the area function of the disk. A(R, u) is included in Equation (6) to account for the non-uniform spacing in r and z of the computational grid (i.e., Δr and Δz both increase with R). The exact nature of this weighting matters less than the fact that all the models have been averaged in the same way. Comparing these plots illustrates the effects of rotation and density on the vertical structure of these disks.

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In these plots, the mid-plane of the disk is at u = 0. All vertical temperature profiles show a temperature maximum occurring between one and two scale heights above the mid-plane. The location of this maximum moves higher above the mid-plane with both increasing density and increasing rotation. Rotation causes the width of the temperature peak to increase and become broader, while increasing density causes width of the peak to become narrower and sharper. Low-density models without rotation have mid-plane regions either at the same temperature or hotter than the upper disk. Increasing the density causes the development of a cool region in the mid-plane. Rotation causes this cool region to form in the mid-plane at lower densities. At high rotation rates, pronounced cool regions are seen even in the least dense models.

There is little variation in the temperature of the upper edge of the disk with either rotation or density in the B5V models and no significant change with density for the non-rotating B0V models. The upper edges of the disk increase in temperature with rapid rotation in the B2V and B3V models and have a minimum value at v_frac = 0.80. In the B2V, B3V, and the rotating B0V models, the upper edges of the disk are hotter for lower densities. While there is a change in the temperature of the upper edges of the disk with density in the rotating B0V models, which increases in size with rotation, the median of this range is not significantly affected by rotation and remains at ≈13, 000 K.

Figure 13 shows the two-dimensional temperature structure for a selection of eight B0V models. Two models are shown for each density, one without rotation and one for v_frac = 0.95 v_crit. Clearly, the temperature in the equatorial plane becomes cooler as the density increases; this has been noted many times. What is interesting is that this cool region appears at smaller densities when rotation is included.

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Finally, a useful way to illustrate the effect of density and rotation on the range of disk temperature is to construct histograms of T(R, z) for each model. Figures 14 and 15 each show a set of histograms for increasing rotation. Only two sets of histograms are shown for brevity, the B2V model with ρ_o = 5.0 × 10⁻¹² g cm⁻³ (the lowest density considered) and another with ρ_o = 5.0 × 10⁻¹¹ g cm⁻³ (the highest density considered). In Figure 14, the model without rotation is fairly warm and as rotation increases, a low-temperature tail forms in the distribution. In addition, the fraction of disk temperatures in the highest temperature bins also increases at high rotation rates. In Figure 15, the non-rotating model has two strong peaks, one quite cool and the second middle peak. There is also a large and long high-temperature tail and a weak high-temperature peak because this model is dense enough to possess a cool region in the equatorial plane, with warmer regions above. As rotation increases, the middle-temperature peak weakens and eventually disappears, while the low-temperature peak becomes stronger and cooler. The weak, higher temperature peak becomes stronger. The distribution of temperatures in the fastest rotating model has two strong peaks at the temperature extremes. Overall, these histograms demonstrate that gravitational darkening increases the amount of very cool and very hot material in the disk and decreases the amount of disk material of intermediate temperatures.

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It is often useful in the study of circumstellar material to compare the temperatures found in the disk to the effective temperature of the star. However, when the star is a rapid rotator, there is a range of surface temperatures, and some representative value must be found. The most basic definition of effective temperature is from the Stephan–Boltzmann law, namely T_eff ≡ (L/σA)^1/4 where A is the surface area of the star. Extending this to a rotating star gives T_eff(v_frac) = (L/σA(v_frac))^1/4. If the luminosity is considered unaffected by rotation we have

$$\begin{equation} T_{\rm {eff}}(v_{\rm {frac}})={T_{\rm {eff}}(v_{\rm {frac}}=0)} \left(\frac{A(v_{\rm {frac}}=0)}{A(v_{\rm {frac}})}\right)^{1/4}. \end{equation} \tag{ 7 }$$

Figure 16 shows the density-weighted average temperatures divided by the stellar effective temperatures defined by Equation (7) versus rotation rate for all models. This plot summarizes the predicted global temperatures of the disks around rapidly rotating stars of a wide range of disk densities. The temperature ratio of the disk to the star is between 0.40 and 0.65 for all models. Denser disks are proportionally cooler. The disks around cooler stars all have temperatures which are larger fractions the stellar effective temperatures than those around hot stars. The ratio between the density-weighted average temperatures in the disk and the effective stellar temperatures drops with moderate rotation. Approach to critical rotation causes an increase in the ratios with rotation in both cooler stars and less dense disks. The slope of the ratio flattens out for rapid rotation in hot and dense disks. In all cases, the trends are dominated by the variations in disk density, not rotation.

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Table 3 gives the density-weighted temperatures for all density scales and spectral types without rotation and at maximum rotation (v = 0.99 v_crit). This table shows the full range of temperature changes that occur with rotation for different densities. Increases in either density or rotation cause decreases in the density-weighted temperatures. For the B0V models, these two effects strengthen each other; denser disks are more strongly affected by increasing rotation and disks surrounding critically rotating stars are more strongly affected by density changes than disks surrounding non-rotating stars. The density-weighted average temperatures change by −6% as rotation varies from zero to 0.99 v_crit for the least dense models, and change by −21% for the densest model over the same range in velocity. When we look at the results with rotation rates fixed, we find that the density-weighted average temperatures change by −14% from ρ_o = 5.0 × 10⁻¹² g cm⁻³ to ρ_o = 5.0 × 10⁻¹¹ g cm⁻³ without rotation, and change by −28% for the fastest rotating models. In the B2V and B3V models, the effect of rotation is nearly the same for all ρ_o, and the percent differences range between −13% and −15% for the B2V models and between −15% and −16% for the B3V models. The effect of density changes is also nearly the same on models without rotation and those with a rotation rate of 0.99 v_crit, with the percent differences ranging between −23% and −26% for the B2V models and remaining unchanged at −24% for the B3V models. By B5V these two effects weaken each other. The least dense models are more strongly affected by rotation and the non-rotating models are most affected by density changes. The density-weighted average temperatures change by −20% from zero to 0.99 v_crit for the least dense models, and change by −12% for the densest model. Holding rotation rates fixed, we find that the density-weighted average temperatures change by −24% from ρ_o = 5.0 × 10⁻¹² g cm⁻³ to ρ_o = 5.0 × 10⁻¹¹ g cm⁻³ without rotation, and change by −16% for the fastest rotating models.

Table 3. Density-weighted Average Temperatures without Rotation and with v_frac = 0.99

Type	ρo	$\bar{T}_{\rho }(0.00)$	$\bar{T}_{\rho }(0.99)$	% Δ(vfrac)
	(g cm−3)	(K)	(K)
B0V	5.0 × 10−12	15200.	14250.	−6.
	1.0 × 10−11	15180.	13920.	−9.
	2.5 × 10−11	14520.	13010.	−11.
	5.0 × 10−11	13280.	10780.	−21.
	% Δ(ρ)	−14.	−28.
B2V	5.0 × 10−12	11620.	10220.	−13.
	1.0 × 10−11	11090.	9690.	−13.
	2.5 × 10−11	10200.	8800.	−15.
	5.0 × 10−11	9190.	7870.	−15.
	% Δ(ρ)	−23.	−26.
B3V	5.0 × 10−12	11000.	9520.	−15.
	1.0 × 10−11	10590.	9110.	−15.
	2.5 × 10−11	9670.	8230.	−16.
	5.0 × 10−11	8690.	7460.	−15.
	% Δ(ρ)	−24.	−24.
B5V	5.0 × 10−12	9710.	7970.	−20.
	1.0 × 10−11	9290.	7760.	−18.
	2.5 × 10−11	8450.	7230.	−16.
	5.0 × 10−11	7660.	6760.	−12.
	% Δ(ρ)	−24.	−16.

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Taking all the results into consideration, we see that while the bulk of the disk becomes cooler, there is evidence of heating in some parts of the disk (either from the volume-weighted average temperatures or the maximum temperatures or both). Heating of the upper edge of the disk is seen in the vertical profiles of the B2V and the B3V models, as shown is Figures 10 and 11. This was also seen in Figures 8, 9, and 13–15 in McGill et al. (2011). When density changes are combined with rotation, we see that the cooling associated with increasing density occurs at lower ρ_o at higher rotation rates. This suggests that if rotation is not taken into account, the densities of these disks could be overestimated.

In this section, we describe the effects of rotation on disk models that have been hydrostatically converged, as described in Sigut et al. (2009). The vertical structure of classical Be star disks are believed to be in hydrostatic equilibrium. Equation (1) governs the density structure of a hydrostatically supported isothermal disk, but real Be star disks are not isothermal. Temperatures throughout the disks typically vary by factors of 2–3, as seen in Figures 1–4 and Figures 14 and 15. This means that Equations (1) and (2) are inconsistent with the detailed T(R, Z) distributions for the models computed from radiative equilibrium. Vertical pressure support in the disk requires that the vertical density profile should follow −dP/dz = ρ(z, r)g_z, in which g_z is the vertical component of the star's gravitational acceleration. The local pressure is strongly affected by the local temperature of the disk via the equation of state, P = ρkT/μ. This means that in order to create a density profile consistent with the temperature solution, an additional loop is required within the Bedisk code to enforce hydrostatic equilibrium in each column, as described by Sigut et al. (2009). Because of the extra loop, a converged model takes approximately 10 times longer to calculate than an unconverged model.

Figure 17 shows the ratio of the disk scale height to the disk radius, H(R)/R, versus disk radius, R, in the inner disk (r ⩽ 5r_p) for B0V models of both the fixed density structure described by Equation (1) and those that have been hydrostatically converged. H(R) is the height where the density drops by a factor of 1/e. For an isothermal disk, H(R)/R is equal to the ratio of the sound speed, $c_{\rm {sound}}=\sqrt{P/\rho }$, to the Keplerian orbital speed, $v_{\rm {orbit}}=\sqrt{{\it G M} /R}$. Because of the nature of the fixed isothermal density structure, it always produces the same H(R)/R profile that increases as R^1/2 regardless of ρ_o or n, as it is a function of only T_iso. In the current set of models, H(R)/R is always larger for the fixed structure compared to the hydrostatically converged structure because we have adopted the usual result that T_iso = 0.6T_eff. However, this temperature is often higher than the temperatures actually found in the mid-plane of dense disks. Therefore, Equation (1) overestimates the pressure support of the mid-plane (see Sigut et al. 2009 for more details). The difference between T_iso and the actual disk temperature becomes smaller as the disk radius increases and this is why the difference in H(R)/R between the hydrostatically converged models and those with the fixed density structure decreases with increasing radius. The difference in H(R)/R between the fixed structure and the hydrostatically converged models also increases as ρ_o increases because the temperatures in the mid-plane decrease strongly as ρ_o increases. Note that there is no single choice for T_iso that would be able to match the correct vertical density structures of the hydrostatically converged models shown in Figure 17.

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We now demonstrate that rotation, in addition to density, affects the scale heights computed for these disks. Figure 17 demonstrates that H(R)/R for v = 0.99 v_frac is always lower than that for v = 0 for the same radius and density. This is because rapidly rotating models have cooler temperatures in the mid-plane, resulting in smaller scale heights: rapid rotation causes models with thinner disks. The model with ρ_o = 5.0 × 10⁻¹¹ g cm⁻³ and v = 0.99 v_crit has a very thin disk with a maximum value in the inner disk of H(R)/R = 0.033 at 2.6 stellar radii and a minimum of 0.027 at 3.75 stellar radii.

Table 4 shows the density-weighted average temperatures without rotation and with rotation at 0.99 v_frac for the fixed density structure and the hydrostatically converged structure for the four different densities shown in Figure 17. Increasing density or rotation causes a decrease in the density-weighted average temperature with or without hydrostatic equilibrium. Requiring self-consistent hydrostatic equilibrium reduces the effects of changing rotation and density and also increases the temperatures found in the dense B0V models.

As described in van Belle (2012), gravitational darkening has been interferometrically confirmed, but the difference in brightness over the stellar disk is not as strong as that predicted by the standard formulation of gravitational darkening presented by Collins (1963). This formulation states that

$$\begin{equation} F=C_{\omega } g^{\, 4 \beta }, \end{equation} \tag{ 8 }$$

where F is the local radiative flux and C_ω is constant across a star and determined by the luminosity such that

$$\begin{equation} C_{\omega }= L_{\omega }\big/{\int g^{4 \beta } dA}, \end{equation} \tag{ 9 }$$

where the integration is over the surface of the star. The luminosity, L_ω, can be treated as a constant or as a function of rotation. The canonical value of β, defined in von Zeipel (1924), is 1/4, but observations suggest a range of values between 0.25 and 0.19 for B type stars (van Belle 2012). Claret (2012) presents a calculation for the variation of β with local temperature and optical depth in a rotating star of 4 M_☉ and describes departures from von Zeipel in the upper layers where β can be as low as ∼0.18 when energy transport is radiative. Figure 18 shows the variation in temperature with rotation for β = 0.18 and 0.25 at four different stellar co-latitudes. The constant C_ω was recomputed in each case. Using β = 0.18 weakens both the temperature increase in the polar region and the temperature decrease in the equatorial regions, reducing temperature difference between the equator and the pole.

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We choose β = 0.18 as a lower limit and re-ran Bedisk for the four highest rotation rates assuming the B2V star surrounded by a disk with ρ_o = 5.0 × 10⁻¹¹ g cm⁻³. The results are shown in Figure 19. Lowering β reduces the strength of gravitational darkening and the corresponding effect of the stellar flux on the temperatures in the disk. The density-weighted average temperature decreases with rotation and this decrease is less with β = 0.18. The volume-weighted average temperatures in the disk have a more complex behavior, initially decreasing then increasing with rapid rotation, as β is reduced to 0.18. The initial drop in temperature becomes smaller and the increase in temperature at higher rotation rates is also smaller. The value of β does not affect the maximum or minimum temperatures found in the disk.

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The discrepancy in β between theory and observation has motivated a closer look at gravitational darkening and possible refinements to the theory. Espinosa Lara & Rieutord (2011) present such a refinement. Both Espinosa Lara & Rieutord (2011) and the canonical method begin with the expression for radiative energy transport. The equation of hydrostatic equilibrium is used to relate the radiative flux to the local gravity, resulting in expressions of the form F_rad∝g, which becomes T_eff∝|g|^1/4 using the Stephan–Boltzmann law (von Zeipel 1924; Clayton 1983). The difference in these approaches is how the other terms in the equation are handled. Von Zeipel (1924) treats the equipotentials as isobaric surfaces which implies that all other terms in the equation are constant over the surface of a rotating star (for this type of argument see Section 6.8 of Clayton 1983). This leads to a contradiction: the gas temperature is taken as a constant across the surface of a rotating star, while the effective temperature decreases with increasing stellar co-latitude. Although this is not truly a paradox (as the effective temperature and the gas temperature are not the same quantities), it would be reasonable for a higher radiative flux at the poles to produce higher local temperatures.

Alternatively, Espinosa Lara & Rieutord (2011) make no assumptions about the other terms necessarily being constant, letting F_rad = f(r, θ)g. The unknown function, f(r, θ), is found by requiring that ∇ · F_rad = 0. Using the Roche model and solving the resulting partial differential equation give

$$\begin{equation} T_{\rm {eff}} ={ \left(\frac{L_{\omega }}{\sigma {\it GM}}\right)}^{1/4} \sqrt{\frac{\tan \theta _w}{\tan \theta }} {g^{1/4}}, \end{equation} \tag{ 10 }$$

where θ_w is defined by the requirement that

$$\begin{equation} \cos \theta _w+\ln {\tan \frac{\theta _w}{2}}= \frac{1}{3} x^3 w^2 \cos ^3 \theta +\cos \theta +\ln {\tan \frac{\theta }{2}}. \end{equation} \tag{ 11 }$$

In Equation (11), θ is the spherical coordinate, x is a scaled radius r/r_eq(ω_star), and w is a different fractional angular velocity given by $\omega _{\rm star} \sqrt{\vphantom{A^A}\smash{\hbox{${r^3_{\rm eq} (\omega _{\rm star})/{\it G M}}$}}}$ which is not ω_frac. Equations of this form which describe a variable f(θ) are sometimes written in the equivalent form of $T=f_o g^{\beta _o +\delta (\theta) }$, where f_o is now constant with all variation accounted for in δ(θ). Gravity darkening laws expressed in this form are found in Claret (2012) and Zorec et al. (2011).

Equation (10) predicts lower polar temperatures and higher equatorial temperatures than von Zeipel (1924). This is similar to, but not the same as, reducing β. At slow rotation rates, there is very little difference between the temperatures predicted by Espinosa Lara & Rieutord (2011) and von Zeipel (1924), but the differences increase with rotation and the ratio between them actually diverges at the equator for critical rotation because while both functions find a temperature of zero at the equator for critical rotation, the prediction of Espinosa Lara & Rieutord (2011) goes to zero with rotation more slowly (see Figure 20). Figure 21 shows the effect of using the Equation (10) compared to traditional gravity darkening of von Zeipel (1924) and Collins (1963). It is similar to Figure 19, predicting a weakening of the effects of gravitational darkening. However, Equation (10) reduces the heating of the pole more than the cooling of the equator, in comparison to simply reducing β. This is why using Equation (10) does not affect the density-weighted average temperatures as much as setting β = 0.18. Using Equation (10) has a similar effect on the volume-weighted average temperatures as setting β = 0.18, reducing the size of the initial drop in temperatures as well as reducing the heating at high rotation rates but the effect of setting β to 0.18 is stronger.

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