CONSTRAINING THE PHYSICS OF AM CANUM VENATICORUM SYSTEMS WITH THE ACCRETION DISK INSTABILITY MODEL

A calculation of the vertical structure of geometrically thin disks reveals the steady state physics underlying the limit cycle behavior: the vertical structure as effective temperature ${{T}_{{\rm eff}}}$ versus surface density ${\Sigma}$ (Meyer & Meyer-Hofmeister 1981; Cannizzo & Wheeler 1984). One finds a hysteretic relation between midplane temperature ${{T}_{{\rm mid}}}$ and surface density ${\Sigma}$, or equivalently, between effective temperature ${{T}_{{\rm eff}}}$ and ${\Sigma}$. Each radius r and viscosity parameter α value has its own S-curve. A parameterization of the results of these calculations allows a determination of the surface densities at the maxima and minima in the S-curve, ${{{\Sigma}}_{{\rm max} }}(r,\alpha )$ and ${{{\Sigma}}_{{\rm min} }}(r,\alpha )$, as well as other physical quantities associated with these extrema, like midplane and effective temperature. One relates the local accretion rate to ${{T}_{{\rm eff},{\rm max} }}$ and ${{T}_{{\rm eff},{\rm min} }}$ using the standard Shakura & Sunyaev (1973) relation

$$\begin{eqnarray}&&\sigma T_{{\rm eff}}^{4}=\frac{3}{8\pi }\frac{G\dot{M}{{M}_{1}}}{{{r}^{3}}}.\end{eqnarray} \tag{ 1 }$$

In the DIM, gas accumulates in quiescence and accretes onto the central object in outburst (e.g., Cannizzo 1993a; Lasota 2001 for reviews). The phases of quiescence and outburst are mediated by the action of heating and cooling fronts that transverse the disk and bring about phase transitions between low and high states, consisting of neutral and ionized gas, respectively. During quiescence, when the surface density ${\Sigma}(r)$ at some radius within the disk exceeds a critical value ${{{\Sigma}}_{{\rm max} }}(r)$, a transition to the high state is initiated; during outburst, when ${\Sigma}(r)$ drops below a different critical value ${{{\Sigma}}_{{\rm min} }}(r)$, a transition to the low state is initiated. Low $\to$ high transitions can begin at any radius, whereas high $\to$ low transitions begin at the outer disk edge.

This situation comes about because of the following: In quiescence the disk is very non-steady so that mass accumulates in the outer regions. The surface density distribution is bounded by ${{{\Sigma}}_{{\rm max} }}(r)$, which increases with radius. (Both ${{{\Sigma}}_{{\rm max} }}(r)$ and ${{{\Sigma}}_{{\rm min} }}(r)$ scale close to linearly with r.) In the outburst disk, however, ${\Sigma}(r)\underset{\thicksim}{\propto }{{r}^{-3/4}}$. The accretion disk mass is conserved, i.e., the disk mass accumulated by the end of quiescence ${\Delta}{{M}_{{\rm cold}}}$ is the same as that in the hot disk ${\Delta}{{M}_{{\rm hot}}}$ immediately after the heating transition has occurred. Therefore, there must be a substantial redistribution of ${\Sigma}(r)$–from a profile $\underset{\thicksim}{\propto }r$ in quiescence to $\underset{\thicksim}{\propto }{{r}^{-3/4}}$ in outburst. Since ${\Sigma}(r)$ is smallest at large radii in the outbursting disk, and since ${\Sigma}(r)\lt {{{\Sigma}}_{{\rm min} }}(r)$ is the condition for the cooling transition to begin, cooling fronts are always initiated in the outer disk.

One can define a "maximum mass" ${\Delta}{{M}_{{\rm max} }}\equiv \int 2\pi rdr{{{\Sigma}}_{{\rm max} }}(r)$ that the disk could possibly achieve during the accumulation phase, i.e., quiescence. Obviously one cannot have ${\Sigma}(r)\gt {{{\Sigma}}_{{\rm max} }}(r)$ in quiescence or else the instability would have already been triggered. In practice, time-dependent calculations show that the true disk mass at the end of quiescence ${\Delta}{{M}_{{\rm cold}}}$ is typically ∼$1/10-1/3$ of the "maximum mass" (e.g., Cannizzo 1993b). Therefore, one may write ${\Delta}{{M}_{{\rm cold}}}=f{\Delta}{{M}_{{\rm max} }}$, with $f\simeq 1/10-1/3$.

Levitan et al. (2015) restrict their attention to superoutbursts in their outburst statistics. These are analogous to long outbursts in DNe above the period gap (van Paradijs 1983); most of the mass accumulated during quiescence accreted onto the primary before the disk shuts off. For normal, "short" outbursts, only a few percent of the stored gas accretes onto the central object: the thermal timescale of the thin disk is short compared to the viscous timescale, and the cooling front that is launched from the outer edge of the disk almost as soon as the disk enters into outburst traverses the disk and reverts it back to quiescence. For disks that have been "filled" to a higher level with respect to ${\Delta}{{M}_{{\rm max} }}$, the surface density in the outer disk can significantly exceed the critical surface density ${{{\Sigma}}_{{\rm min} }}$. In order for the cooling front to begin, however, the outer surface density ${\Sigma}({{r}_{{\rm outer}}})$ must drop below ${{{\Sigma}}_{{\rm min} }}({{r}_{{\rm outer}}})$. Disks in this state generate much longer outbursts, with slower "viscous" plateaus, because the entire disk must remain in its high, completely ionized state until enough mass has been lost onto the primary for the condition ${\Sigma}({{r}_{{\rm outer}}})\lt {{{\Sigma}}_{{\rm min} }}({{r}_{{\rm outer}}})$ to be satisfied.

Various studies have investigated the DIM in AM CVn systems (Smak 1983a; Cannizzo 1984; Tsugawa & Osaki 1997 = TO97; El-Khoury & Wickramasinghe 2000; Menou et al. 2002; Lasota et al. 2008 = LDK; Kotko et al. 2012 = KLDH). These investigations generally consist of first calculating the steady-state accretion disk structure by integrating the vertical structure equations to parameterize the S-curve relation between surface density ${\Sigma}(r,\alpha )$ and effective temperature ${{T}_{{\rm eff}}}(r,\alpha )$, and then using scalings for the steady-state physics as input into a time-dependent model to calculate light curves. TO97 did not solve the full set of equations for the vertical structure but rather prescribed a functional form for the flux F(z). We restrict our consideration of the scalings to the two most recent studies LDK and KLDH, which integrate the complete set of structure equations and present a complete set of scalings for both the local minima and maxima in Σ.

As a model for the mass-losing secondaries in AM CVns, Deloye et al. (2005) calculate pseudo-evolutionary sequences for donors with varying degrees of degeneracy. In their Figure 2 they calculate tracks in the ${{\dot{M}}_{T}}$–${{P}_{{\rm orb}}}$ plane for four isotherms based on central donor temperatures T_c ranging from 10⁴ to 10⁷ K, assuming a constant primary mass $0.6\;{{M}_{\odot }}$. They also indicate the upper and lower bounds of the instability strip for the DIM taken from TO97. For their two lowest tracks, ${{T}_{c}}={{10}^{4}}$ and 10⁶ K, the instability strip spans roughly the correct (i.e., observed) period range. Their tracks do not represent true evolutionary sequences since T_c is taken to be constant along a track.

The range of mass transfer from the secondary star feeding into the outer disk ${{\dot{M}}_{T}}$ that allows for unstable behavior, i.e., DN outbursts, is set by the local stability criteria at the inner and outer edge of the accretion disk.

KLDH calculate many S-curves for accretion disks relevant for ultracompact binaries with no hydrogen, X = 0. The basic finding is that the steady-state scalings for the DIM are shifted to higher surface densities and temperatures. They present three sets of scalings for (i) Y = 1, (ii) Y = 0.98, Z = 0.02, and (iii) Y = 0.96, Z = 0.04. Their results are given in terms of α, r, and M₁. The range for unstable disk behavior is determined by the S-curve for the inner and outer disk radii. If ${{\dot{M}}_{T}}\gt {{\dot{M}}_{T,2}}\equiv {{\dot{M}}_{{\rm min} ,{\rm outer}}}$, the rate of accretion associated with the minimum in Σ at ${{r}_{{\rm outer}}}$, the disk will be stable in the high, ionized state. If ${{\dot{M}}_{T}}\lt {{\dot{M}}_{T,1}}\equiv {{\dot{M}}_{{\rm max} ,{\rm inner}}}$, the rate of accretion associated with the maximum in Σ at ${{r}_{{\rm inner}}}$, the disk will be stable in the low state.

The KLDH scalings are not convenient; for comparison with observations we must replace disk radius r with orbital period ${{P}_{{\rm orb}}}$ in the KLDH scaling for ${{\dot{M}}_{T,2}}$, and with the primary radius in the ${{\dot{M}}_{T,1}}$ scaling. We follow TO97 in making these conversions:

1. For the outer scaling, ${{\dot{M}}_{T,2}}$, we relate the outer disk radius ${{r}_{{\rm outer}}}$ to orbital period using Figure 5 from van van Haaften et al. (2012). We identify the Roche lobe radius R_L1 with ${{r}_{{\rm outer}}}$ and fit a simple power law to ${{R}_{L1}}/a$ as a function of the mass ratio $q={{m}_{2}}/{{m}_{1}}$.⁵ The semimajor axis is a. Over the relevant range $0.01\lesssim q\lesssim 0.03$ we fit ${{R}_{L1}}/a\simeq 0.85{{(q/0.01)}^{-0.06}}$. Based on the results of van Haaften et al. (2012), we adopt a fiducial secondary star ${{m}_{2}}({{P}_{{\rm orb}}})$ relation

$$\begin{eqnarray}&&{{m}_{2}}=0.038{{\left( \frac{{{P}_{{\rm orb}}}}{1000\;{\rm s}} \right)}^{-1.3}}\end{eqnarray} \tag{ 2 }$$

(see their Section 2.3), corresponding to ${{R}_{2}}\underset{\thicksim}{\propto }M_{2}^{-0.18}$, which gives, using Kepler's law,

$$\begin{eqnarray}&&{{r}_{o,10}}=1.18m_{1}^{0.39}{{(1+q)}^{1/3}}{{\left( \frac{{{P}_{{\rm orb}}}}{1000\;{\rm s}} \right)}^{0.74}},\end{eqnarray} \tag{ 3 }$$

where ${{r}_{o,10}}={{r}_{{\rm outer}}}/{{10}^{10}}$ cm.

2. For the inner scaling, ${{\dot{M}}_{T,1}}$, we adopt the standard inner boundary zero torque condition (Shakura & Sunyaev 1973) for which the maximum effective temperature

$$\begin{eqnarray}&&{{T}_{{\rm eff}}}({\rm max} )=0.488{{\left( \frac{3G{{M}_{{\rm wd}}}\dot{M}}{8\pi \sigma {{R}_{{\rm wd}}}} \right)}^{1/4}}\end{eqnarray} \tag{ 4 }$$

is reached at 49/36 times the inner edge, taken to be the primary radius. (The "max" in this equation refers to the local radial maximum in ${{T}_{{\rm eff}}}$ in a steady-state disk due to the inner boundary condition, not the local maximum in Σ in the S-curve.) Thus, ${{\dot{M}}_{T,1}}$ is enhanced by a factor ${{(0.488)}^{-4}}$ and evaluated at $(49/36){{R}_{{\rm wd}}}$.

For the primary mass–radius relation, we fit a power law to Eggleton's scaling of a zero-temperature WD, i.e., for which the ratio of atomic number to atomic weight $Z/A=1/2$ (Rappaport et al. 1987); see their Equation (19), with ${{M}_{{\rm Ch}}}=1.44\;{{M}_{\odot }}$). Taking the scaling relevant for a He WD (van Haaften et al. 2012); see their Equation (25), with ${{M}_{p}}=5.66\times {{10}^{-4}}\;{{M}_{\odot }}$), we tabulate values for $x={\rm log} {{M}_{{\rm WD}}}$ and $y={\rm log} {{R}_{{\rm WD}}}$ over the range of interest, $0.5\lt {{m}_{1}}\lt 0.7$, and fit a least-squares power law ${{R}_{{\rm WD}}}={{10}^{8.80}}$ cm $m_{1}^{-0.62}$. Over the fit range this relation gives a maximum deviation $\lt 1.5\%$ from the Eggleton scaling.

Applying these conversions to the scalings given in KLDH (and ignoring the weak α dependencies), we obtain scalings for ${{\dot{M}}_{T,1}}$ and ${{\dot{M}}_{T,2}}$ relevant for their three compositions (i) Y = 1, (ii) Y = 0.98, Z=0.02, and (iii) Y = 0.96, Z = 0.04. Adopting the general forms

$$\begin{eqnarray}&&{{\dot{M}}_{T,1}}={{\dot{M}}_{T,1,0}}\ {{m}_{1}}{{({{P}_{2}})}^{{{\epsilon }_{1m1}}}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{\dot{M}}_{T,2}}={{\dot{M}}_{T,2,0}}\ {{m}_{1}}{{({{P}_{1}})}^{{{\epsilon }_{2m1}}}}{{(1+q)}^{{{\epsilon }_{q}}}}{{\left( \frac{{{P}_{{\rm orb}}}}{1000\;{\rm s}} \right)}^{{{\epsilon }_{p}}}},\end{eqnarray} \tag{ 6 }$$

we may now calculate the coefficients. These are given in Table 1.

Table 1.  KLDH-based Coefficients for Local Extrema

Composition	${{\dot{M}}_{T,1,0}}$ $({{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}})$	${{\dot{M}}_{T,2,0}}$ $({{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}})$	${{\epsilon }_{1m1}}$	${{\epsilon }_{2m1}}$	${{\epsilon }_{q}}$	${{\epsilon }_{p}}$
(i)	$1.29\times {{10}^{-11}}$	$2.49\times {{10}^{-9}}$	−2.54	0.16	0.89	2.0
(ii)	$9.19\times {{10}^{-12}}$	$1.53\times {{10}^{-9}}$	−2.50	0.16	0.89	1.99
(iii)	$8.03\times {{10}^{-12}}$	$1.17\times {{10}^{-9}}$	−2.49	0.16	0.88	1.97

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Levitan et al. (2015) presents a list of AM CVn systems ordered by ${{P}_{{\rm orb}}}$, with outburst properties indicated. The rate of mass transfer ${{\dot{M}}_{T}}$ decreases sharply with ${{P}_{{\rm orb}}}$, so that the shortest ${{P}_{{\rm orb}}}$ systems have disks in permanent outburst, those with intermediate ${{P}_{{\rm orb}}}$ exhibit outbursts, and those with the longest ${{P}_{{\rm orb}}}$ have disks in permanent low states. The dividing point between high state systems and outbursting systems lies at ${{P}_{1}}\approx 20$ minutes. The dividing point between outbursting systems and low state systems, P₂, is not as straightforward. A block of systems starting at ${{P}_{{\rm orb}}}=44.3$ minutes are listed as not showing outbursts,⁶ but two systems among these do show outbursts, at ${{P}_{{\rm orb}}}=47.3$ and 48.3 minutes. The expressions for ${{\dot{M}}_{T,1}}$ indicate a steep inverse scaling with m₁; therefore, it seems probable that these systems have somewhat high primary masses. Therefore, we set ${{P}_{2}}=44$ minutes. Furthermore, we adopt ${{m}_{1}}({{P}_{1}})={{m}_{1}}({{P}_{2}})=0.6$. We note that the m₁ values for AM CVn systems are not well constrained observationally from dynamical measurements.

We may now use the observed instability strip for AM CVn systems to constrain the secondary mass transfer rate ${{\dot{M}}_{T}}$. Let us assume a power law ${{\dot{M}}_{T}}=A{{\left( {{P}_{{\rm orb}}}/1000\;{\rm s} \right)}^{n}}$. For simplicity we adopt the convention that ${{\dot{M}}_{T}}\gt 0$. Since we expect ${{\dot{M}}_{T}}$ to decrease with orbital period, we set ${{\dot{M}}_{T,1}}={{\dot{M}}_{T}}({{P}_{2}})$ and ${{\dot{M}}_{T,2}}={{\dot{M}}_{T}}({{P}_{1}})$, which gives two equations in two unknowns. We may solve for the normalization constant A and dependence on orbital period $n=d{\rm ln} {{\dot{M}}_{T}}/d{\rm ln} {{P}_{{\rm orb}}}$. This yields the general solution

$$\begin{eqnarray}&&n=\frac{{\rm log} [({{{\dot{M}}}_{T,2,0}}/{{{\dot{M}}}_{T,1,0}})g{{({{P}_{1}}/1000\;{\rm s})}^{{{\epsilon }_{p}}}}]}{{\rm log} ({{P}_{1}}/{{P}_{2}})}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&A={{\dot{M}}_{T,1,0}}{{m}_{1}}{{({{P}_{2}})}^{{{\epsilon }_{1m1}}}}{{\left( \frac{{{P}_{2}}}{1000\;{\rm s}} \right)}^{-n}},\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&g={{(1+q)}^{{{\epsilon }_{q}}}}m_{1}^{{{\epsilon }_{2m1}}-{{\epsilon }_{1m1}}}.\end{eqnarray} \tag{ 9 }$$

Table 2 gives the values for A and n for the three KLDH compositions, adopting ${{P}_{1}}=20$ minutes, ${{P}_{2}}\;=$ 44 minutes, and ${{m}_{1}}=0.6\pm 0.05$. The magnitude of the putative assigned error on m₁ was propagated through to n and A in order to indicate the strength of the dependency. The main uncertainty entering into n and A is the assumed primary mass. For instance, taking ${{m}_{1}}=1$ would give $n\simeq -7$, a much steeper ${{\dot{M}}_{T}}({{P}_{{\rm orb}}})$ relation.

Table 2.  Coefficients for ${{\dot{M}}_{T}}=A{{({{P}_{{\rm orb}}}/1000\;{\rm s})}^{n}}$ with ${{m}_{1}}=0.6\pm 0.05$

Composition	$A$ $({{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}})$	n
(i)	$8.82\pm 0.6\times {{10}^{-9}}$	−5.38 ± 0.3
(ii)	$5.27\pm 0.4\times {{10}^{-9}}$	−5.23 ± 0.3
(iii)	$3.88\pm 0.3\times {{10}^{-9}}$	−5.06 ± 0.3

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Since precise abundances for AM CVns as a group are not known, we adopt a representative fiducial scaling ${{\dot{M}}_{T}}\simeq 5\times {{10}^{-9}}\;{{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}$ ${{({{P}_{{\rm orb}}}/1000\;{\rm s})}^{-5.2}}$, relevant for the middle range of scalings given in KLDH.

How does this ${{\dot{M}}_{T}}$ law compare to theoretical expectations? As mentioned above, in AM CVn systems the driving of the mass transfer is most likely the AML due to GWR. For that case one can derive the expected scaling of mass transfer rate with orbital period as is done in Warner (1995b). We begin by expressing the mass transfer rate in terms of the AML

$$\begin{eqnarray}&&\frac{\dot{{{M}_{2}}}}{{{M}_{2}}}\propto {{\left( \frac{{\dot{J}}}{{{J}_{{\rm orb}}}} \right)}_{{\rm GWR}}}\propto \frac{{{M}_{1}}{{M}_{2}}({{M}_{1}}+{{M}_{2}})}{{{a}^{4}}}\end{eqnarray} \tag{ 10 }$$

(e.g., see Savonije et al. 1986; Marsh et al. 2004; Equation (6) of van Haaften et al. 2012), where ${{\dot{M}}_{2}}=-{{\dot{M}}_{T}}$. Assuming that M₁ remains constant and ${{M}_{2}}\ll {{M}_{1}}$ and using Kepler's law to replace orbital separation with orbital period, we find

$$\begin{eqnarray}&&\frac{\dot{{{M}_{2}}}}{{{M}_{2}}}\propto {{M}_{2}}P_{{\rm orb}}^{-8/3}.\end{eqnarray} \tag{ 11 }$$

We then can use the fact that the size of the Roche lobe for a given period depends only very weakly on the mass of the accretor (the well-known period–mean density relation) to find how ${{P}_{{\rm orb}}}$ scales with M₂, given the mass–radius relation for the donor ${{R}_{2}}\propto M_{2}^{\zeta }$. Hence,

$$\begin{eqnarray}&&{{P}_{{\rm orb}}}\propto {{\left( \frac{R_{2}^{3}}{{{M}_{2}}} \right)}^{1/2}}\propto M_{2}^{(3\zeta -1)/2}\end{eqnarray} \tag{ 12 }$$

and therefore

$$\begin{eqnarray}&&\dot{{{M}_{2}}}\propto M_{2}^{2}P_{{\rm orb}}^{-8/3}\propto P_{{\rm orb}}^{4/(3\zeta -1)-8/3}.\end{eqnarray} \tag{ 13 }$$

For $\zeta =-1/3$ we find an exponent⁷ $n=-14/3$. In van Haaften et al. (2012) the values of ζ are plotted in their Figure 2. For low masses $\zeta \gt -1/3$, yielding a larger absolute value of the exponent. Indeed, the fit they make to the dependence of mass transfer rate on orbital period (their Appendix A) gives $n=-5.32$ for a 1.4 ${{M}_{\odot }}$ accretor. Warner (1995b) uses $\zeta =-0.19$, based on Savonije et al. (1986), and finds $n=-5.21$. We conclude that the exponent for the scaling we infer from the DIM based on the observed instability strip for AM CVns is in good agreement with expectations from stellar structure if we adopt ${{m}_{1}}\simeq 0.6$.