8.2 Resonant Excitation of Modes
9.1.2 Tidal Heating: Bombs vs. Duds
As modes are excited resonantly during the inspiral of an eccentric white dwarf- compact object binary, the energy transfer to the white dwarf may be sufficient to raise its temperature to the point where runaway thermonuclear burning disassembles the star, producing a Type Ia supernova. For this to occur, the modes must damp on a time-scale shorter than the inspiral time to tidal disruption. We shall revisit this point later, but for the moment we take it as given. In addition, sufficient energy
must be transferred during passage through a sequence of resonances to attain the relevant temperatures. To investigate the plausibility of this scenario, we consider the resonant excitation of the ` = m = 2 f-mode during inspiral, with orbital ec- centricities of less than 0.5. There are two reasons for this: (i) as we have argued above, initial eccentricities of∼0.5 are typical of what we expect from binary forma- tion mechanisms, and (ii) our formalism, as developed in Chapter 6, is not valid for high eccentricities. In addition, the resonant excitation off-modes is unimportant for companion masses &106 M, as the last stable orbits then correspond to high-order harmonics (&40; see Section 9.2 below). Because the location of the last stable orbit is proportional to the companion mass (for large q), it follows that the excitation of f-modes is only of interest for companion masses . 105 M. Accordingly, we focus on this regime.
The heat capacity of a white dwarf is essentially dominated by the ions (e.g., Hansen & Kawaler, 1994), and is therefore given approximately by the ideal gas heat capacity:
CV = 3 2kB
M∗
µmu
, (9.1)
where µis the molecular mass.1 The heat capacities and binding energies for several white dwarfs are shown in Table 9.1. We note that to raise the temperature by
∼ 108 K requires ∼ 1–5 percent of the binding energy. The heat capacities allow us to identify mode amplitudes with effective temperature differences, which is a convenient characterization for the present application:
∆T = Mjωj2 CV
A2j (9.2)
(for modes with m = 0, the right hand side of the above equation has a factor of 1/2). Figure 9.1 shows the correspondence for the` =m = 2 f-mode. An important observation is that a temperature difference of about 108 K corresponds to mode amplitudes of around 0.45–0.65, which are expected to be in the non-linear regime.
1For low temperatures (. 107 K), crystallization of the ions changes the heat capacity signifi- cantly, but this does not affect our calculations as the temperatures we are concerned with are quite
Mass Radius B.E. CV
(M) (108 cm) (1050 erg) (1040 erg K−1)
0.6 8.83 0.43 2.12
1.0 5.71 1.6 3.52
1.4 1.98 5.1 4.93
Table 9.1: The binding energies (B.E.) and heat capacities (CV) for several Chan- drasekhar white dwarfs, assumed to be equal carbon-oxygen mixtures. For helium, the heat capacities are a factor of 7/2 higher.
0 0.2 0.4 0.6 0.8 1
Af22 0
1 2 3 4 5
∆T (108 K)
Figure 9.1: The correspondence between `=m= 2 f-mode amplitudes and effective temperature differences for the white dwarf models listed in Table 9.1. The solid line corresponds to the 0.6M model, and the long and short dashed lines correspond to the 1.0 and 1.4 M models, respectively.
Figure 9.2 shows plots of several gravitational inspiral trajectories in the eccentricity- harmonic plane for the `=m= 2 f-mode with different white dwarf and companion masses. The resonant energy transfer, assuming a zero initial mode amplitude at each resonance, has been used to plot contours of constant ∆T. Also shown are the tidal limit and contours corresponding to constant inspiral times to tidal disruption.
Stellar evolution calculations indicate ignition temperatures of about 2.5×108 K and 8×107 K for thermonuclear burning of carbon and helium, respectively (Kippenhahn
& Weigert, 1990). Thus, we expect that the probability of a detonation becomes sig- nificant for a carbon-oxygen white dwarf if its temperature approaches 2.5×108 K.
It is interesting to note that for a helium white dwarf of identical mass, the ignition temperature is lower, but the heat capacity is higher by a factor of 7/2, so that the required amount of energy for ignition is only a factor of about 1.1 higher than for the carbon-oxygen case.
If we assume that the mode is damped completely between resonances, then the heating of the white dwarf along an inspiral trajectory is given simply by adding up the values of ∆T for each resonance before tidal disruption. In this way, we can identify trajectories which are potentially viable for detonating the white dwarf. It is immediately obvious from Figure 9.2 that, regardless of the white dwarf mass, tidal detonation through resonant excitation of f-modes is not a possibility with a companion mass of 1.4M or less, as the required rise in the temperature cannot be attained before tidal disruption. Therefore, when the companion is either a neutron star or another white dwarf, we can assert that we have a ‘dud’ rather than a ‘bomb.’2 For companion masses of 103 and 105 M, Figure 9.2 provides a rough estimate of the limiting inspiral tracks that separate trajectories for which detonation is a theo- retical possibility from those where detonation can be ruled out. We parametrize the limiting trajectories by their orbital periods at an eccentricity of 0.5. The results are summarized in Table 9.2 (companion masses of 10 and 100 M are also provided for reference). Trajectories with periods longer than those listed in Table 9.2 at an eccen-
a bit higher.
2One can argue that the possibility of detonation with a 1.4M companion still exists if the white dwarf is initially very hot. We assume that this is not the case.
100 200 300
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k
3 3.5 4
6 7
8.4 M0=1.4
M∗=0.6
100 200 300
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k 1
1.5 2
6 7
8.4 M0=103
M∗=0.6
100 200 300
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k - 0.5
0 0.5
6 7
8.4 M0=105 M∗=0.6
20 40 60 80 100
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k 1.5
2 2.5
6 7
8.4 M0=1.4
M∗=1.0
20 40 60 80 100
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k - 0.5
0 0.5
6 7
8.4 M0=103
M∗=1.0
20 40 60 80 100
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k
- 1.5 - 1 - 0.5
6 7
8.4 M0=105 M∗=1.0
5 10 15
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k
- 1 - 0.5 0
6 7
8.4 M0=1.4
M∗=1.4
5 10 15
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k
- 2.5 - 2 - 1.5
6 7
8.4 M0=103
M∗=1.4
5 10 15
Orbital Period (s)
0 0.1 0.2 0.3 0.4 0.5
e 5
10 15 20 25
k
- 4 - 3.5 - 3
6 7 8.4 M0=105 M∗=1.4
Figure 9.2: Several inspiral trajectories for different white dwarf and companion masses (in units of M) are shown in the eccentricity-harmonic plane. In each plot, the solid lines correspond to the trajectories, the short dashed lines are contours of constant ∆T (in Kelvins, and labeled with base-10 logarithms), and the long dashed lines are contours of constant inspiral time to tidal disruption (measured in years, and also labeled with base-10 logarithms). The tidal disruption limit is denoted by the dotted line. The curve corresponding to our assumed threshold for carbon ignition is log(∆T /K) = 8.4 (i.e., ∆T = 2.5×108 K).
H HH
HH
M∗ H
M0
1.4 M 10M 102 M 103 M 105 M
0.6 M - 253 s 415 s 571 s 2008 s
1.0 M - 113 s 198 s 265 s 517 s
1.4 M - 19 s 27 s 43 s 72 s
Table 9.2: Approximate orbital periods at an eccentricity of 0.5 for gravitational radiation inspiral tracks that delineate trajectories for which detonation via resonant excitation of quadrupolar f-modes is a theoretical possibility. For a given pair of white dwarf and companion masses, trajectories with longer periods than the given value are expected to be ‘duds.’ Most trajectories with shorter periods are potential
‘bombs.’ For a companion mass of 1.4M, tidal detonation is ruled out.
tricity of 0.5 are expected to be duds, where as most trajectories with shorter periods are potential bombs. We say ‘most’ rather than ‘all’ because the variation in the tidal limit for different trajectories can introduce strips in the eccentricity-harmonic plane which are duds despite meeting the criterion of Table 9.2. For reference, Figure 9.3 shows the orbital period as a function of eccentricity for gravitational inspiral. The period is shown in units of the period at an eccentricity of 0.5. Note that this plot is scale-free in the sense that it applies to all inspiral trajectories.