the Mount Stromlo and Siding Spring Observatories. The Australian National University. ACT 0200, AUSTRALIA. 2 ANU Astrophysical Theory Centre.
Planets Around White Dwarfs Jianke Li1 , Lilia Ferrario2 & Dayal Wickramasinghe2
ANU Astrophysical Theory Centre Department of Mathematics, Faculty of Science & the Mount Stromlo and Siding Spring Observatories The Australian National University ACT 0200, AUSTRALIA
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ANU Astrophysical Theory Centre Department of Mathematics The Australian National University ACT 0200, AUSTRALIA 2
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ABSTRACT
The fate of a planetary system like our own, as the parent star expands through the red giant phase and becomes a white dwarf has been a topic of some discussion. While it appears certain that the outer giant planets will survive this phase, for an Earthlike inner planet, only the conducting core is likely to remain intact. We argue that a planetary core in orbit around a white dwarf may reveal its presence through its interaction with the magnetosphere of the white dwarf. As the planet moves through the magnetosphere, electrical currents will be generated, which will heat the atmosphere of the white dwarf near its magnetic poles leading to the ejection of plumes of hot gas which may be detected in the optical as H emission. Ohmic dissipation will result in the slow decay of the planetary orbit, and such a planet will merge with the white dwarf in less than a Hubble time, unless the initial orbital separation is greater than 10 solar radii. We propose that the peculiar emission line white dwarf GD356 may be a system in the process of such a merger.
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1. Introduction The future of our solar system as the sun evolves through the red giant phase and becomes a white dwarf is an intriguing question which remains unresolved. Of the many possible scenarios, the following appears most plausible. Calculations suggest that the Sun has its maximum radius at the rst thermal pulse as it evolves up the asymptotic giant branch. Depending on mass loss rate, the outer envelope could engulf the innermost terrestrial planets including the Earth1 . The subsequent evolution of the orbits of the inner planets depends on two competing e ects. The total mass of the system is reduced due to the heavy mass loss su ered by the red giant, and as a result the planets tend to move outwards. On the other hand, the drag force encountered by the planet as it moves hypersonically through the envelope could be signi cant2 . Through frictional and other processes, the planet would tend to become ablated, and drift inwards towards the degenerate stellar core. The nal orbit, and the degree of ablation, is dicult to predict, but it is likely that for an Earth type planet, the orbit will shrink, and the outer mantle will be stripped, with only the Fe core surviving. We may thus expect to nd the cores of terrestrial planets, surviving the red giant phase, in close orbit around white dwarfs. Such planets will have a negligible mass in comparison to that of a white dwarf, and they will therefore be dicult to detect through purely dynamical e ects - for instance through radial velocity variations of the spectral lines from the white dwarf. Given the similar radii of planets and white dwarfs, photometric searches for eclipses of the white dwarf by the planet may be more successful. However, only nearly edge on systems would eclipse, and then only for a small fraction of the orbit, so that the probability of detection is small. We propose that there may be another mechanism through which a white dwarf may reveal the presence of a close planetary companion. Roughly 5 percent of isolated white dwarfs are expected to have magnetic elds in the range 105 ? 109 G 3. 3
and all white dwarfs are expected to be magnetic at some level. If a white dwarf is magnetic, then, under certain circumstances, a current system may develop linking the white dwarf and the planet, much like in the Io-Jupiter interaction 4 . The associated electromagnetic processes may have an appreciable e ect on the white dwarf and the planet core. Since the white dwarf can be almost an ideal conductor and the iron core also suciently conducting, closed current loops which connect the two objects can result in a direct heating on the white dwarf atmosphere where the conductivity could be lowest.
2. The model Consider a planetary core of mass Mp in orbit around a magnetic white dwarf of polar magnetic eld B0 and mass Mw . For the sake of de niteness, we assume that the planet has an Earth-like iron core of radius 3500 km and a mean density of between 9.4 g cm?3 and 13 g cm?3. The parameters of our model, where subscripts \w" and \p" are used to denote to the white dwarf and the planet respectively, are as follows: Mw = 0:6M , Rw = 10?2R , B0 = 13 MG, Rp = 0:55R = 3:5 108 cm, Mp = 0:39M = 2:3 1027 g. In addition, we assume that the white dwarf has an aligned dipole and the planet core has a circular orbit around the white dwarf. For simplicity, the white dwarf is assumed to have negligible spin rotation. Fig. 1 describes the model. An Earth-like inner iron core is an electrical conductor, and a motion around the white dwarf (WD) induces an electromotive force, resulting from cutting eld lines, E = ?Vrot B=c. The maximum potential across the planet core is U0 = Rp Rw B0 (Rw =r)2 =c, where c is the velocity of the light and is the angular velocity of the planet's Keplerian motion which is 2 = GMw =r3 , where G is the gravitational constant. The white dwarf magnetosphere is as usual assumed to be a highly conducting medium, and only eld aligned current will be important. A eld-aligned conductivity 4
in a fully ionised hydrogen plasma is = 2 107 T 3=2, where T is the temperature of the plasma. However, the conductivity of the atmosphere depends on the temperature. The conductivity in the electric eld direction across a eld line is ? = =f1+(!e e)2 g, where e is the collision time for the electrons and !e is the Larmour angular frequency of electrons. To show that we adopt the plasma data from a WD atmosphere model constructed for the uniquely peculiar magnetic white dwarf GD356 which appears to exhibit chromospheric type activity 5 . At zero optical depth, the ionised hydrogen number density and temperature are respectively N = 1:1 1013cm?3 , and T = 6:1 103K. These numbers lead to =0 = 9:6 1012 esu and (!e e) =0 = 6:8 105. At optical depth 10, we have N = 4:7 1017cm?3 and T = 1:4 104 K, and they lead to =10 = 3:4 1013 and (!e e ) =10 = 80. It is apparent that in the region = 0 to 10, the electric eld-aligned conductivity across the eld lines is negligible. (Also the Hall conductivity is not important.) We note that the eld aligned conductivity increases as entering deeper into the atmosphere but it does not change dramatically across the atmosphere in the same region. Further deeper, !e becomes smaller and collisions will result in an isotropic conductivity. The current will thus connect in the deep atmosphere, forming a current circuit. If there is a voltage in the system, it will be on either the upper atmosphere or on the planet core. Once a circuit is formed, heating exists in general for both the planet and the atmosphere. If we simply assumed a constant conductivity and a constant current density, the heating rate in the planet Wp = 4I 2=32Rp p . where I is the total current pass through the planet core. Current sheets will be formed near the magnetic poles on the atmosphere, as shown in Fig.2 . To a rst approximation, the current sheets map from the dipole eld lines on the core surface, and the heating rate becomes Ww = I 2 H=4aw d, (see Fig. 2). Since a ' (Rw =r)3=2 Rp =2, we have 8 ( d )( w )( Rw )3=2: Wp =2 = Ww 32 H p r
(1)
The conductivity of a planet core may be estimated by using values deduced for the 5
Earth. We use the conservative value = 1013 esu, and so p ' w . (Such a conductivity results in a magnetic eld decay time scale comparable to 103 years.) d and H are more dicult to estimate precisely, although we expect that d a and H must be smaller than the thickness of the total atmosphere. For Rw =r 100?1, we have a Rp=2000 = 1:75 km. Also, from = 0 to = 10 the thickness is 0.6 km. Thus, we may adopt d=H 1. Putting these together, we expect (Wp =2)=Ww 1, showing that the heating within the planet is negligible compared to the heating in the WD atmosphere. The planet core thus acts like a battery without internal resistivity. This result is partly due to the high conductivity of the iron core, and partly due to the convergence of the magnetic ux tube at the white dwarf surface. The total current in half sphere can be estimated as I = U0 w a(d=H ), and the corresponding total heating rate associated with the current for half atmosphere is Rp2 Rp 2 d R Ww = ( 2 )( ) B0 (GMw ) w ( )( w )17=2 8c Rw H r
(2)
In practice, we set d=H = 1 due to the reason stated previously. The ultimate source of the electromagnetic energy which is dissipated in the above process is the gravitational potential energy of the WD-Planet system. We expect a slow inward drift of the planet mediated by Lorentz torques, where half of the gravitational energy dissipated, and the other half remains as kinetic energy due to Keplerian motion. Since the heating would be on both poles of the white dwarf, we can also write Ww =
1 d ( GMw Mp ) = ? 1 ( GMw Mp )( Rw2 )2v : 2 dt 2r 4 Rw r r
(3)
Here vr is the inward drift velocity of the planet. As noted, the spin increase of a slowly rotating white dwarf as the planet core moves inwards is negligible. Our model predicts that there could be a signi cant input of energy into the atmosphere of the white dwarf. In order to have an impact on observations, we require that the heating luminosity in half atmosphere be a signi cant fraction f of the white dwarf luminosity Lwd . From equation (2), and adopting w = 1013 we estimate that a 6
luminosity fLwd would result in a planet core distance ( Rr ) =200(B0=1:3 107)4=17(Rp=3:5 108 )6=17(Rw =0:01R )?2=17 w
(Mw =0:6M )2=17(w =1013)2=17(Lw =1030)?2=17(f=0:1)?2=17
(4)
Using the parameters given above, an Earth-type core at twice of the solar radius would produce a magnetic heating will be formed rate 1029 erg s?1 will be formed. Conversely, if the heating rate for half atmosphere is the same, the predicted orbit of the core is 2 solar radii. At such a orbit, the orbital period is 609 minutes or 10.2 hours. The inward drift velocity vr can be predicted by (3) given a heating. For an orbital period 609 minutes and a heating rate 1029 erg s?1 , the predicted value is jvr j = 4:23 10?2 cm s?1 = 13:3km yr?1 . The characteristic time for the planet to plunge into the surface surface of the white dwarf is therefore 105 years. If we assume that the conductivity w and d=H does not change during the evolution, the heating power is Ww / (Rw =r)17=2. It follows that a planet with an orbital separation signi cantly greater than 200 Rw will not be subject to signi cant heating. Furthermore, from (3) the drift velocity jvr j / (Rw =r)13=2 so that a planet with an orbital radius signi cantly greater than 200 Rw at the time of white dwarf formation will not quickly merge with the white dwarf. In contrast, if the separation is signi cantly smaller than 200 Rw , the heating is so large that the planet will merge with the white dwarf in a time scale much shorter than 105 years. However, for any system within about 1000 Rw = 10 R , the planet will merge with the white dwarf in less than the Hubble time. The current sample of white dwarfs is expected to have some 100 stars with elds of greater than 10 KG. If one adopts a typical cooling age for a the white dwarf of 109 years, the probability of detecting a system in this phase of high externally heated luminosity is 0.01. We may also conclude that most planetary cores in orbit around white dwarfs are likely to have orbital separations of greater than 1000 Rw and unlikely to be easily detected. 7
3. Application to a possible candidate GD 356 is a magnetic white dwarf with a eld strength of 13 MG and a mass 0.6 M . No rotation has been yet detected nor is there any trace of a companion star. It is a unique system in showing resolved Zeeman triplets of H and H in emission 5;6. Detailed modelling has shown that the region giving rise to emission lines is restricted only to a small fraction of the white dwarf surface, and is remarkably uniform in eld. An interpretation in terms of chromospheric activity appears unsatisfactory given the evidence that the emission region is localised on the white dwarf surface, and that no such phenomenon is observed in other stars with similar physical parameters. Magnetic heating of the polar regions through the interaction with a planetary companion is an attractive possibility which deserves consideration. In this model heating could be achieved deep in the photosphere through Ohmic dissipation. As the planet sweeps around the white dwarf, plumes of hot gas may rise above the photosphere, and emit H line emission as it cools and falls back onto the photosphere. The evidence for a nearly uniform eld in the line emission region nds an easy explanation on this model. The region of activity is expected to be narrowly con ned to the polar regions. Furthermore, the hot plasma would be con ned by gravity to possibly only a few atmospheric scale heights in the vertical direction, and the vertical magnetic eld spread will therefore also be small. The luminosity of the White dwarf in GD356 is about 1030 erg s?1 , and the emission lines contribute about 10% of the total luminosity. The estimates made in the previous section are therefore directly applicable to this star. For a viable model, the companion planet will need to be in a very close orbit with a period of about 600 minutes. Variability in the line emission at this period would be strong con rmation of this model.
4. Conclusions We have shown that a planetary core in orbit around a white dwarf may reveal 8
its presence through its interaction with the magnetosphere of the white dwarf. Such an interaction will generate electrical currents which will directly heat the atmosphere near its magnetic poles leading to the ejection of plumes of hot gas. This gas may be detected in the optical as H emission, and may have been seen in the peculiar white dwarf GD356. The dissipation mechanism that we have identi ed will result in the decay of the planetary orbit, and only planets initially in orbits with radii greater than 10 solar radii will fail to merge with the white dwarf in a Hubble time.
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REFERENCES 1 2 3 4 5
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Figure 1 caption:
The illustration of a white dwarf-planet core system. The eld-aligned electric currents are generated by the orbital motion of a conducting planet core, and a circuit is formed, resulting heating the atmosphere.
Figure 2 Caption:
The geometry of the current sheets on the atmosphere. In this simpli ed model, the current sheets have a thickness d, a radius a and an e ective heating depth H .
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