We propose a scenario to form low-mass, single, slow rotating white dwarfs
from a solar-like star accompanied by a massive planet, or a brown dwarf, in a
relatively close orbit (e.g. HD 89707). Such white dwarfs were recently found
by Maxted & Marsh (1998). When the solar-like star ascends the giant branch it
captures the planet and the subsequent spiral-in phase expels the envelope of
the giant leaving a low-mass helium white dwarf remnant. In case the planet
evaporizes, or fills its own Roche-lobe, the outcome is a single undermassive
white dwarf. The observed distribution of planetary systems supports the
applicability of this scenario.Comment: 4 pages, accepted for publication in A&A letter

Nelemans, G.

Tauris, T. M.

English

arXiv

Nelemans, G.A.

Tauris, Th.M.

NARCIS 

Formation of undermassive single white dwarfs and the influence of planets on late stellar evolution

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Formation of undermassive single white dwarfs and the influence of planets onlate stellar evolutionNelemans, G.A.; Tauris, Th.M.Publication date1998Published inAstronomy & AstrophysicsLink to publicationCitation for published version (APA):Nelemans, G. A., & Tauris, T. M. (1998). Formation of undermassive single white dwarfs andthe influence of planets on late stellar evolution. Astronomy & Astrophysics, 335, L85-L88.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:11 Mar 2023Astron. Astrophys. 335, L85–L88 (1998) ASTRONOMYANDASTROPHYSICSLetter to the EditorFormation of undermassive single white dwarfsand the influence of planets on late stellar evolutionG. Nelemans and T.M. TaurisAstronomical Institute “Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The NetherlandsReceived 18 May 1998 / Accepted 1 June 1998Abstract. We propose a scenario to form low-mass, single,slow rotating white dwarfs from a solar-like star accompaniedby a massive planet, or a brown dwarf, in a relatively close orbit(e.g. HD 89707). Such white dwarfs were recently found byMaxted & Marsh (1998). When the solar-like star ascends thegiant branch it captures the planet and the subsequent spiral-inphase expels the envelope of the giant leaving a low-mass he-lium white dwarf remnant. In case the planet evaporizes, or fillsits own Roche-lobe, the outcome is a single undermassive whitedwarf. The observed distribution of planetary systems supportsthe applicability of this scenario.Key words: stars: giant, mass-loss, planetary systems – bina-ries: evolution – white dwarfs: formation1. IntroductionRecent searches for double degenerates (two white dwarfs ina binary; Marsh 1995; Marsh, Dhillon & Duck 1995) have re-sulted in the discovery of two single, low-mass helium whitedwarfs – cf. Table 1. Similar undermassive white dwarfs (<∼0.5M) are usually found in binaries and can not be formedfrom normal, single star evolution which leaves a>∼ 0.6MC-O white dwarf as a remnant. Any potential single, low-massprogenitor star of these newly discovered undermassive whitedwarfs can be excluded, since they would have a main sequencelifetime exceeding the age of our Milky Way. A scenario hasbeen proposed (Iben, Tutukov & Yungelson 1997) in which adouble degenerate has merged, due to the emission of gravita-tional wave radiation. According to Maxted & Marsh (1998),this scenario predicts high (∼ 1000 km s−1) rotational veloci-ties for the remnant of the merged objects in contradiction withtheir measurements of a maximum projected rotational veloc-ity of only ∼ 50 km s−1. Therefore the merger scenario seemsquestionable – unless there is an extremely efficient removal ofangular momentum in the merging process, or the inclinationangles for both these systems are extremely small.Send offprint requests to: gijsn@astro.uva.nlTable 1.Properties of the two recently discovered undermassive whitedwarfs – cf. Marsh, Dhillon & Duck 1995; Maxted & Marsh 1998.Name Mass (M) vrot sin i ( km s−1) d (pc)WD 1353+409 0.40 < 50 130WD 1614+136 0.33 < 50 180In this letter we suggest a different, simple solution to the for-mation of these single, low-mass (undermassive) white dwarfsby investigating the influence of massive planets, or browndwarfs, in relatively close orbits around solar-like stars (Sect. 2).A short discussion of the consequences of our planetary scenariois given in Sect. 3.2. Planets around solar-like stars2.1. IntroductionWe propose a scenario in which a solar-like star is surroundedby a massive planet, or a brown dwarf, in a relatively close or-bit. When the star evolves on the giant branch it will becomebig enough to capture its planet via tidal forces (cf. Rasio et al.1996; Soker 1996). The planet spirals into the envelope of thegiant and a so-called common envelope phase is initiated. Thefrictional drag on the planet, arising from its motion through thecommon envelope, will lead to loss of its orbital angular mo-mentum (spiral-in) and deposit of orbital energy in the envelope.The orbital energy is converted into thermal and kinetic energyof the envelope which is therefore being ejected. The result ofthis common envelope evolution is determined by the energybalance and the fate of the planet. As a result of friction, andthe large temperature difference between the envelope of thegiant and the equilibrium temperature of the planet, low-massplanets evaporize due to heating. In case the planet evaporizescompletely, the outcome will be a single star with a rotating andreduced envelope – otherwise we end up with a planet orbitingthe naked core of a giant. The destiny of this white dwarf-planetsystem is determined by the orbital separation.In this letter we first present the expected outcome of a com-mon envelope evolution between a giant and a planet; thereafterLETTERL86 G. Nelemans & T.M. Tauris: Formation of undermassive white dwarfswe look at the important question of the onset of this evolution.We will closely follow the treatment of Soker (1996; 1998),focusing on the cases where (most of) the envelope is lost ina common envelope, leaving a undermassive white dwarf. Re-search in this field has been carried out to explain elliptical andbipolar planetary nebulae (Soker 1996) and the morphology ofthe Horizontal Branch in clusters (Soker 1998).2.2. The outcome of the common envelope phaseBelow we outline our scenario in somewhat more detail. Bysimply equating the difference in orbital energy to the bindingenergy of the envelope of the giant we can compute the ratioof final to initial separation (Webbink 1984). Letηce describethe efficiency of ejecting the envelope,i.e.of converting orbitalenergy into the kinetic energy that provides the outward motionof the envelope:∆Ebind ≡ ηce ∆Eorb or (usingmp  Menv):af ' ηce λ2Mcore mpM MenvRg = f(χ1 − χ)mpMRg (1)whereRg is the radius of the giant star at the onset of the spiral-in phase,λ is a weighting factor (< 1.0) for the binding energyof the core and envelope of the giant star,χ ≡ Mcore/M , mpis the planetary mass andMcore, Menv andaf are the mass ofthe helium core and hydrogen-rich envelope of the evolved star(M = Mcore + Menv), and the final separation after all theenvelope is expelled, respectively. In our calculations we choseλ = 0.5 and ηce = 4 (cf. Tauris 1996; Portegies Zwart &Yungelson 1998) and hencef = 1.To model the effect of planetary evaporation we followSoker (1998) and equate the local sound speed in the giantsenvelope to the escape velocity from the (gaseous) planet sur-face in order to find the approximate location of evaporation:c2s ≈ v2esc ⇐⇒ γkBTµ mu≈ 2 G mpα rp(2)We use a temperature profile for evolved solar-like stars (cf.Fig. 1) ofT ≈ 1.78× 106 (r/R)−0.85K, in the entire intervalof Rcore < r < Rg, whereRcore is the radius of the He-core.During the spiral-in the radius of a giant-gas planet,rp, mayexpand slightly (α rp, α > 1) even though only a small amountof mass (< 0.1 mp) is believed to be accreted (Hjellming &Taam 1991).Solving Eq. (2), with the temperature dependence givenabove and assumingγ = 5/3 and Pop.I chemical abundances(X=0.7; Z=0.02), yields the location of the evaporation:aevap =[10 α(MJmp)]1.18R (3)whereMJ = 0.001M (≈ a Jupiter mass) and we have as-sumedrp = 0.1R, which is a reasonable assumption forall planets and brown dwarfs in the mass range0.0001 <mp/M < 0.08 (Hubbard 1994).For a given stellar structure (i.e.core and envelope mass andradius) the final outcome of the common envelope phase is de-termined only by the mass of the planet. We can easily computeFig. 1. The temperature profiles for 1M evolved stars on the RedGiant Branch. Notice, that logT is approximately a linear function oflog r at all evolutionary stages from the beginning until the tip of theRed Giant Branch (just before the helium flash).the critical planetary mass for which the planet evaporizes justat the moment the envelope is completely expelled,i.e. whenaevap = af . The mass associated with this critical mass (mcrit)is found from Eqs. 1 and 3 (α = 1):mcrit = 10[(1 − χχ) (MM) (Rg100 R)]0.46MJ (4)Planets more massive thanmcrit survive the spiral-in. However,in order to avoid a destructive mass transfer to the white dwarfafter the spiral-in, it must have a radius smaller than its Roche-lobe given by (Paczynski 1971):aRLO =α rp0.462(MWDmp)1/3R (5)whereMWD = Mcore.If af > aevap andaf > aRLO, the planet will survive and theentire envelope is lost from the giant leaving a low-mass heliumwhite dwarf remnant with a planetary companion. However, ifthe final separation is small enough, the planetary orbit willdecay due to emission of gravitational waves on a timescalegiven by:τgwr ≈ (af/60 RWD)4(MWD/M)2 (mp/MJ)5.0 × 109 yr (6)Hence, also in this case the final outcome of the evolution mighteventually be a single undermassive white dwarf.Planets less massive thanmcrit will evaporate (or overflowtheir Roche-lobe ifaRLO > aevap) before the envelope is ex-pelled completely. However heavy planets deposit significantorbital angular momentum in the envelope of the giant, causingenhanced mass loss due to rotation. This could lead to ejectionof the envelope by planets somewhat less massive thanmcrit.The change in structure of the star may alter the further evo-lution of the giant considerably. Soker (1998) suggests that suchLETTERG. Nelemans & T.M. Tauris: Formation of undermassive white dwarfs L87an evolution could explain the morphology of the HorizontalBranch in clusters.For the evolution of the giant we used the relations of Iben &Tutukov (1984) for the structure of a (Pop.I) giant on the RGB:Rg = 103.5M4core, L = 105.6M6.5core, Ṁcore = 10−5.36M6.6core.These equations are valid on the RGB for a low-mass star (0.8 ≤M/M ≤ 2.2).2.3. The onset of the common envelope phase:tidal forces and mass loss on the RGBThe moment the common envelope starts is determined by tidalforces. In the absence of any significant tidal interaction thedonor star is only able to capture planets, via Roche-lobe over-flow, out to a distance,amaxi ≈ 1.6 Rg. Taking tidal effects intoaccount using the equilibrium tide model (Zahn 1977; Verbunt& Phinney 1995) we find, following Soker (1996):amaxi '2.4 Rg(1 − χχ9)1/12 (MM)−11/12 (mp10MJ)1/8(7)where we have used the equations for the structure of the giantas given above. In our calculations (see below) we have alsoincluded mass loss, which amounts to as much as|∆M |/M ≈0.20 at the tip of the RGB. The mass is lost as a fast isotropicwind with the specific angular momentum of the giant causingthe orbital separation of the planet to increase by the same ratioas the total mass of the system decreases. We modeled this effectaccording to the Reimers formula (Kudritzki & Reimers 1978)with η = 0.6 (cf. Rasio et al. 1996).2.4. ResultsWe will now demonstrate an approximate picture for the fate ofstars with planets of different masses and separations to illustratethe applicability of this scenario for producing undermassivesingle white dwarfs as observed in nature. For the evolution ofa solar-like star on the Red Giant Branch we will investigate towhich separation (or alternatively which orbital period) a givenplanet will be captured by the star and compute the outcome ofthe spiral-in process for different planetary masses.In Fig. 2 we have plotted the different critical separationsdiscussed above as a function of planetary mass. Our exampleis based on a 1.0M star with a core-mass of 0.33M (cf. WD1614+136 in Table 1). We findmcrit = 21 MJ. Less massiveplanets expel only part of the envelope (e.g. a planet withmp =15 MJ will only expel half of the envelope, neglecting enhancedmass loss of the giant due to the spin-up of the envelope). Planetswith masses between 15 and25 MJ are presumably disruptedas they fill their Roche-lobe during/after the spiral-in1. Planetsmore massive than∼ 25 MJ survive the spiral-in and will ejectthe entire envelope. However ifmp < 32 MJ, the planet willspiral in, due to emission of gravitational waves, and hence fillits Roche-lobe within 5 Gyr.1 The final fate of the planet depends on its adiabatic exponent, oractually (∂ ln r/∂ ln m) and requires detailed calculations.Fig. 2.Separations of interest (in units ofRWD = 10 000 km) after thespiral-in phase for a 1M star with a core of 0.33M as a functionof planetary mass. The solid line gives the separation for which theliberated orbital energy is equal to the binding energy of the envelope(dotted line for ejecting half of the envelope). The dashed line gives theseparation below which the planet fills its Roche-lobe. The dash-dottedline gives the separation at which the planet is evaporated. A minimumplanetary mass of∼ 21 MJ is needed to expel the entire envelope.Planets lighter than this value are seen to be evaporated. However, for15 < mp/MJ < 25 the planet fills its Roche-lobe and is likely tobe disrupted as a result. Planets more massive than∼ 25 MJ survivethe common envelope phase but will later spiral in due to gravitationalwave radiation (shaded area indicates a spiral-in timescale of less than5 Gyr). Above 0.08M (80 MJ), the companions are heavy enoughto ignite hydrogen as stars (hatched region).In Fig. 3 (top) we calculated the final outcome of theevolution of a planet orbiting a 1M star as a functionof planetary mass and initial orbital period. We also plot-ted some of the known planetary and brown dwarf systemswith solar-like stars (0.70–1.20M). Data were taken from“The Extrasolar Planets Encyclopaedia” (wwwusr.obspm.-fr/departement/darc/planets/encycl.html ). We no-tice that, of the observed systems HD 89707 and HD 140913are the best candidates for producing single undermassive whitedwarfs. In HD 217580, HD 18445 and HD 29587 the planet isexpected to survive the ejection of the envelope. In HD 114762and 70 Vir they are captured already early on the RGB, wherethe binding energy of the envelope is too large to be expelled,so these planets will evaporate shortly after contact with theevolved donor star. The solitary white dwarfs resulting fromthese two systems will therefore be normal C-O white dwarfs.3. DiscussionWe must bear in mind the uncertainties at work in our scenario,and it is possible that future detailed studies of the interactionsbetween a planet and a common envelope may change the masslimits derived in this letter. Also notice that the two undermas-sive white dwarfs in Table 1 might very well have substellarLETTERL88 G. Nelemans & T.M. Tauris: Formation of undermassive white dwarfsFig. 3. a Final outcome of the common envelope phase for differentplanetary masses and initial periods around a 1M star. The solid lineindicates the critical mass,mcrit, below which the planet will evaporizeduring the spiral-in. Above the solid line the planet survives the spiral-in phase and the outcome is an undermassive white dwarf with a planetorbiting it – unless the initial period is sufficiently short leading to adisruption of the planet as it fills its Roche-lobe (left shaded area) afterthe spiral-in. The dash-dotted line indicates the limiting initial periodsbelow which the planet will fill its Roche-lobe, after the spiral-in, inless than 5 Gyr due to gravitational wave radiation. The dotted lineyields the planetary mass for which half of the envelope is ejected– neglecting rotation (see text). Also indicated in the figure are theobserved extrasolar planets and brown dwarfs. In the shaded area tothe right, the planet is too far away from the giant to be engulfed in itsenvelope during evolution on the Red Giant Branch.b Final mass of thewhite dwarf in case all of the envelope is expelled (i.e.mp > mcrit).companions (brown dwarfs or planets) below the observationalthreshold mass of∼ 0.1 M.3.1. The final mass of the white dwarfIn the lower panel of Fig. 3 we give the final white dwarf massin case all of the envelope is expelled. We see that white dwarfswith masses between 0.20–0.45M can in principle be formedwith this scenario. If the common envelope phase initiates whilethe donor is on the Asymptotic Giant Branch, a C-O white dwarfwill be formed.3.2. Rotation of the white dwarfThe final rotational period of the white dwarf is essentially de-termined only by the rotation of the core of the giant: the planettransfers almost all of its angular momentum to the giants enve-lope which is expelled. The rotation of the core strongly dependson the coupling between the core and the envelope of the giant(Spruit 1998), but is in any case in agreement with the the mea-sured upper-limits for the white dwarfs as given in Table 1.3.3. A white dwarf ejected from a binary?An other possibility for the formation of single undermassivewhite dwarfs is a binary origin. Consider a compact system witha giant star (the progenitor of the undermassive white dwarf)and a normal white dwarf companion. When the giant fills itsRoche-lobe it transfers its envelope to the companion leavinga low-mass helium white dwarf as a remnant. The companionmay be subsequently lost either because it exploded as a type IaSNe, or formed a (high velocity) neutron star from an accretioninduced collapse – also leading to disruption of the binary.Acknowledgements.We would like to thank Frank Verbunt for dis-cussions and Bart Bisscheroux for providing Fig. 1. This research wassupported in part by the NWO Spinoza-grant SPI 78-327. T.M.T. ac-knowledges the receipt of a Marie Curie Research Grant from the Eu-ropean Commision.ReferencesHjellming M. S., Taam R. E., 1991, ApJ. 370, 709Hubbard W., 1994, in: The Equation of State in Astrophysics, IAUColloq. Vol. 147, eds: G. Chabrier & E. Schatzman, Cambridge,p. 443Iben I. Jr., Tutukov A. V., 1984, ApJS, 54, 335Iben I. Jr., Tutukov A. V., Yungelson L. R., 1997, ApJ. 475, 291Kudritzki R. P., Reimers D., 1978, A&A 70, 227Marsh T.R., 1995, MNRAS 275, L1Marsh T.R., Dhillon V. S., Duck S., 1995, MNRAS 275, 828Maxted P., Marsh, T.R., 1998, astro-ph/9803203Paczynski B., 1971, ARA&A 9, 183Portegies Zwart S. F., Yungelson L. R., 1998, A&A 332, 173Rasio F. A., Tout, C. A., Lubow, S. H., Livio, M., 1996, ApJ. 470, 1187Soker N., 1996, ApJ. 460, L53Soker N., 1998, astro-ph/9803223Spruit H. C., 1998, A&A 333, 603Tauris T. M., 1996, A&A 315, 453Verbunt F., Phinney E. S., 1995, A&A 296, 709Webbink R. F., 1984, ApJ. 277, 355Zahn J.-P., 1977, A&A 57,383; erratum 67,162LETTER

UvA-DARE

We propose a scenario to form low-mass, single, slow rotating white dwarfs from a solar-like star accompanied by a massive planet, or a brown dwarf, in a relatively close orbit (e.g. HD 89707). Such white dwarfs were recently found by Maxted &amp; Marsh (1998). When the solar-like star ascends the giant branch it captures the planet and the subsequent spiral-in phase expels the envelope of the giant leaving a low-mass helium white dwarf remnant. In case the planet evaporizes, or fills its own Roche-lobe, the outcome is a single undermassive white dwarf. The observed distribution of planetary systems supports the applicability of this scenario...

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Formation of undermassive single white dwarfs and the influence of planets onlate stellar evolutionNelemans, G.A.; Tauris, Th.M.Publication date1998Published inAstronomy & AstrophysicsLink to publicationCitation for published version (APA):Nelemans, G. A., & Tauris, T. M. (1998). Formation of undermassive single white dwarfs andthe influence of planets on late stellar evolution. Astronomy & Astrophysics, 335, L85-L88.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:09 Mar 2023Astron. Astrophys. 335, L85–L88 (1998) ASTRONOMYANDASTROPHYSICSLetter to the EditorFormation of undermassive single white dwarfsand the influence of planets on late stellar evolutionG. Nelemans and T.M. TaurisAstronomical Institute “Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The NetherlandsReceived 18 May 1998 / Accepted 1 June 1998Abstract. We propose a scenario to form low-mass, single,slow rotating white dwarfs from a solar-like star accompaniedby a massive planet, or a brown dwarf, in a relatively close orbit(e.g. HD 89707). Such white dwarfs were recently found byMaxted & Marsh (1998). When the solar-like star ascends thegiant branch it captures the planet and the subsequent spiral-inphase expels the envelope of the giant leaving a low-mass he-lium white dwarf remnant. In case the planet evaporizes, or fillsits own Roche-lobe, the outcome is a single undermassive whitedwarf. The observed distribution of planetary systems supportsthe applicability of this scenario.Key words: stars: giant, mass-loss, planetary systems – bina-ries: evolution – white dwarfs: formation1. IntroductionRecent searches for double degenerates (two white dwarfs ina binary; Marsh 1995; Marsh, Dhillon & Duck 1995) have re-sulted in the discovery of two single, low-mass helium whitedwarfs – cf. Table 1. Similar undermassive white dwarfs (<∼0.5M) are usually found in binaries and can not be formedfrom normal, single star evolution which leaves a>∼ 0.6MC-O white dwarf as a remnant. Any potential single, low-massprogenitor star of these newly discovered undermassive whitedwarfs can be excluded, since they would have a main sequencelifetime exceeding the age of our Milky Way. A scenario hasbeen proposed (Iben, Tutukov & Yungelson 1997) in which adouble degenerate has merged, due to the emission of gravita-tional wave radiation. According to Maxted & Marsh (1998),this scenario predicts high (∼ 1000 km s−1) rotational veloci-ties for the remnant of the merged objects in contradiction withtheir measurements of a maximum projected rotational veloc-ity of only ∼ 50 km s−1. Therefore the merger scenario seemsquestionable – unless there is an extremely efficient removal ofangular momentum in the merging process, or the inclinationangles for both these systems are extremely small.Send offprint requests to: gijsn@astro.uva.nlTable 1.Properties of the two recently discovered undermassive whitedwarfs – cf. Marsh, Dhillon & Duck 1995; Maxted & Marsh 1998.Name Mass (M) vrot sin i ( km s−1) d (pc)WD 1353+409 0.40 < 50 130WD 1614+136 0.33 < 50 180In this letter we suggest a different, simple solution to the for-mation of these single, low-mass (undermassive) white dwarfsby investigating the influence of massive planets, or browndwarfs, in relatively close orbits around solar-like stars (Sect. 2).A short discussion of the consequences of our planetary scenariois given in Sect. 3.2. Planets around solar-like stars2.1. IntroductionWe propose a scenario in which a solar-like star is surroundedby a massive planet, or a brown dwarf, in a relatively close or-bit. When the star evolves on the giant branch it will becomebig enough to capture its planet via tidal forces (cf. Rasio et al.1996; Soker 1996). The planet spirals into the envelope of thegiant and a so-called common envelope phase is initiated. Thefrictional drag on the planet, arising from its motion through thecommon envelope, will lead to loss of its orbital angular mo-mentum (spiral-in) and deposit of orbital energy in the envelope.The orbital energy is converted into thermal and kinetic energyof the envelope which is therefore being ejected. The result ofthis common envelope evolution is determined by the energybalance and the fate of the planet. As a result of friction, andthe large temperature difference between the envelope of thegiant and the equilibrium temperature of the planet, low-massplanets evaporize due to heating. In case the planet evaporizescompletely, the outcome will be a single star with a rotating andreduced envelope – otherwise we end up with a planet orbitingthe naked core of a giant. The destiny of this white dwarf-planetsystem is determined by the orbital separation.In this letter we first present the expected outcome of a com-mon envelope evolution between a giant and a planet; thereafterLETTERL86 G. Nelemans & T.M. Tauris: Formation of undermassive white dwarfswe look at the important question of the onset of this evolution.We will closely follow the treatment of Soker (1996; 1998),focusing on the cases where (most of) the envelope is lost ina common envelope, leaving a undermassive white dwarf. Re-search in this field has been carried out to explain elliptical andbipolar planetary nebulae (Soker 1996) and the morphology ofthe Horizontal Branch in clusters (Soker 1998).2.2. The outcome of the common envelope phaseBelow we outline our scenario in somewhat more detail. Bysimply equating the difference in orbital energy to the bindingenergy of the envelope of the giant we can compute the ratioof final to initial separation (Webbink 1984). Letηce describethe efficiency of ejecting the envelope,i.e.of converting orbitalenergy into the kinetic energy that provides the outward motionof the envelope:∆Ebind ≡ ηce ∆Eorb or (usingmp  Menv):af ' ηce λ2Mcore mpM MenvRg = f(χ1 − χ)mpMRg (1)whereRg is the radius of the giant star at the onset of the spiral-in phase,λ is a weighting factor (< 1.0) for the binding energyof the core and envelope of the giant star,χ ≡ Mcore/M , mpis the planetary mass andMcore, Menv andaf are the mass ofthe helium core and hydrogen-rich envelope of the evolved star(M = Mcore + Menv), and the final separation after all theenvelope is expelled, respectively. In our calculations we choseλ = 0.5 and ηce = 4 (cf. Tauris 1996; Portegies Zwart &Yungelson 1998) and hencef = 1.To model the effect of planetary evaporation we followSoker (1998) and equate the local sound speed in the giantsenvelope to the escape velocity from the (gaseous) planet sur-face in order to find the approximate location of evaporation:c2s ≈ v2esc ⇐⇒ γkBTµ mu≈ 2 G mpα rp(2)We use a temperature profile for evolved solar-like stars (cf.Fig. 1) ofT ≈ 1.78× 106 (r/R)−0.85K, in the entire intervalof Rcore < r < Rg, whereRcore is the radius of the He-core.During the spiral-in the radius of a giant-gas planet,rp, mayexpand slightly (α rp, α > 1) even though only a small amountof mass (< 0.1 mp) is believed to be accreted (Hjellming &Taam 1991).Solving Eq. (2), with the temperature dependence givenabove and assumingγ = 5/3 and Pop.I chemical abundances(X=0.7; Z=0.02), yields the location of the evaporation:aevap =[10 α(MJmp)]1.18R (3)whereMJ = 0.001M (≈ a Jupiter mass) and we have as-sumedrp = 0.1R, which is a reasonable assumption forall planets and brown dwarfs in the mass range0.0001 <mp/M < 0.08 (Hubbard 1994).For a given stellar structure (i.e.core and envelope mass andradius) the final outcome of the common envelope phase is de-termined only by the mass of the planet. We can easily computeFig. 1. The temperature profiles for 1M evolved stars on the RedGiant Branch. Notice, that logT is approximately a linear function oflog r at all evolutionary stages from the beginning until the tip of theRed Giant Branch (just before the helium flash).the critical planetary mass for which the planet evaporizes justat the moment the envelope is completely expelled,i.e. whenaevap = af . The mass associated with this critical mass (mcrit)is found from Eqs. 1 and 3 (α = 1):mcrit = 10[(1 − χχ) (MM) (Rg100 R)]0.46MJ (4)Planets more massive thanmcrit survive the spiral-in. However,in order to avoid a destructive mass transfer to the white dwarfafter the spiral-in, it must have a radius smaller than its Roche-lobe given by (Paczynski 1971):aRLO =α rp0.462(MWDmp)1/3R (5)whereMWD = Mcore.If af > aevap andaf > aRLO, the planet will survive and theentire envelope is lost from the giant leaving a low-mass heliumwhite dwarf remnant with a planetary companion. However, ifthe final separation is small enough, the planetary orbit willdecay due to emission of gravitational waves on a timescalegiven by:τgwr ≈ (af/60 RWD)4(MWD/M)2 (mp/MJ)5.0 × 109 yr (6)Hence, also in this case the final outcome of the evolution mighteventually be a single undermassive white dwarf.Planets less massive thanmcrit will evaporate (or overflowtheir Roche-lobe ifaRLO > aevap) before the envelope is ex-pelled completely. However heavy planets deposit significantorbital angular momentum in the envelope of the giant, causingenhanced mass loss due to rotation. This could lead to ejectionof the envelope by planets somewhat less massive thanmcrit.The change in structure of the star may alter the further evo-lution of the giant considerably. Soker (1998) suggests that suchLETTERG. Nelemans & T.M. Tauris: Formation of undermassive white dwarfs L87an evolution could explain the morphology of the HorizontalBranch in clusters.For the evolution of the giant we used the relations of Iben &Tutukov (1984) for the structure of a (Pop.I) giant on the RGB:Rg = 103.5M4core, L = 105.6M6.5core, Ṁcore = 10−5.36M6.6core.These equations are valid on the RGB for a low-mass star (0.8 ≤M/M ≤ 2.2).2.3. The onset of the common envelope phase:tidal forces and mass loss on the RGBThe moment the common envelope starts is determined by tidalforces. In the absence of any significant tidal interaction thedonor star is only able to capture planets, via Roche-lobe over-flow, out to a distance,amaxi ≈ 1.6 Rg. Taking tidal effects intoaccount using the equilibrium tide model (Zahn 1977; Verbunt& Phinney 1995) we find, following Soker (1996):amaxi '2.4 Rg(1 − χχ9)1/12 (MM)−11/12 (mp10MJ)1/8(7)where we have used the equations for the structure of the giantas given above. In our calculations (see below) we have alsoincluded mass loss, which amounts to as much as|∆M |/M ≈0.20 at the tip of the RGB. The mass is lost as a fast isotropicwind with the specific angular momentum of the giant causingthe orbital separation of the planet to increase by the same ratioas the total mass of the system decreases. We modeled this effectaccording to the Reimers formula (Kudritzki & Reimers 1978)with η = 0.6 (cf. Rasio et al. 1996).2.4. ResultsWe will now demonstrate an approximate picture for the fate ofstars with planets of different masses and separations to illustratethe applicability of this scenario for producing undermassivesingle white dwarfs as observed in nature. For the evolution ofa solar-like star on the Red Giant Branch we will investigate towhich separation (or alternatively which orbital period) a givenplanet will be captured by the star and compute the outcome ofthe spiral-in process for different planetary masses.In Fig. 2 we have plotted the different critical separationsdiscussed above as a function of planetary mass. Our exampleis based on a 1.0M star with a core-mass of 0.33M (cf. WD1614+136 in Table 1). We findmcrit = 21 MJ. Less massiveplanets expel only part of the envelope (e.g. a planet withmp =15 MJ will only expel half of the envelope, neglecting enhancedmass loss of the giant due to the spin-up of the envelope). Planetswith masses between 15 and25 MJ are presumably disruptedas they fill their Roche-lobe during/after the spiral-in1. Planetsmore massive than∼ 25 MJ survive the spiral-in and will ejectthe entire envelope. However ifmp < 32 MJ, the planet willspiral in, due to emission of gravitational waves, and hence fillits Roche-lobe within 5 Gyr.1 The final fate of the planet depends on its adiabatic exponent, oractually (∂ ln r/∂ ln m) and requires detailed calculations.Fig. 2.Separations of interest (in units ofRWD = 10 000 km) after thespiral-in phase for a 1M star with a core of 0.33M as a functionof planetary mass. The solid line gives the separation for which theliberated orbital energy is equal to the binding energy of the envelope(dotted line for ejecting half of the envelope). The dashed line gives theseparation below which the planet fills its Roche-lobe. The dash-dottedline gives the separation at which the planet is evaporated. A minimumplanetary mass of∼ 21 MJ is needed to expel the entire envelope.Planets lighter than this value are seen to be evaporated. However, for15 < mp/MJ < 25 the planet fills its Roche-lobe and is likely tobe disrupted as a result. Planets more massive than∼ 25 MJ survivethe common envelope phase but will later spiral in due to gravitationalwave radiation (shaded area indicates a spiral-in timescale of less than5 Gyr). Above 0.08M (80 MJ), the companions are heavy enoughto ignite hydrogen as stars (hatched region).In Fig. 3 (top) we calculated the final outcome of theevolution of a planet orbiting a 1M star as a functionof planetary mass and initial orbital period. We also plot-ted some of the known planetary and brown dwarf systemswith solar-like stars (0.70–1.20M). Data were taken from“The Extrasolar Planets Encyclopaedia” (wwwusr.obspm.-fr/departement/darc/planets/encycl.html ). We no-tice that, of the observed systems HD 89707 and HD 140913are the best candidates for producing single undermassive whitedwarfs. In HD 217580, HD 18445 and HD 29587 the planet isexpected to survive the ejection of the envelope. In HD 114762and 70 Vir they are captured already early on the RGB, wherethe binding energy of the envelope is too large to be expelled,so these planets will evaporate shortly after contact with theevolved donor star. The solitary white dwarfs resulting fromthese two systems will therefore be normal C-O white dwarfs.3. DiscussionWe must bear in mind the uncertainties at work in our scenario,and it is possible that future detailed studies of the interactionsbetween a planet and a common envelope may change the masslimits derived in this letter. Also notice that the two undermas-sive white dwarfs in Table 1 might very well have substellarLETTERL88 G. Nelemans & T.M. Tauris: Formation of undermassive white dwarfsFig. 3. a Final outcome of the common envelope phase for differentplanetary masses and initial periods around a 1M star. The solid lineindicates the critical mass,mcrit, below which the planet will evaporizeduring the spiral-in. Above the solid line the planet survives the spiral-in phase and the outcome is an undermassive white dwarf with a planetorbiting it – unless the initial period is sufficiently short leading to adisruption of the planet as it fills its Roche-lobe (left shaded area) afterthe spiral-in. The dash-dotted line indicates the limiting initial periodsbelow which the planet will fill its Roche-lobe, after the spiral-in, inless than 5 Gyr due to gravitational wave radiation. The dotted lineyields the planetary mass for which half of the envelope is ejected– neglecting rotation (see text). Also indicated in the figure are theobserved extrasolar planets and brown dwarfs. In the shaded area tothe right, the planet is too far away from the giant to be engulfed in itsenvelope during evolution on the Red Giant Branch.b Final mass of thewhite dwarf in case all of the envelope is expelled (i.e.mp > mcrit).companions (brown dwarfs or planets) below the observationalthreshold mass of∼ 0.1 M.3.1. The final mass of the white dwarfIn the lower panel of Fig. 3 we give the final white dwarf massin case all of the envelope is expelled. We see that white dwarfswith masses between 0.20–0.45M can in principle be formedwith this scenario. If the common envelope phase initiates whilethe donor is on the Asymptotic Giant Branch, a C-O white dwarfwill be formed.3.2. Rotation of the white dwarfThe final rotational period of the white dwarf is essentially de-termined only by the rotation of the core of the giant: the planettransfers almost all of its angular momentum to the giants enve-lope which is expelled. The rotation of the core strongly dependson the coupling between the core and the envelope of the giant(Spruit 1998), but is in any case in agreement with the the mea-sured upper-limits for the white dwarfs as given in Table 1.3.3. A white dwarf ejected from a binary?An other possibility for the formation of single undermassivewhite dwarfs is a binary origin. Consider a compact system witha giant star (the progenitor of the undermassive white dwarf)and a normal white dwarf companion. When the giant fills itsRoche-lobe it transfers its envelope to the companion leavinga low-mass helium white dwarf as a remnant. The companionmay be subsequently lost either because it exploded as a type IaSNe, or formed a (high velocity) neutron star from an accretioninduced collapse – also leading to disruption of the binary.Acknowledgements.We would like to thank Frank Verbunt for dis-cussions and Bart Bisscheroux for providing Fig. 1. This research wassupported in part by the NWO Spinoza-grant SPI 78-327. T.M.T. ac-knowledges the receipt of a Marie Curie Research Grant from the Eu-ropean Commision.ReferencesHjellming M. S., Taam R. E., 1991, ApJ. 370, 709Hubbard W., 1994, in: The Equation of State in Astrophysics, IAUColloq. Vol. 147, eds: G. Chabrier & E. Schatzman, Cambridge,p. 443Iben I. Jr., Tutukov A. V., 1984, ApJS, 54, 335Iben I. Jr., Tutukov A. V., Yungelson L. R., 1997, ApJ. 475, 291Kudritzki R. P., Reimers D., 1978, A&A 70, 227Marsh T.R., 1995, MNRAS 275, L1Marsh T.R., Dhillon V. S., Duck S., 1995, MNRAS 275, 828Maxted P., Marsh, T.R., 1998, astro-ph/9803203Paczynski B., 1971, ARA&A 9, 183Portegies Zwart S. F., Yungelson L. R., 1998, A&A 332, 173Rasio F. A., Tout, C. A., Lubow, S. H., Livio, M., 1996, ApJ. 470, 1187Soker N., 1996, ApJ. 460, L53Soker N., 1998, astro-ph/9803223Spruit H. C., 1998, A&A 333, 603Tauris T. M., 1996, A&A 315, 453Verbunt F., Phinney E. S., 1995, A&A 296, 709Webbink R. F., 1984, ApJ. 277, 355Zahn J.-P., 1977, A&A 57,383; erratum 67,162LETTER

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Formation of undermassive single white dwarfs and the influence of planets on late stellar evolution

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