Stellar growth by disk accretion: the effect of disk irradiation on the protostellar evolution
aa r X i v : . [ a s t r o - ph ] N ov Draft version December 5, 2018
Preprint typeset using L A TEX style emulateapj v. 02/09/03
STELLAR GROWTH BY DISK ACCRETION: THE EFFECT OF DISK IRRADIATION ON THEPROTOSTELLAR EVOLUTION.
Roman R. Rafikov Draft version December 5, 2018
ABSTRACTYoung stars are expected to gain most of their mass by accretion from a disk that forms aroundthem as a result of angular momentum conservation in the collapsing protostellar cloud. Accretioninitially proceeds at high rates of 10 − − − M ⊙ yr − resulting in strong irradiation of the stellarsurface by the hot inner portion of the disk and leading to the suppression of the intrinsic stellarluminosity. Here we investigate how this luminosity suppression affects evolution of the protostellarproperties. Using simple model based on the energy balance of accreting star we demonstrate thatdisk irradiation causes only a slight increase of the protostellar radius, at the level of several per cent.Such a weak effect is explained by a minor role played by the intrinsic stellar luminosity (at the timewhen it is significantly altered by irradiation) in the protostellar energy budget compared to the stellardeuterium burning luminosity and the inflow of the gravitational potential energy brought in by thefreshly accreted material. Our results justify the neglect of irradiation effects in previous studies ofthe protostellar growth via disk accretion. Evolution of some other actively accreting objects such asyoung brown dwarfs and planets should also be only weakly sensitive to the effects of disk irradiation. Subject headings: stars: formation – accretion, accretion disks – planets and satellites: formation –solar system: formation introduction. It is currently established that circumstellar disks arequite ubiquitous around young stellar objects of allmasses (Muzerolle et al. 2003; Cesaroni et al. 2007).They represent an important ingredient of the star for-mation since initially protostars must be growing pre-dominantly by accretion through the disk: angular mo-mentum conservation forces the infalling protostellarcloud material to form a centrifugally supported diskwhich then accretes onto a star. It is quite likely thatonly a small fraction of the final stellar mass gets ac-quired by the direct infall onto the protostellar surface, sothat almost all of the stellar mass gets processed throughthe disk.This picture of star formation implies very high initialmass accretion rates in the disk, at the level of 10 − − − M ⊙ yr − as the Solar-type stars are thought to gainmost of their mass during the first several 10 yr. Thepresence of such a high- ˙ M accretion flow just outside theprotostar immediately raises an issue of its possible effecton the protostellar properties.There are several ways in which disk accretion affectsthe protostar. First, star gains mass from the disk whichincreases its binding energy and tends to make the starmore compact. Second, accreting gas brings in someamount of thermal energy with it which contributes tothe pressure support in the star. The exact amount ofheat advected into the star with the accreted materialis unknown but it seems likely that because of the diskgeometry the accreted gas would have enough time toradiate away most of its thermal energy and would jointhe convective interior of the star with temperature muchsmaller than the stellar virial temperature.Third, intense energy dissipation taking place in the in- Department of Astrophysical Sciences, Princeton University,Ivy Lane, Princeton, NJ 08540, USA; [email protected] nermost parts of the accretion disk leads to strong irradi-ation of the stellar surface by the disk (Frank & Shu 1986;Popham 1997). It has been recently realized (Rafikov2007) that irradiation can act to suppress the internalluminosity of the protostar similar to the suppression ofthe cooling of hot Jupiters by the radiation of their par-ent stars (Guillot et al. 1996; Burrows et al. 2000; Baraffeet al. 2003; Chabrier et al. 2004). Disk accretion at rates˙ M ∼ − − − M ⊙ yr − can easily reduce internalstellar luminosity by a factor of several which may haveimportant implications for the early stellar evolution.The goal of the present paper is to assess these impli-cations by following the evolution of accreting protostarproperly taking into account effects of disk irradiation. method of calculation. We consider a protostar of mass M and radius R grow-ing by accretion of gas at rate ˙ M from the circumstellardisk. In this work ˙ M is specified as an explicit functionof time so that M ( t ) is also known. We assume the diskto extend all the way to the stellar surface as even a1 kG magnetic field (typical value measured in the ma-ture T Tauri systems, see Bouvier et al. 2007) would notbe able to truncate the disk accreting at a high rate of˙ M ∼ − − − M ⊙ yr − . At the stellar surface accret-ing gas passes through the boundary layer (Popham etal. 1993) in which its speed is reduced from the Keplerianvelocity to the velocity of the stellar surface.To evaluate the effect of disk irradiation on the proto-stellar evolution we use an approach based on the energyconservation which was developed in Hartmann et al.(1997). In this approach the convective part of the starcomprising most of its mass is assumed to behave likea polytrope with index n = 3 /
2, so that inside the starpressure P is related to the density ρ via P ∝ ρ / . Thisapproximation works very well in highly ionized, dense,and fully convective interiors of young stars. The totalenergy of such a star (a sum of its thermal and gravita-tional energies ) is E tot = − (3 / GM /R (Kippenhahn& Weigert 1994). Evolution of protostellar properties– luminosity L , radius R – as a function of time [or,equivalently, stellar mass M ( t )] is then governed by thefollowing equation: ddt (cid:18) − GM R (cid:19) = − GM ˙ MR + ˙ E th + L D − L. (1)The l.h.s. of this equation represents the change in thetotal stellar energy, the first and second terms in ther.h.s. are the gravitational potential energy and the ther-mal energy brought in with the accreted material, while L D ( M, R ) is a deuterium luminosity of a protostar. Stel-lar luminosity L is the luminosity carried towards thephotosphere by the convective motions in the stellar in-terior. It is different from the integrated emissivity ofthe stellar surface since the star also intercepts and rera-diates a fraction of energy released in the accretion disk.Rate at which thermal energy gets accreted by the staris ˙ E th = ˙ Mγ − k B Tµ = α GM ˙ MR , (2)where γ is the ratio of specific heats of accreted gas(which can be different from γ = 5 / k B is the Boltzmann constant, µ and T are the mean molecular weight and the temperatureof the accreted gas. Dimensionless parameter α can bewritten as α = 1 γ − TT vir , T vir ≡ µk B GMR , (3)where T vir is the stellar virial temperature. Gas accret-ing from the disk experiences strong dissipation in theboundary layer near the stellar surface. In this layergas temperature can become an appreciable fraction of T vir . However, the cooling time in the boundary layerand the outermost layers of the star is very short so thatthe accreted gas cools efficiently and should ultimatelyjoin stellar interior with temperature T which is muchlower than T vir [unless ˙ M is extremely high, in excessof 10 − M ⊙ yr − , see Popham (1997)]. Thus, under theconditions considered in this work one expect α ≪ α = 0 for simplicity . In this case equation(1) can be rewritten as˙ RR = 73 RGM ( L D − L ) −
13 ˙
MM . (4)This is an evolution equation for R and can be easilyintegrated numerically once the dependencies of L D and L on stellar parameters are known. In this work we neglect stellar rotation. Prialnik & Livio (1985) and Hartmann et al. (1997) have previ-ously investigated the effect of the variation of α on the protostellarevolution. For L D we adopt the expression obtained in Stahler(1998) by integrating the rate of energy release due to Dburning within the n = 3 / L D = f D [D / H]L D , (cid:18) MM ⊙ (cid:19) . (cid:18) RR ⊙ (cid:19) − . , (5)where L D, = 1 . × L ⊙ and f D is the fractionalD abundance relative to the initial D number abundance[D/H] taken to be 2 × − . Parameter f D is not constantin time – it evolves since D burns in the stellar interiorwhile the new D is being brought in with the accretingmaterial (with the initial abundance [D/H]). As a result,one finds (Stahler 1988; Hartmann et al. 1997) ddt ( f D M ) = ˙ M − L D β D , (6)where β D = 9 . × ergs g − is the energy released byD fusion per gram of stellar material (assuming [D/H]=2 × − ).The most important aspect of this work which distin-guishes it from Hartmann et al. (1997) is the calculationof L . In the approximation adopted by Hartmann et al.(1997) L is a function of R and M only. In our case sit-uation is different: irradiation of the stellar surface givesrise to an outer convectively stable layer below the stel-lar photosphere (Rafikov 2007), similar to the radiativelayer that form in the atmospheres of the close-in giantplanets irradiated by their parent stars (Guillot et al.1996; Burrows et al. 2000). This external radiative zonesuppresses the local radiative flux coming from stellar in-terior and this changes the integrated stellar luminosity,which becomes a function of irradiation intensity. As aresult, in irradiated case L depends not only on R and M but also on ˙ M (which determines the strength of theirradiation flux).Rafikov (2007) has demonstrated that for a given opac-ity behavior at the stellar surface (parametrized in hiscase to be a power-law function of gas pressure P andtemperature T , κ ∝ P α T β ) the degree of luminosity sup-pression depends only on the so-called irradiation param-eter L = L χ (Λ) , Λ ≡ π GM ˙ MRL , (7)which is (up to a constant factor) the ratio of the accre-tion luminosity of the disk GM ˙ M /R to the luminosity L that a star would have had in the absence of irra-diation. Suppression factor χ (Λ) → ≪ χ (Λ) . ≫
1. In a simple case consideredby Rafikov (2007) the dependence of χ on the opacitybehavior comes only through the parameter ξ = β + (1 + α ) / ∇ ad , (8)where ∇ ad is the adiabatic temperature gradient near thestellar surface.In the strongly irradiated case the temperature of thestellar surface varies as a function of latitude: equatorialbelt is strongly heated by the hot inner parts of the diskwhile the polar regions of the star are virtually unaffected For convection to set in at some depth below the outer radiativezone a condition ξ > by irradiation and preserve their temperature at the levelof T = ( L / πR σ ) / . In this situation one may won-der whether it is reasonable to assume a fixed opacitylaw (as was done in Rafikov 2007) for the calculation ofluminosity suppression given that the behavior of κ canbe different between the polar and the equatorial regionsof the star. However, is was shown in Rafikov (2007)that for the opacity behavior typical for stellar photo-spheres in the temperature interval from ∼ . × Kto 10 K the cooling of irradiated stars occurs mainlythrough their polar caps (even though the opacity scal-ing with temperature and pressure changes quite drasti-cally within this temperature interval at around 5000 K).As a result, no matter how hot the equatorial parts of thestar become and how complicated the opacity behavior isin this portion of the stellar surface, the integrated stel-lar luminosity L does not depend on these details verystrongly but is rather determined by the properties of thepolar regions of the star: the size of the cool polar capsin which the photospheric temperature is preserved atthe level of T , and the opacity behavior in the adjacentparts of the stellar surface. This provides motivation forusing the stellar luminosity prescription represented byequation (7). In this paper we adopt κ characteristic forthe temperature interval 2 . × K . T . × K(Bell & Lin 1994): κ ≈ × − P / T / . (9)This scaling should be reasonable in the polar regionsof irradiated stars where the photospheric temperature T is not strongly affected by irradiation and is closeto the photospheric temperature that an isolated non-accreting star would have possessed in the Hayashi phase.The opacity law (9) corresponds to ξ = 6 .
5, assuming ∇ ad = 2 / γ = 5 / χ (Λ) (whichwe take from Rafikov [2007], see the curve correspondingto ξ = 6 . L .One remaining ingredient of the calculation is thechoice of L ( M, R ) – the luminosity of a non-irradiatedstar. Hartmann et al. (1997) have adopted the followingfit to the stellar evolution tracks of D’Antona & Mazz-itelli (1994): L ( M, R ) = 1 L ⊙ (cid:18) M . M ⊙ (cid:19) . (cid:18) R R ⊙ (cid:19) . . (10)Evolution tracks in D’Antona & Mazzitelli (1994) havebeen calculated using the equation of state from Magni& Mazzitelli (1979) that has been superceded by themore refined treatments (Saumon et al. 1995). Also,D’Antona & Mazzitelli’s treatment of convection is basedon Canuto & Mazzitelli (1991) which has previouslyraised some concerns (Demarque et al. 1999; Nordlund &Stein 1999). Despite these deficiencies, we have chosento adopt the prescription (10) in our work because of itssimplicity and also to allow direct comparison with theresults of Hartmann et al. (1997). Situation is different in the case of giant planets irradiated bythe circumplanetary disks, see Rafikov (2007) for details. We do not require perfect knowledge of L since our primarygoal is to evaluate the importance of disk irradiation. Fig. 1.—
Evolution of the protostellar properties for ˙ M =4 × − M ⊙ yr − . Initially M = 0 . ⊙ and R = 1 . ⊙ .We display the runs of irradiation parameter Λ ( a ), suppressionfactor χ (Λ) ( b ), ratio L D /L of the D burning luminosity to theluminosity given by Eq. (10) ( c ), stellar radius R ( d ), and the Dabundance f D relative to its initial value ( e ), as functions of stellarmass M . In the last three panels we display the correspondingquantities both in irradiated ( solid ) and non-irradiated ( dashed )cases. Equations (4), (5), (6), (7), & (10) supplemented with dM/dt = ˙ M ( t ) and the dependence χ (Λ) from Rafikov(2007) fully determine the evolution of an accreting pro-tostar irradiated by its own disk. This system of equa-tions is then evolved numerically assuming that the pre-scription for ˙ M ( t ) is given. results. Here we present the results of our calculations. Asinitial conditions we choose M = 0 . ⊙ and f D = 1.We vary R and the prescription for ˙ M ( t ) to see theireffect on the evolution of stellar properties.In Figure 1 we display protostellar evolution for ini-tial R = 1 . ⊙ and a uniform ˙ M = 4 × − M ⊙ yr − with and without the effects of disk irradiation included.One can see that in both irradiated and non-irradiatedcases evolution is pretty much the same: star initiallycontracts until its central density and temperature be-come high enough for the deuterium to ignite. Rightafter that D burning strongly dominates over the stellarluminosity and D abundance f D starts going down. Re-sulting energy release in the stellar interior causes starto expand out to 2 . ⊙ and D luminosity L D decreasesappreciably (but still exceeds L by a factor of several).These results are in full agreement with the calculationsof Hartmann et al. (1997).In the top two panels of Figure 1 we show quanti-ties unique for the irradiated case: run of the irradia-tion parameter Λ and the suppression factor χ (Λ). As R initially decreases Λ goes up to ≈ . × since Fig. 2.—
Relative difference between the stellar radii in theirradiated R and non-irradiated R cases ( panel c ) for a protostaraccreting at a uniform rate ˙ M = 2 × − M ⊙ yr − . Differentcurves correspond to different initial radii: 1 . ⊙ ( solid ), 1 . ⊙ ( dashed ), and 2 . ⊙ ( dotted ). Panels a and b display runs ofirradiation parameter Λ and suppression factor χ as a function ofstellar mass. Λ ∝ ˙ M M . R − . for L given by equation (10). As aresult, the internal stellar luminosity is appreciably sup-pressed and χ reaches ≈ .
55 demonstrating the impor-tance of disk irradiation in regulating L .At the same time, although the effect of irradiation on L is of order unity, Figure 1 clearly demonstrates thatirradiation affects other stellar properties such as R and f D only very weakly. As expected, R in irradiated caseis larger than in the non-irradiated case (because L sup-pression allows more heat to be retained inside the star)but only by several per cent.To check that this result is not an artefact of our initialconditions and assumed ˙ M we have additionally calcu-lated protostellar evolution for uniform ˙ M = 2 × − M ⊙ yr − and ˙ M = 10 − M ⊙ yr − and different initial R . Results are presented in Figures 2 and 3 in which wedisplay the evolution of ∆ R/R = ( R − R ) /R , where R and R are the values of the protostellar radius with andwithout disk irradiation taken into account. It is quiteclear from these plots that despite the rather severe lu-minosity suppression ( χ reaching 0 . M casefor the initial R = 1 R ⊙ ) the relative stellar radius in-crease due to irradiation is always rather small. Notethat at M ∼ M ⊙ we find ∆ R/R ≈
4% when ˙ M = 10 − M ⊙ yr − (Figure 2) which is larger than in the higher ˙ M case shown in Figure 3 (when ∆ R/R ≈ M : in Figure 4 we plot ∆ R/R for differentinitial R and ˙ M = aM with a chosen in such a waythat a protostar grows to 1 M ⊙ in 10 yr. This growthtime is close to the time needed to reach 1 M ⊙ in the Fig. 3.—
Same as Fig 2 but for a uniform ˙ M = 10 − M ⊙ yr − . case of constant ˙ M = 10 − M ⊙ yr − , see Figure 3. Fornon-uniform ˙ M one again finds that irradiated protostardiffers in radius from the non-irradiated star by only afew per cent (∆ R/R ≈
3% at M ∼ M ⊙ ). It is alsoobvious from the comparison of Figures 3c and 4c thatroughly the same growth time translates into very similarbehavior of ∆ R/R in the two cases independent of how˙ M ( t ) evolves. We thus conclude that irradiation by thecircumstellar accretion disk has rather small effect onthe protostellar evolution and that this outcome is rathergeneric. discussion. Weak sensitivity of R to disk irradiation given thestrong effect that irradiation has on the internal lumi-nosity L of a protostar may seem surprising. However,one should bear in mind that besides L there are othercontributors to the stellar energy budget, namely the Dburning luminosity L D and the gravitational potentialenergy gained with the accreted material. It turns outthat these contributions dominate the energy budget over L .To see this we rewrite equation (4) using definition (7)in the following form:˙ RR = 28 π − ˙ MM (cid:20) L D L − χ (Λ) − Λ28 π (cid:21) . (11)In the right-hand side of this equation the first term inbrackets describes the relative role of D burning in thetotal energy budget, second term represents intrinsic stel-lar luminosity, while the third term is due to the inflowof the gravitational potential energy GM ˙ M /R – the ra-tio of this energy inflow to L differs from Λ only by aconstant factor.Suppression of L is largest when Λ is highest which iseasy to see by inspecting Figures 1-4. But this automat- Fig. 4.—
Same as Fig 2 but for ˙ M ∝ M and the growth time10 yr to reach M = 1 M ⊙ . ically means that the maximum deviation of the sup-pression factor χ from unity occurs precisely when theinflow of the gravitational potential energy far exceedsthe stellar luminosity. Apparently, under these circum-stances L is a subdominant contribution to the stellarenergy budget and thus even a significant reduction of L compared to L is going to be negligible compared to thegravitational energy influx.Moreover, Figure 1 demonstrates that Λ reaches itsmaximum when R is at its minimum while f D is still veryclose to unity. At this point vigorous D burning com-mences inside the star giving rise to very high L D /L .As a result, at the evolutionary stage when χ is minimal L is subdominant in comparison to not only GM ˙ M /R but also L D . This additionally downplays the role of theluminosity suppression by irradiation in the early proto-stellar evolution.This line of reasoning also explains why at M ∼ ⊙ we have found ∆ R/R to be larger for lower ˙ M (see § M means lower Λ so that the ratio of L tothe gravitational energy inflow rate GM ˙ M /R in the low˙ M case is larger than in the high ˙ M case. Also, at M ∼ M ⊙ one generally finds f D ≪ L D is mainly due to the burning of the freshly accreted D(rather than the D that remained in the protostar fromprevious accretion). Since in the low ˙ M case less fresh D is supplied to the protostar L D must also be lowerthan in the high ˙ M case. As a result, in the lower ˙ M case L plays a more significant role compared to L D (inwhich case ∆ R/R should be more sensitive to changes in L caused by irradiation) than in the high ˙ M case.This conclusion immediately raises the following ques-tion: since ∆ R/R increases as ˙ M decreases would onefind ∆ R/R ∼ M ? The answer is no,and it has to do with the fact that χ appreciably differs Fig. 5.—
Plot of Λ (the value of Λ at which the suppressionfactor χ = 0 .
5) as a function of parameter ξ characterizing theopacity law in the outer layers of irradiated protostar. Note thatΛ never decreases below several hundred. from unity (obviously, a necessary condition for getting∆ R/R ∼
1) only at rather high Λ. This is a genericfeature of disk irradiation which is illustrated in Fig-ure 5 where we display Λ – the value of Λ at which χ (Λ ) = 0 . ξ , see equation (8). One cansee that Λ & for all ξ >
4, meaning that significantluminosity suppression requires rather high ˙ M . This in-efficiency of irradiation in suppressing L is caused by thespecific geometry of disk irradiation in which the irradia-tion flux is a very sensitive function ( ∝ θ ) of the latitudeat the stellar surface θ , see Rafikov (2007). Because ofthat stellar polar caps can stay cool even at rather high˙ M allowing unsuppressed flux to be emitted over a sig-nificant portion of the stellar surface.In the case of ξ = 6 . = 2 . × which accord-ing to equation (7) immediately implies that GM ˙ M /R ≈ L when χ = 0 .
5. Clearly, in this case stellar luminos-ity should have small effect on the protostellar evolution.If ˙ M is reduced so that GM ˙ M /R ∼ L (and Λ ∼ χ would be very closeto unity (see Rafikov 2007) and the L suppression byirradiation would be negligible. Thus, under no circum-stances should one expect ∆ R/R larger than several percent, meaning that quite generally the irradiation by ac-cretion disk is unlikely to play a significant role in theevolution of the protostellar properties.Looking at this conclusion under slightly different an- At small enough ˙ M ( ˙ M . − M ⊙ yr − ) protostellar mag-netic field is likely to disrupt accretion flow (K¨onigl 1991) thusinvalidating our assumption of the direct disk accretion onto thestar. gle, our results also imply that when considering evo-lution of the protostars accreting at ˙ M ∼ − − − M ⊙ yr − one may completely neglect L in the stellar en-ergy budget and still get rather decent description of theprotostellar evolution.Our results are obtained assuming a specific value of ξ typical for the low-mass stars. However, one expectsour major conclusions to remain valid also in the caseof other accreting objects such as young brown dwarfsand giant planets. Although these objects are likely tobe characterized by values of ξ different from 6 .