Effects Of The Isothermal Region In Protoplanetary Disks And The Protostar Irradiation On The Disk Instability Model For Giant Planet Formation

During the last two decades,the observation of exoplanets has been an active field in astronomy.This field developed rapidly,the number of the detected exoplanets increases,and the parameter space of the exoplanets for the study expands.Due to the wide parameter space,the study of the giant planets can now be more completed.The mass,radius,and other properties can be studied by combining the radial velocity and the transits.Extra constraints can be obtained by direct imaging,astrometry,and gravitational microlensing.Besides,the properties of the host star can be studied and associated with the occurrence of exoplanets to obtain more clues about planet formation.There are two qualitatively different models for giant planet formation.Core accretion model and disk instability model.In the core accretion model for giant planet formation,the rock and ice core of a giant planet forms first and the acquisition of the massive gas envelope is the final phase.The time scale of the giant planet formation depends on the time that the core is assembled and on the time that the gas in the envelope cools and is accreted onto the core.In the disk instability model,via the fragmentation of a gravitationally unstable protoplanetary disk,giant planets can form rapidly.Fragmentation requires that the time scale on which the disk cools is relatively short.This time scale is comparable to the orbital time scale.The main question is that whether these conditions can be satisfied in a disk.Disk instability model for giant planet formation is based on an assumption that the gaseous protoplanetary disk is massive enough,so that the disk can be unstable against its own gravity.This instability leads to disk fragmentation into massive planets.In contrast to core accretion model,in disk instability model,the solid components of the disk only play an indirect role in planet formation.The discussions of disk instability model as a mechanism for planet formation precedes core accretion model in history.In spite of the long history,recently the computational methods progress to the extent that the viability of the disk instability model can be reliably evaluated.In this paper,we focus on the disk instability model for giant planet formation.In disk instability model,when Toomre parameter,Q,satisfies Q<Q_crit,a disk becomes gravitationally unstable.But satisfying this condition does not guarantee that a gravitationally unstable disk can fragment.Numerous numerical studies show that gravitationally unstable disk can develop spiral structure.This non-axisymmetric perturbations emerge due to the disk instability.These perturbations will produce shocks and torques to redistribute the disk angular momentum and mass,and heat the disk in the gravitationally unstable region.Therefore,the spiral structure can stabilize the disk through heating the disk and spreading the disk mass.This self-limiting process means that there can be two different outcomes of gravitationally unstable disks.The first one is a state with stable angular momentum transport.In this case,the heating due to the disk instability balances the radiative cooling and the disk settles into a quasi-steady state.The second one is the fragmentation.To determine whether the gravitationally unstable disk can fragment,the cooling time,t_cool,is usually used.If the cooling time is shorter than the local dynamical timescale,Ω^-1,the disk will fragment into bound,self-gravitating clumps.Then they contract to form giant planets.In a real disk,cooling time depends on the opacity and vertically energy transport mechanism.Using a specific disk model and combining the gravitationally instability and the cooling time criterion,one can estimate the fragmentation region in the disk.Previous studies show that fragmentation is unlikely within several tens of AU from the central protostar.For standard opacities,fragmentation is expected to occur at quite large radii of the order of 50 AU or 100 AU.At small radii,cooling time is very long compared with the local dynamical timescale.The disk may still be gravitationally unstable at these radii but the instability will saturate.Therefore,the disk will settle into the first state mentioned above instead of fragmentation.The idea that cooling time criterion can be used to determine whether the disk can fragment is useful but somewhat simple.At 50-100 AU where fragmentation can happen,the protostar irradiation can not be ignore and the fragmentation criterion should be changed.When disk temperature is controlled by external irradiation,the disk is isothermal.For an isothermal disk,previous studies found that disk fragments when the condition,Q<Q_frag,is satisfied.By using this different fragmentation criterion,the results of the estimated fragmentation region in the disk may be changed.In this paper,we construct an analytical model of gravitationally unstable protoplanetary disks.This disk model consists of three regions:the inner region,the intermediate region,and the outer region.In the inner region,the internal dissipation dominates the heating of the disk.In the intermediate region,the protostar irradiation dominates.In the outer region,the background irradiation dominates.We also use an evolutionary disk model to calculate disk evolution numerically.We use this analytical model and the calculated results of the evolutionary disk to calculate the cooling time and the isothermal region in the disk.We investigate the effects of isothermal region in a disk on disk instability model for giant planet formation.The main conclusions are summarized as follows:（1）The fragmentation region derived from previous work is in the isothermal region of a disk.Therefore,the fragmentation criterion for the isothermal disk is applicable instead of the cooling time criterion.（2）When the isothermal region is considered,by using the fragmentation criterion for the isothermal disk,the fragmentation region extends inward to²0 AU.（3）The protostar irradiation can increase the disk surface temperature.If this increase in surface temperature can be included in disk cooling rate,the region where the cooling time criterion can be satisfied expands.The inner boundary of this region extends inward to²6 AU.Therefore,the fragmentation region defined by the cooling time criterion can also extend inward to²6 AU.（4）The region where the cooling time criterion can be satisfied tends to be isothermal.（5）Even if the contribution of the protostar irradiation to the disk surface temperature is included in the disk cooling rate,by using the cooling time criterion,the region where the disk can fragment is still contained in the isothermal region of the disk most of the time.Even though the isothermal region can not contain the whole fragmentation region during a short period of time,the isothermal region can still contain most of it.（6）In previous studies,the inner boundary of the region where the cooling time criterion can be satisfied,R_cooling,is considered to be the inner boundary of the fragmentation region.Differently,we found that there are three radii which determine the inner boundary of the fragmentation region:the inner boundary of the isothermal region,R_ii,the inner boundary of the gravitationally unstable region,R_Q,and the inner boundary of the region where the cooling time criterion can be satisfied,R^*_cooling,when the possible effects of the protostar irradiation on the disk cooling rate is considered.They separately determine the inner boundary of the fragmentation region at different times.