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The serturbed potential associated with the plauct excites spiral deusitv waves in the disk. which propagate away )oth inwards anc outwards. with a pattern frequency equal to the planet orbital frequency.
The perturbed potential associated with the planet excites spiral density waves in the disk, which propagate away both inwards and outwards, with a pattern frequency equal to the planet orbital frequency.
The spiral waves interact with the disk aud give it the angular momoeutunm hey removed from the planet. and eventually open a gap centered on the planet orbit. provided the planet mass is high enough (Papaloizou Liu. 1981).
The spiral waves interact with the disk and give it the angular momentum they removed from the planet, and eventually open a gap centered on the planet orbit, provided the planet mass is high enough (Papaloizou Lin, 1984).
We present a mun with a one solar mass primary. one Jupiter mass xotoplanet initially on a fixed circular orbit at ry=5 ATU, enibedded.: in a standard protoplanetary nebula whose xumnueters have been mentioned above.
We present a run with a one solar mass primary, one Jupiter mass protoplanet initially on a fixed circular orbit at $r_0=5$ A.U. embedded in a standard protoplanetary nebula whose parameters have been mentioned above.
The erid las an inner radius at 2 AU. and an outer radius at 12.5 ALU. The sequence 1)νι] is equally spaced. with NV,=19: The eric has Ny=1125 sectors. it is fixed iu a rou-Calilean non-rotating frame centered on the primary.
The grid has an inner radius at $2$ A.U. and an outer radius at $12.5$ A.U. The sequence $(R_i)_{i\in[0,N_r]}$ is equally spaced, with $N_r=49$; The grid has $N_s=143$ sectors, it is fixed in a non-Galilean non-rotating frame centered on the primary.
Its outer boundary is rigid and its ner boundary allows outfiow but no inflow.
Its outer boundary is rigid and its inner boundary allows outflow but no inflow.
The disk aspect ratio is set to 1:10? evervwhere.
The disk aspect ratio is set to $4\cdot 10^{-2}$ everywhere.
The planet perturbed potential is s«noothed on a leugth scale which amounts to 10 of the Roche radius.
The planet perturbed potential is smoothed on a length scale which amounts to 40 of the Roche radius.
In eq. (119)
In eq. \ref{eqn:shearlimit}) )
we choose Cy=0.5.
we choose $C_0=0.5$.
We plot ou fig.
We plot on fig.
2. the quautity €i; after 2.86 orbits.
\ref{fig:eij} the quantity $e_{ij}=\frac{v^{\theta a}_{ij}\Delta t}{\Delta y_i}$ after $2.86$ orbits.
This quantity represcuts the effective GiCFL ratio.
This quantity represents the effective CFL ratio.
With the standard transport aleorithin this ratio is hounded by Cy.
With the standard transport algorithm this ratio is bounded by $C_0$.
We see that the innermost ring sweeps almost four cells on oue timestep. hence the use of the FARGO transport algoritlan iu this case results iun a speed-up by a factor ~8 of the computation.
We see that the innermost ring sweeps almost four cells on one timestep, hence the use of the FARGO transport algorithm in this case results in a speed-up by a factor $\sim 8$ of the computation.
One cau note that the difference iu 6j; between the iunerinost ring aud its imauuediate ucighbor is 0.5. which is the maxima allowed by ec. OLD.
One can note that the difference in $e_{ij}$ between the innermost ring and its immediate neighbor is $0.5$, which is the maximum allowed by eq. \ref{eqn:shearlimit}) ).
Indeed the timestep in this run is shear-limited. aud the constraint on the residual velocities only would lead to au even bigecr timestep. since as one can see the residuals of the distance swept over oue timestep amounts to far ess than 1/2. even in the vicinity of the planet.
Indeed the timestep in this run is shear-limited, and the constraint on the residual velocities only would lead to an even bigger timestep, since as one can see the residuals of the distance swept over one timestep amounts to far less than $1/2$, even in the vicinity of the planet.
Tn order to see how umuerical viscosity affects the disk response iu both cases. we plot on the fig.
In order to see how numerical viscosity affects the disk response in both cases, we plot on the fig.
3 the disk deusitv after 28.6 orbits. obtained from differen algoritlius.
\ref{fig:diskdens} the disk density after $28.6$ orbits, obtained from different algorithms.
The lef plot corresponds to à non-rotating frame standard trauspor run. while the middle plo represents a non-rotatine frame FARGO trausport run. and the right plot represents a standard transport run in a frame corotating with the planet (heuce the plane is fixed with respect to the eid. so we expect from the results of section L the deusitv response in the vicinity of the planet to be given with a high accuracy).
The left plot corresponds to a non-rotating frame standard transport run, while the middle plot represents a non-rotating frame FARGO transport run, and the right plot represents a standard transport run in a frame corotating with the planet (hence the planet is fixed with respect to the grid, so we expect from the results of section \ref{sec:1d} the density response in the vicinity of the planet to be given with a high accuracy).
Note tha special care has to be devoted to the treatment of the Coriolis force iu that case in order ο COliserve exactV TLC anenlar niomeutunm aud then to avoid a spurious outwards fralspor in the disk (IXlev. 1998).
Note that special care has to be devoted to the treatment of the Coriolis force in that case in order to conserve exactly the angular momentum and then to avoid a spurious outwards transport in the disk (Kley, 1998).
We clearly see that the elobal spiral pattern excited by the protoplanet in the disk is identical in the three cases. though the response in the inunediate vicinity of the planet is much more spread out in the frame standard accretion case (left plot). aud that the most sharply peaked response is achieved through the use of a corotating frame (right plot). as expected.
We clearly see that the global spiral pattern excited by the protoplanet in the disk is identical in the three cases, though the response in the immediate vicinity of the planet is much more spread out in the frame standard accretion case (left plot), and that the most sharply peaked response is achieved through the use of a corotating frame (right plot), as expected.
Indeed. we plot ou Πο,
Indeed, we plot on fig.
5. a cut of the disk density at the planet radius in the three cases.
\ref{fig:denspeak} a cut of the disk density at the planet radius in the three cases.
The solid line represents the FARGO transport result. and the dot-dashed line the corotating frame result.
The solid line represents the FARGO transport result, and the dot-dashed line the corotating frame result.
They both have the same width. though the iiaxiumua of the density iu the corotating case is higher.
They both have the same width, though the maximum of the density in the corotating case is higher.
The dashed curve represcuts the result of the standard transport in a frame.
The dashed curve represents the result of the standard transport in a frame.
Its width is about twice as large as the other curves’ width. aud we also see that uunucerical effects in that case lead to additional leadiug aud trailing material (uear cells πο 65 ancl 77). aud to a smaller density peak value.
Its width is about twice as large as the other curves' width, and we also see that numerical effects in that case lead to additional leading and trailing material (near cells number $65$ and $77$ ), and to a smaller density peak value.
Gould&Kollmeier(2004) (from now on GKOA) carefully combine SDSS Data Release | (SDSS DRI} and USNO-B proper motions to produce a catalogue of 390476 objects with proper motions pe20 mas * and magnitudes +20 mag.
\citet{gou2004} (from now on GK04) carefully combine SDSS Data Release 1 (SDSS DR1) and USNO-B proper motions to produce a catalogue of 390476 objects with proper motions $\mu \geq 20$ mas $^{-1}$ and magnitudes $r \leq 20$ mag.
SDSS DRI proper motions are based on matches of SDSS to USNO-A2.0 (Monetetal. 1998)). which suffer from mismatches causing spurious high proper-motion objects. and systematic trends in proper motion.
SDSS DR1 proper motions are based on matches of SDSS to USNO-A2.0 \citealt{mon1998}) ), which suffer from mismatches causing spurious high proper-motion objects, and systematic trends in proper motion.
However. by cross-correlating SDSS DRI and USNO-B. GKO4 successfully removed the vast majority of spurious proper-motion stars.
However, by cross-correlating SDSS DR1 and USNO-B, GK04 successfully removed the vast majority of spurious proper-motion stars.
Furthermore. by considering the set of spectroscopically contirmed quasars in SDSS DRI. GKO4 calibrate out the position-dependent astrometric biases. using a very similar method to our recalibration of SDSS astrometry presented in Section 2.4.
Furthermore, by considering the set of spectroscopically confirmed quasars in SDSS DR1, GK04 calibrate out the position-dependent astrometric biases, using a very similar method to our recalibration of SDSS astrometry presented in Section 2.4.
We compare our proper motions in the HLC to those derived by GKO4 for the 30546 stars from their catalogue with an unambiguous positional match in the HLC using a match radius of 0.
We compare our proper motions in the HLC to those derived by GK04 for the 30546 stars from their catalogue with an unambiguous positional match in the HLC using a match radius of .
7".. In Figure 9.. we plot the running 3o-clipped mean difference between our HLC propermotions and those of GKO4 (middle
In Figure \ref{fig:comp_usno}, we plot the running $\sigma$ -clipped mean difference between our HLC propermotions and those of GK04 (middle
whole cosmic evolution in the case of the high Q,, SCDM cosmology.
whole cosmic evolution in the case of the high $\Omega_{m}$ SCDM cosmology.
While we do see rise of the FP thickness before Gezp<0.5 in the ACDMF2a and ACDMC2 cosmologies, after that time the increase levels off and may even flatten completely.
While we do see a rise of the FP thickness before $a_{exp}<0.5$ in the $\Lambda$ CDMF2 and $\Lambda$ CDMC2 cosmologies, after that time the increase levels off and may even flatten completely.
Note, however, that these simulations do not attain sufficient halo mass resolution at higher redshifts: in these cosmologies halos still are low mass objects at these epochs.
Note, however, that these simulations do not attain sufficient halo mass resolution at higher redshifts: in these cosmologies halos still are low mass objects at these epochs.
The one exceptional cosmology is that of the low Qn Universe ACDMOJ2.
The one exceptional cosmology is that of the low $\Omega_{m}$ Universe $\Lambda$ CDMO2.
Except for a rather abrupt and sudden jump in FP thickness at ae;5~0.3, there is no noticeable change at later epochs.
Except for a rather abrupt and sudden jump in FP thickness at $a_{exp}\sim0.3$, there is no noticeable change at later epochs.
By ἄεαρ=0.3 nearly all its clusters are in place and define a Fundamental Plane that does not undergo any further evolution.
By $a_{exp}=0.3$ nearly all its clusters are in place and define a Fundamental Plane that does not undergo any further evolution.
In summary, the trend seems to be for initial increase of the FP thickness followed by a convergence to a nearly constant value.
In summary, the trend seems to be for initial increase of the FP thickness followed by a convergence to a nearly constant value.
The epoch of convergence is later for higher values of Qm: while the thickness remains constant for the low Qm ACDMO2 cosmology, it involves a slow but continuous increase in the SCDM cosmology.
The epoch of convergence is later for higher values of $\Omega_m$ : while the thickness remains constant for the low $\Omega_m$ $\Lambda$ CDMO2 cosmology, it involves a slow but continuous increase in the SCDM cosmology.
On the basis of their study of galaxy merging, Nipotial.(2003) argued that the disposition of galaxies in the Fundamental Plane is not simply a realization of the virial theorem, but contains additional information on galaxy structure and dynamics.
On the basis of their study of galaxy merging, \cite{nipoti03} argued that the disposition of galaxies in the Fundamental Plane is not simply a realization of the virial theorem, but contains additional information on galaxy structure and dynamics.
This should be reflected in the location of the halo population within the Fundamental Plane.
This should be reflected in the location of the halo population within the Fundamental Plane.
Figs.
Figs.
12 and 13 show how the location of the clusters within the plane shifts as time proceeds.
\ref{fig:Lambdaplanes} and \ref{fig:SCDMplanes} show how the location of the clusters within the plane shifts as time proceeds.
The color scheme is the same as for Fig. 2..
The color scheme is the same as for Fig. \ref{fig:radiusrel}.
Fig.
Fig.
12 shows the location of theclusters in the ACDMF2 cosmology in the Fundamental Plane inferred for the current epoch, ie.
\ref{fig:Lambdaplanes} shows the location of theclusters in the $\Lambda$ CDMF2 cosmology in the Fundamental Plane inferred for the current epoch, ie.
at redshift z—0, To locate their position within the Fundamental Plane, we use the (artificial) coordinates PF, and F» of the halo points with respect to two mutually perpendicular normalized vectors in the Fundamental Plane atz= 0, wrt.
at redshift $z=0$, To locate their position within the Fundamental Plane, we use the (artificial) coordinates $F_1$ and $F_2$ of the halo points with respect to two mutually perpendicular normalized vectors in the Fundamental Plane at$z=0$ , wrt.
the coordinate system defined by the FP quantities
the coordinate system defined by the FP quantities
where q=E»ο”...J - a sound velocity.
where ${\displaystyle a=\left(\frac{\partial P}{\partial \rho}\right)^{1/2}_T}$ - a sound velocity.
∙⊽ When obtaining∙∙ equation| (17))i ib was asstuned that the wind is isothermal T=const. and P=a7p.
When obtaining equation \ref{eqn1}) ) it was assumed that the wind is isothermal $T={\rm const.}$ and $P=a^2\rho$.
Introducing nondimmensional variables: where r, is (he critical point radius aud ος ds the velocity of matter at the critical point.
Introducing nondimmensional variables: where $r_c$ is the critical point radius and $v_c$ is the velocity of matter at the critical point.
Taking into account. (13)). equation (17)) reads: p= —and P= TM
Taking into account \ref{nondimvar}) ), equation \ref{eqn1}) ) reads: where ${\displaystyle p\equiv \frac{dv}{dx}}$ and ${\displaystyle \Gamma\equiv\frac{L\sigma_e}{4\pi GM c}}$.
Lo, simplicitv hereafter⋅⋅⋅ we⋅⋅ omit tilde.
For simplicity hereafter we omit tilde.
where. The following nondimmentional Forparameters were introduced: | 6 ⋋≼≻↥↩↥↥↥≀↧↴↥↽≳↾∶⊣⇃↓≀↧↴∐≺⇂⊔∐↲↕⋅≼↲↕⋟∖⊽∪∐↥⋡∖↽∪∐≼↲∐∐⇂≼↲↕↽≻≼↲∐≺⇂≼↲↕∐↕↽≻≀↕↴↕⋅≀↕↴∐∐↲∩↲↕⋅∩↓⋅↼≚ ⋅
The following nondimmentional parameters were introduced: Note that ${\displaystyle \beta=\frac{c}{a} a_1}$ and there is only one independent parameter $a_1$.
⋅ constant jis determined according to relation: Equation (19)) is non linear with respect to p.
A constant $\mu$ is determined according to relation: Equation \ref{dmless1}) ) is non linear with respect to $p$.
The point where the speed of sound is equal to velocity of the flow is no longer the singular point of the equation of motion.
The point where the speed of sound is equal to velocity of the flow is no longer the singular point of the equation of motion.
To (reat such equations a special lechuique must be applied.
To treat such equations a special technique must be applied.
We are interested of the solution that starts subsonic near DII is continuous through the singular point ancl goes supersonically to infinity.
We are interested of the solution that starts subsonic near BH is continuous through the singular point and goes supersonically to infinity.
Singular point is defined by the condition:
Singular point is defined by the condition:
time separation bin, ~15 minutes.
time separation bin, $\sim$ 15 minutes.
While sparsely sampled, the Hydra data reinforce this interpretation.
While sparsely sampled, the Hydra data reinforce this interpretation.
In Figure 8aa, both the SDSS and Hydra exposure times are longer than the underlying characteristic timescale for the high-variability data, and therefore all measurement pairs are randomly sampling the intrinsic equivalent width variations, leading to flat structure function amplitudes.
In Figure \ref{strfunc}a a, both the SDSS and Hydra exposure times are longer than the underlying characteristic timescale for the high-variability data, and therefore all measurement pairs are randomly sampling the intrinsic equivalent width variations, leading to flat structure function amplitudes.
The longer exposure time for the Hydra data (an hour compared to 10-15 minutes for SDSS) damps the stochastic variations between exposures, and the amplitude of the Hydra structure function is correspondingly smaller.
The longer exposure time for the Hydra data (an hour compared to 10-15 minutes for SDSS) damps the stochastic variations between exposures, and the amplitude of the Hydra structure function is correspondingly smaller.
The low-variability Hydra sample in Figure 8bb has a characteristic timescale that is apparently comparable to the exposure time, and thus the structure function amplitude is flat over the 1-3 hour time separations that are sampled.
The low-variability Hydra sample in Figure \ref{strfunc}b b has a characteristic timescale that is apparently comparable to the exposure time, and thus the structure function amplitude is flat over the 1–3 hour time separations that are sampled.
In this case, the structure function amplitude is consistent between the SDSS and Hydra samples.
In this case, the structure function amplitude is consistent between the SDSS and Hydra samples.
'The increase in variability amplitude with time seen in our low-variability (high activity) sample is reminiscent of the ? result that there was a much larger number of Ha emission events that lasted at least 30 minutes compared to the number of shorter events they observed.
The increase in variability amplitude with time seen in our low-variability (high activity) sample is reminiscent of the \citet{Lee2010} result that there was a much larger number of $\alpha$ emission events that lasted at least 30 minutes compared to the number of shorter events they observed.
'Their sample was primarily composed of very active stars which is comparable to this subset of our data.
Their sample was primarily composed of very active stars which is comparable to this subset of our data.
Following the discussion in 833.2, we interpret the timescale results as follows.
Following the discussion in 3.2, we interpret the timescale results as follows.
The low-variability (high activity) stars are covered with spots, so that small variations are not easily visible against the high background.
The low-variability (high activity) stars are covered with spots, so that small variations are not easily visible against the high background.
In order to see a noticeable variation, a rather significant event must occur.
In order to see a noticeable variation, a rather significant event must occur.
Since higher-energy events are known to last longer (e.g.??),, this leads to a longer timescale for events that are strong enough to be detected above the persistent activity.
Since higher-energy events are known to last longer \citep[e.g.][]{lme,hp1991}, this leads to a longer timescale for events that are strong enough to be detected above the persistent activity.
For the high-variability (low activity) stars, small events will cause a noticeable change, and these occur on short timescales, below the ~15 minute limit of our sampling cadence.
For the high-variability (low activity) stars, small events will cause a noticeable change, and these occur on short timescales, below the $\sim$ 15 minute limit of our sampling cadence.
We used two spectroscopic samples of active M dwarfs to examine Ha emission variability on timescales from minutes to weeks.
We used two spectroscopic samples of active M dwarfs to examine $\alpha$ emission variability on timescales from minutes to weeks.
We found that the Ho variability measured using the Rew metric remains relatively constant as a function of spectral type, in contrast to previous results.
We found that the $\alpha$ variability measured using the $R_{EW}$ metric remains relatively constant as a function of spectral type, in contrast to previous results.
However, using our fractional variability metric (cEwHA/(EWHA)), we showed that the higher-variability stars have relatively low activity strength (Lita/Loot), and conversely the low-variability stars have high activity strength.
However, using our fractional variability metric $\sigma_{EWHA}$ $\langle$ $\rangle$ ), we showed that the higher-variability stars have relatively low activity strength $_{\rm{H}\alpha}$ $_{bol}$ ), and conversely the low-variability stars have high activity strength.
This naturally leads to an apparent dependence on spectral type because the later type stars typically have lower activity strength.
This naturally leads to an apparent dependence on spectral type because the later type stars typically have lower activity strength.
We speculated that the physical reason for the higher variability in the low activity stars is that small changes in Ha emission, e.g. due to small-scale flaring or the appearance of a new active region on the stellar surface, would have a relatively larger effect on the less active stars.
We speculated that the physical reason for the higher variability in the low activity stars is that small changes in $\alpha$ emission, e.g. due to small-scale flaring or the appearance of a new active region on the stellar surface, would have a relatively larger effect on the less active stars.
We investigated the timescales of Ha emission variability using structure functions.
We investigated the timescales of $\alpha$ emission variability using structure functions.
The observed Ha variations for low-variability stars (cogwgA/(EWHA) < 0.16) occur on a timescale longer than an hour, while the high-variability M. dwarfs exhibit a timescale that is shorter than our sampling time of ~15 minutes.
The observed $\alpha$ variations for low-variability stars $\sigma_{EWHA}$ $\langle$ $\rangle<$ 0.16) occur on a timescale longer than an hour, while the high-variability M dwarfs exhibit a timescale that is shorter than our sampling time of $\sim$ 15 minutes.
Neither the low nor high variability samples show significant changes in the structure function on timescales longer than a day.
Neither the low nor high variability samples show significant changes in the structure function on timescales longer than a day.
We speculated that the low-variability sample, which is mainly composed of the higher activity strength stars, requires more energetic and hence longer-lived emission events in order to be detected above the strong persistent surface activity, leading to a longer characteristic variability timescale.
We speculated that the low-variability sample, which is mainly composed of the higher activity strength stars, requires more energetic and hence longer-lived emission events in order to be detected above the strong persistent surface activity, leading to a longer characteristic variability timescale.
Better time resolution in spectroscopic M dwarf monitoring, as well as a wider and better-sampled range of time separations, will allow for a more accurate determination of the timescales of Ho variability in low-mass dwarfs and may lead to a better understanding of the physical mechanisms that cause these changes.
Better time resolution in spectroscopic M dwarf monitoring, as well as a wider and better-sampled range of time separations, will allow for a more accurate determination of the timescales of $\alpha$ variability in low-mass dwarfs and may lead to a better understanding of the physical mechanisms that cause these changes.
We acknowledge NSF grant AST 08-07205, and thank David Schlegel for help obtaining the individual component SDSS spectra.
We acknowledge NSF grant AST 08-07205, and thank David Schlegel for help obtaining the individual component SDSS spectra.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.
The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions.
The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions.
The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Planck Institute for Astronomy the Max- Institute for Astrophysics (MPA), (MPIA),New Mexico State University, Ohio State University, University
The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck Institute for Astronomy (MPIA), the Max-Planck Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University
an entropy eracicnt (Paardekooper&Papaloizou2008). the sign of the total torque can change reversing in this way the direction of the migration.
an entropy gradient \citep{paapap08} the sign of the total torque can change reversing in this way the direction of the migration.
For high-mass planets the disc response becomes non linear and a gap forms in the disc around the planet orbit.
For high-mass planets the disc response becomes non linear and a gap forms in the disc around the planet orbit.
Lf the gap is very clean ancl the disce is stationary. the evolution of the planet is determined by the radial velocity drift in the disc (Lin&Papaloizou1986).. namely The migration> time of the planet can be estimated. as (Lin&Papaloizou1993) llowever. recent. numerical simulations (edgar2007) showed that the migration rate depends also on the planet mass and on the disc surface density profile. which can be Wrillen as The latter estimation gives a dependence on the surface density which is in good agreement with hvdrocdynamic simulations. although the variation. with the disc viscosity appears to be much weaker than expected. from analytical predictions (I:dear2007.2008).
If the gap is very clean and the disc is stationary, the evolution of the planet is determined by the radial velocity drift in the disc \citep{linpap86}, namely The migration time of the planet can be estimated as \citep{linpap93} However, recent numerical simulations \citep{edgar} showed that the migration rate depends also on the planet mass and on the disc surface density profile, which can be written as The latter estimation gives a dependence on the surface density which is in good agreement with hydrodynamic simulations, although the variation with the disc viscosity appears to be much weaker than expected from analytical predictions \citep{edgar,edgar2008}.
. For intermeciate-mass planets which open the gap only partially. tvpe HE migration has been proposed. (Masset.&Papaloizou 2003).
For intermediate-mass planets which open the gap only partially, type III migration has been proposed \citep{maspap}.
. Fhis tvpe of migration occurs if the disc mass is much higher than the mass of the planet.
This type of migration occurs if the disc mass is much higher than the mass of the planet.
Lt is than required to employ. numerical simulations in order to determine how the relative migration rate of the planets varies in details with their masses and with the parameters specifving the properties of the disc.
It is than required to employ numerical simulations in order to determine how the relative migration rate of the planets varies in details with their masses and with the parameters specifying the properties of the disc.
Dilferent rates of planetary. migration can lead. to the resonant capture of the planets as it was shown for two giant planets in Whey(2000) or Whey.Peitz&Brveen (2004).
Different rates of planetary migration can lead to the resonant capture of the planets as it was shown for two giant planets in \citet{kley} or \citet{kbp04}.
. Also low-mass planets may undergo convergent migration and form a resonant structure (PapaloizouSzuszkiewiez 2005).
Also low-mass planets may undergo convergent migration and form a resonant structure \citep{papszusz}.
. In à previous paper of ours (Podlewska&Szuszkiewiez 2008).. we have considered a system with low- and high-mass planets (à Super-Earth on the interna orbit and a Jupiter-like planet on the external one) am we have found that they are captured into 3:2 or 4:3 mean motion resonances if the cise properties allow for convergen migration.
In a previous paper of ours \citep{paperI}, we have considered a system with low- and high-mass planets (a Super-Earth on the internal orbit and a Jupiter-like planet on the external one) and we have found that they are captured into 3:2 or 4:3 mean motion resonances if the disc properties allow for convergent migration.
Without drawing a too close analogy to the formation of the Solar System. it has been pointed out tha the Hilda and "Fhule groups of asteroids are in the interior 3:2 and 4:3 resonances with Jupiter.
Without drawing a too close analogy to the formation of the Solar System, it has been pointed out that the Hilda and Thule groups of asteroids are in the interior 3:2 and 4:3 resonances with Jupiter.
Similar configurations with a Super-Earth instead of an asteroid might be presen in extrasolar planetary systems.
Similar configurations with a Super-Earth instead of an asteroid might be present in extrasolar planetary systems.
In. order to draw such a conclusion. a long-term. stability analysis of the obtaine resonant structures should be performed.
In order to draw such a conclusion, a long-term stability analysis of the obtained resonant structures should be performed.