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Such lowmass objects are probably not the result of gravitational instabilities. but are more likely formed through the accumulation of a metal rich core followed by (he capture of a more or less massive envelope of gas (Lissauer 1993 and references {herein). | Such low–mass objects are probably not the result of gravitational instabilities, but are more likely formed through the accumulation of a metal rich core followed by the capture of a more or less massive envelope of gas (Lissauer 1993 and references therein). |
In this model. (he core is believed to be formed through the solidbody accretion of kilometresized planetesimals. | In this model, the core is believed to be formed through the solid–body accretion of kilometre–sized planetesimals. |
For tvpical disc parameters. when the core reaches about 0.1 earth mass. it starts binding the gas of the nebula in which it is embedded ancl a gaseous atmosphere forms around it. | For typical disc parameters, when the core reaches about 0.1 earth mass, it starts binding the gas of the nebula in which it is embedded and a gaseous atmosphere forms around it. |
As long as the core is less massive than the socalled.mass. Lhe energy radiated from the atmosphere is compensated for by the exavitational enerev (hat the planetesimals entering the atmosphere release when (μον collide with the surface of the core. | As long as the core is less massive than the so–called, the energy radiated from the atmosphere is compensated for by the gravitational energy that the planetesimals entering the atmosphere release when they collide with the surface of the core. |
During this stage of evolution. the atmosphere is therefore in and thermal equilibrium and grows slowly in mass along with the core (Perri Cameron 1974. Mizuno 1980). | During this stage of evolution, the atmosphere is therefore in quasi--static and thermal equilibrium and grows slowly in mass along with the core (Perri Cameron 1974, Mizuno 1980). |
By the time the core reaches the critical mass. the atmosphere has become too massive lo be supported at equilibriumn by the energy. released by the planetesimals. | By the time the core reaches the critical mass, the atmosphere has become too massive to be supported at equilibrium by the energy released by the planetesimals. |
At that point. it starts contracting and the subsequent runaway. aceretion of gas leads to the formation of giant planets (Boclenheimer Pollack 1956. Pollack et al. | At that point, it starts contracting and the subsequent runaway accretion of gas leads to the formation of giant planets (Bodenheimer Pollack 1986, Pollack et al. |
1996). | 1996). |
Al the same time as they form. protoplanets migrate through the nebula as a result of tidal interaction with the surrounding gas (Goldreich and Tremaine 1979. 1930. Lin and Papaloizou 1979. 1993 and references therein. Papaloizou Lin 1984. Ward 1936. 1997). | At the same time as they form, protoplanets migrate through the nebula as a result of tidal interaction with the surrounding gas (Goldreich and Tremaine 1979, 1980, Lin and Papaloizou 1979, 1993 and references therein, Papaloizou Lin 1984, Ward 1986, 1997). |
Calculations performed in isothermal discs show (hat cores of several earth masses migrate inward on a relatively short Gimescale. shorter than the planet formation timescale (Wd 1997. Tanaka et al. | Calculations performed in isothermal discs show that cores of several earth masses migrate inward on a relatively short timescale, shorter than the planet formation timescale (Ward 1997, Tanaka et al. |
2002. Date et al. | 2002, Bate et al. |
2003). | 2003). |
However. more recent caleulations including detailed energy balance in non isothermal disces show that migration can be slower or even outward (Paardekooper Mellema 2006. IXIev Cricda 2008. Paardekooper Mellema 2003. Daruteau Masset 2008. Paardekooper Papaloizou 2008). | However, more recent calculations including detailed energy balance in non isothermal discs show that migration can be slower or even outward (Paardekooper Mellema 2006, Kley Crida 2008, Paardekooper Mellema 2008, Baruteau Masset 2008, Paardekooper Papaloizou 2008). |
Usually. the torque exerted bv the disc on the protoplanet is caleulated by excluding the gas comprised in its Roche lobe. | Usually, the torque exerted by the disc on the protoplanet is calculated by excluding the gas comprised in its Roche lobe. |
Tlowever. it has been suggested by D'Angelo et al. ( | However, it has been suggested by D'Angelo et al. ( |
2003. see also Crida οἱ al. | 2003, see also Crida et al. |
2009) that the eas present in the Roche lobe but not bound to the protoplanet max contribute significantly to the total torque and slow down migration. | 2009) that the gas present in the Roche lobe but not bound to the protoplanet may contribute significantly to the total torque and slow down migration. |
This happens when the atmosphere of the protoplanet. ie. (he volume of gas at hydrostatic equilibrium that is bound to (he core. does not fill in its Roche lobe. and is surrounded by a cold (Bondi type) accretion flow. | This happens when the atmosphere of the protoplanet, i.e. the volume of gas at hydrostatic equilibrium that is bound to the core, does not fill in its Roche lobe, and is surrounded by a cold (Bondi type) accretion flow. |
restrict cosimnological various of the fuudinueutal physical constants at the level of ~1-2 ppin. | restrict cosmological variations of the fundamental physical constants at the level of $\sim$ 1-2 ppm. |
The estimate of fractiona changes in aand bby spectral methods is always a imeasuremieut of the relative Doppler shifts between the line centers of different atous/imolecules aud their comparison with corresponding laboratory values (Savedoff 1956: Bahecall L967: Wolfe 11976: Dzuba 1999. 2002: Levshakov 2001: Ianekar Cheneahw 2001). | The estimate of fractional changes in and by spectral methods is always a measurement of the relative Doppler shifts between the line centers of different atoms/molecules and their comparison with corresponding laboratory values (Savedoff 1956; Bahcall 1967; Wolfe 1976; Dzuba 1999, 2002; Levshakov 2004; Kanekar Chengalur 2004). |
To distinguish the line shifts due to radial motion of the object from those caused bv the variability. of constants Lnes with different sensitivity cocfiicicuts. O. to the variations of p aud/or à are to jeused?. | To distinguish the line shifts due to radial motion of the object from those caused by the variability of constants, lines with different sensitivity coefficients, ${\cal Q}$, to the variations of $\mu$ and/or $\alpha$ are to be. |
. Tt is clear that the Larger the differcuce [AQl )etwoeen two vausitions. the higher the accuracy of such estimates, | It is clear that the larger the difference $|\Delta {\cal Q}|$ between two transitions, the higher the accuracy of such estimates. |
Optical and UV frausifious du atoms. dons ancl nolecular hydrogen IH» have simular scusitivity cocficicuts with |AQ] not exceeding 0.05 (Varshalovich Levshakov 1993: Dzuba 1999. 2002: Porsev 22007). | Optical and UV transitions in atoms, ions and molecular hydrogen $_2$ have similar sensitivity coefficients with $|\Delta Q|$ not exceeding 0.05 (Varshalovich Levshakov 1993; Dzuba 1999, 2002; Porsev 2007). |
For atomic spectra. the estimate of iis given in linear approximation (|Aafa|€ 1) bv (e.gBS. Levshakov 22006): where Y4.15 are the radial velocities of two atomic lines. aud e is the speed of light. | For atomic spectra, the estimate of is given in linear approximation $|\Delta\alpha/\alpha| \ll 1$ ) by (e.g., Levshakov 2006): where $V_1, V_2$ are the radial velocities of two atomic lines, and $c$ is the speed of light. |
It was shown im Molaro ((2008b) that the lamiting accuracy of the wavelenetli scale calibration for the VLT/UVES quasar spectra at any point witlin the whole optical domain is about 20Lo which corresponds to the limitiug relative accuracy between two lines imieasured in different parts of the sue spectrum of iout b+. | It was shown in Molaro (2008b) that the limiting accuracy of the wavelength scale calibration for the VLT/UVES quasar spectra at any point within the whole optical domain is about 30, which corresponds to the limiting relative accuracy between two lines measured in different parts of the same spectrum of about 50. |
.. Taking iuto account that [AQ]20.05, it follows from Eq.(1)) that the limiting accuracy of Aa/a is we ppui. which is the utmost value that can be achieved iu observations of extragalactic οjects with present optical facilities. | Taking into account that $|\Delta Q| \simeq 0.05$, it follows from \ref{eq1}) ) that the limiting accuracy of $\Delta\alpha/\alpha$ is 2 ppm, which is the utmost value that can be achieved in observations of extragalactic objects with present optical facilities. |
A considerable higher seusitivitv to the variation of plysical constants d$ observe in radio rauec. | A considerably higher sensitivity to the variation of physical constants is observed in radio range. |
For exaniple. van Veldhoven (9000) first showed that the inversion frequeney of he (JN)=(1.1) level of the ammonia isotopologue OND: has the sensitivity coefficient Qj=5.6. | For example, van Veldhoven (2004) first showed that the inversion frequency of the $(J,K) = (1,1)$ level of the ammonia isotopologue $^{15}$ $_3$ has the sensitivity coefficient ${\cal Q}_\mu = 5.6$. |
Compared o optical aud UV transitions. the znunonia inethod xoposed by Flambamm Nozloy (2007) provides 35 ines more seusitive estimate of from measurements of the radial velocity offset )etween the NIT; CJ.A)=(1.1) inversion transition at 23.7 GIIz and low-lvineg rotational transitions of other molecules co-spatially distributed with NIT;: The anunonia method was receutlv used to explore possible spatial of plysical constauts from observations of prestellar molecular cores m the Taurus eiut molecular cloud (Levshakoy 22010. hereafter LIO). the Perseus cloud. the Pipe Nebula. and Iufrared dark clouds (Levshakoy 22008b: Molaro 22009). | Compared to optical and UV transitions, the ammonia method proposed by Flambaum Kozlov (2007) provides 35 times more sensitive estimate of from measurements of the radial velocity offset between the $_3$ $(J,K) = (1,1)$ inversion transition at 23.7 GHz and low-lying rotational transitions of other molecules co-spatially distributed with $_3$: The ammonia method was recently used to explore possible spatial of physical constants from observations of prestellar molecular cores in the Taurus giant molecular cloud (Levshakov 2010, hereafter L10), the Perseus cloud, the Pipe Nebula, and Infrared dark clouds (Levshakov 2008b; Molaro 2009). |
Iu. contrast to the meutioned above laboratory coustraints on£emporal variatious. this method reveals a tentative spatial variation of aat the level of = (2.20.LacEO)«10.* (E10). | In contrast to the mentioned above laboratory constraints on variations, this method reveals a tentative variation of at the level of = $(2.2\pm0.4_{\rm stat}\pm0.3_{\rm sys})\times10^{-8}$ (L10). |
The corresponding conservative upper Πιτ iu this case is equa to [Ap/plx3s107. | The corresponding conservative upper limit in this case is equal to $|\Delta \mu/\mu| \leq 3\times10^{-8}$. |
Iu the preseut paper we consider fractional changes of a comibination⋅⋅ of. two constants o> and µ. Fo-ajfg. which are estimated frou the comparison of transition frequencies measured in differeut. physical ensiromneuts of (terrestrial) andlow (uterstellar) densities of barvouic matter. | In the present paper we consider fractional changes of a combination of two constants $\alpha^2$ and $\mu$, $F = \alpha^2/\mu$, which are estimated from the comparison of transition frequencies measured in different physical environments of (terrestrial) and (interstellar) densities of baryonic matter. |
The idea behind this experiment is that sole class of scalar field models so-called chameleonlike fields predict the dependence of both masses aud coupling constaut on the local matter density (Olive Pospelov 2008: Upadliye 22010). | The idea behind this experiment is that some class of scalar field models — so-called chameleon-like fields — predict the dependence of both masses and coupling constant on the local matter density (Olive Pospelov 2008; Upadhye 2010). |
Chameleou-like scalar fields were introduced hy WWhowy Weltiman (200lad) aud w Drax ((2001) to explain uceative results on laboratory searches or the fifth force which should arise inevitably from couplingsσαi between scalar fields aud standard model artices. | Chameleon-like scalar fields were introduced by Khoury Weltman (2004a,b) and by Brax (2004) to explain negative results on laboratory searches for the fifth force which should arise inevitably from couplings between scalar fields and standard model particles. |
The chameleon models assume that a helt scalar field acquires both an effective potential aud effective mass jecauxse of its coupling to matter that depends ou the züubieut matter cdeusityv. | The chameleon models assume that a light scalar field acquires both an effective potential and effective mass because of its coupling to matter that depends on the ambient matter density. |
In tlis wav. the chameleon scalar field may evade local tests of the equivalence principle and fifth force experiments since the range of the scalar-uediated fifth force for the errestrial matter densities is too small to be detected. | In this way, the chameleon scalar field may evade local tests of the equivalence principle and fifth force experiments since the range of the scalar-mediated fifth force for the terrestrial matter densities is too small to be detected. |
Suiuülulv. laboratory tests with atomic clocks for q-variatious are performed uuder couditious of coustaut local density aud. hence. they are tot scusitive to the presence of the chameleon scalar field (Upacdhve 22010). | Similarly, laboratory tests with atomic clocks for $\alpha$ -variations are performed under conditions of constant local density and, hence, they are not sensitive to the presence of the chameleon scalar field (Upadhye 2010). |
This is not the case for space-based tests. where he matter deusity is cousiderably lower. an effective mass of the scalar field is negheible. aud au effective ranec or the scalar-1uediated force is large. | This is not the case for space-based tests, where the matter density is considerably lower, an effective mass of the scalar field is negligible, and an effective range for the scalar-mediated force is large. |
Light scalar fields are usually attributed to a negative pressure substance omueatiug the entire visible Universe aud known as dark energev (Caldwell11998). | Light scalar fields are usually attributed to a negative pressure substance permeating the entire visible Universe and known as dark energy (Caldwell1998). |
This substance is thought to be responsible for | This substance is thought to be responsible for |
contest of the standard model. | context of the standard model. |
The analysis performed In 833 can tentatively be of some relevance to these observations. | The analysis performed in 3 can tentatively be of some relevance to these observations. |
Using. for iustance the example set by Fie. | Using, for instance the example set by Fig. |
as a euide. we can sav that plateaus appear as lone as thevalues of 7,444 aud e. are chosen from the region defined by the two τος lines. | \ref{par_space} as a guide, we can say that plateaus appear as long as thevalues of $\gammamax$ and $\ee$ are chosen from the region defined by the two tilted lines. |
Moving inside this region roni the lower left to the upper melt. plateaus appear at oxoeressivelv louger times. | Moving inside this region from the lower left to the upper right, plateaus appear at progressively longer times. |
Furthermore. a choice of the initial parameters outside this region (for example. from he lef op corner of Fig. 5)) | Furthermore, a choice of the initial parameters outside this region (for example, from the left top corner of Fig. \ref{par_space}) ) |
leads to afterelows without a ateau phase. | leads to afterglows without a plateau phase. |
Fig. | Fig. |
10. shows differeut types of X-ray ight curves obtained using our πιαΊσα code corresponding o points from different regions of the parameter space of Fie. 5.. | \ref{sximatika2} shows different types of X-ray light curves obtained using our numerical code corresponding to points from different regions of the parameter space of Fig. \ref{par_space}. |
Light curves in panels (a) and (b) correspond o points (1) and (1) alreacky shown in Fig. 5.. | Light curves in panels (a) and (b) correspond to points (4) and (1) already shown in Fig. \ref{par_space}, |
while ight curves of panels (ο) aud (d) are obtained using (Ecμμ)=(0.0032.6.3ον107) and (0.01,10%) A tentative comparison of our model Light curves to hose of Fig. | while light curves of panels (c) and (d) are obtained using $(\ee,\gammamax)=(0.0032, 6.3\times10^3)$ and $(0.01, 10^6)$ A tentative comparison of our model light curves to those of Fig. |
9 can be mace. | \ref{sximatika} can be made. |
As the model presented here produces multiwaveleneth spectra at each iustautf. we can use it to calculate the evolution of the expected XN-rav hardness ratio defined as the ratio of counts in the 1.5-10 keV to the counts in the 0.1-1.5 keV band (??). | As the model presented here produces multiwavelength spectra at each instant, we can use it to calculate the evolution of the expected X-ray hardness ratio defined as the ratio of counts in the 1.5-10 keV to the counts in the 0.1-1.5 keV band \citep{Evans09,evans10}. |
Figure 11 shows the time evolution of the harcduess ratio for each of the exanirple cases shown iu Fig. 10.. | Figure \ref{HR} shows the time evolution of the hardness ratio for each of the example cases shown in Fig. \ref{sximatika2}. |
Time evolutiou of the corresponding photon index is also shown iu the iuserts of Fig. 11.. | Time evolution of the corresponding photon index is also shown in the inserts of Fig. \ref{HR}, |
whenever the spectral shape allows its viable calculation at the particular time (for a more detailed discussion on the shape of our X-ray iocel spectra see 822). | whenever the spectral shape allows its viable calculation at the particular time (for a more detailed discussion on the shape of our X-ray model spectra see 2). |
For Naaw light curves with a distinctive platezi phase. we fud that the spectral evolution shows a characteristic treud as the X-ray window is first dominated w the svuchrotron and later by the SSC component. | For X-ray light curves with a distinctive `plateau' phase, we find that the spectral evolution shows a characteristic trend as the X-ray window is first dominated by the synchrotron and later by the SSC component. |
This can be seeu in panel (a) of Fig. Hl... | This can be seen in panel (a) of Fig. \ref{HR}. . |
At very carly times oth the soft aud hard N-vav bands are dominated by the | At very early times both the soft and hard X-ray bands are dominated by the |
The unexpected discovery of elant planets orbiting their host stars with periods of a few davs (Alavor&Queloz!1995:Butlerotal.1997) made it clear that our theories of planet formation aud evolution are incomplete (c.g.Linetal.1996). | The unexpected discovery of giant planets orbiting their host stars with periods of a few days \citep{mayor1995, butler1997} made it clear that our theories of planet formation and evolution are incomplete \citep[e.g.][]{lin1996}. |
Althoueh progress has Όσοι made toward understanding the the presence of short-period giant planets; many details of their formation aud evolution remain uncertain. | Although progress has been made toward understanding the the presence of short-period giant planets, many details of their formation and evolution remain uncertain. |
That short-period planets form from cieumnstellar disks is clear. but the exact mechanisua i which disk inaterial condeuses to form planets has vet to be established. | That short-period planets form from circumstellar disks is clear, but the exact mechanism in which disk material condenses to form planets has yet to be established. |
Jupiter mass planets are thought to forma through one of two mechamisius: core accretion ≺⋀∖∐∑∏∐∪↕⋂≺∖∖∣∶↕≧∪≼∐∖∐∐↸∖↕⋯↸∖↥⋅∙∖↽↕⋟∪↕⋯⊳↨↘↽↕∩≺∖∖⊓∶↕⋟∪∐⋯⊳↨↘↽etal. 1996).. iui which a rocky or dev core reaches a critical mass aud rapidly acerctes an envelope of gas. or gravitational iustability (Boss1997). in which the massive, cool disk fragnmieuts and collapses iuto eiaut planets | Jupiter mass planets are thought to form through one of two mechanisms: core accretion \citep{mizuno1980, bodenheimer1986, pollack1996}, in which a rocky or icy core reaches a critical mass and rapidly accretes an envelope of gas, or gravitational instability \citep{boss1997}, in which the massive, cool disk fragments and collapses into giant planets. |
Core accretion theories have historically had difficulty producing planets of the correct LAsKCs and locations ofthe giaut planets in our solar system within a ~5 Myr protoplauctary disk lifetime (oe.Pollacketal. 1996}.. although recent work- has been more successful (scoMovsliovitzetal.20]audreferences.therein)... | Core accretion theories have historically had difficulty producing planets of the correct masses and locations of the giant planets in our solar system within a $\sim5$ Myr protoplanetary disk lifetime \citep[e.g.][]{pollack1996}, although recent work has been more successful \citep[see][and references therein]{movshovitz2010}. |
Gravitational oeinstability, on the other haud. requires relatively cool. quiescent disk4.- conditions and sufficiently short cooling timescales. both of which may not be physically realistic at the radi at which gas eiauts are thoneht to form (Rafikov2005}. | Gravitational instability, on the other hand, requires relatively cool, quiescent disk conditions and sufficiently short cooling timescales, both of which may not be physically realistic at the radii at which gas giants are thought to form \citep{rafikov2005}. |
It is clear that short-period giant planets cannot formsitu. since protoplanctary disks at a few tenths of an AU are too hot and do not contain sufficient niass to produce a gas giant. | It is clear that short-period giant planets cannot form, since protoplanetary disks at a few tenths of an AU are too hot and do not contain sufficient mass to produce a gas giant. |
Therefore. iu addition to the uucertainties reerdiug the formation mechaisin. we must face additional uncertainties related to the means by which a planet migrates frou its livpothesized birthsite in order to explain HHgas elauts in short-period orbits, | Therefore, in addition to the uncertainties regarding the formation mechanism, we must face additional uncertainties related to the means by which a planet migrates from its hypothesized birthsite in order to explain gas giants in short-period orbits. |
⋅ Iu what is⋅ labeled Type I migration⋅. (V-ard.ves99r:Goldreich&Tremaine 1980).. the planet experiences torques from the massive gaseous disk. a loss of angular momentum. and subsequent inward motion toward the star. | In what is labeled Type I migration \citep{ward1997, goldreich1980}, the planet experiences torques from the massive gaseous disk, a loss of angular momentum, and subsequent inward motion toward the star. |
Sufficieutlv massive planets are capable of opening a eap in the circtuustellay disk. which can also result iu inward drift as the disk accretes outo the star: this is called Type IT 1uigration (Lin&Papaloizou1986:Ward 1997). | Sufficiently massive planets are capable of opening a gap in the circumstellar disk, which can also result in inward drift as the disk accretes onto the star; this is called Type II migration \citep{lin1986, ward1997}. |
. Recently. plauect mieration through plauct-planct scattering and eccentricity pimping due to a stellar ünarv colpanion coupled with tidal dissipation ("Ikozai ↕⊔⋮↰⋝↕⋅⋮↧↾↕⊲⋅⋟⊔.DU) have heen proposed as migration:⋅ us (SCCNagaxawaEal.&Dicchanis2003:Mazchctal. ). | Recently, planet migration through planet-planet scattering and eccentricity pumping due to a stellar binary companion coupled with tidal dissipation (“Kozai migration”) have also been proposed as migration mechanisms \citep[see][]{nagasawa2008,wu2003,mazeh1997}. |
Theet relative2008: wimaportauce Murrayof of these imechianisuuis1991 to the process of planctary eachuigration | The relative importance of each of these mechanisms to the process of planetary migration remains unclear. |
Each of these remainsdifferent unclear.formation. mtechanistus. should mupriut certain signatures onto the resulting planet | | Each of these different formation mechanisms should imprint certain signatures onto the resulting planet population. |
One hopes. with a sifficicnthy large observed. of population,planets. to be to determine the relative sampleinrportance of of these ableformation channels. | One hopes, with a sufficiently large observed sample of planets, to be able to determine the relative importance of each of these formation channels. |
From au 6 point-of-view.each the wav to unclerstand the origin of servationalshext-period planets is to search a nuuber of stars planets aud a correspondinglarge rich sample of for The pr | From an observational point-of-view, the way to understand the origin of short-period planets is to search a large number of stars for planets and produce a correspondingly rich sample of exoplanets. |
oducestatistical properties of hythese planes their exoplauets, frecneney can help to constrain andmodels of planct observed.formation aud evolution. | The statistical properties of these planets and their observed frequency can then help to constrain models of planet formation and evolution. |
then It is therefore esseutial that sample of stars searched. for planets has well characterizedauy properties so that trends among the stars that are found to host planets cau be TOCOS | It is therefore essential that any sample of stars searched for planets has well characterized properties so that trends among the stars that are found to host planets can be recognized. |
Janesχα.∙ suggested that open clusters were ideal. targets for (1996)planet searches since they host such a nuiform and easily characterizable population of stars. | \citet{janes1996} suggested that open clusters were ideal targets for planet searches since they host such a uniform and easily characterizable population of stars. |
Iu | In |
The numerical study of the system presented in Section 2 was performed. using the standard. solar model caleulated ov Christensen-Dalseaard et al. ( | The numerical study of the system presented in Section \ref{sec:model}
was performed using the standard solar model calculated by Christensen-Dalsgaard et al. ( |
1991). as the reference state. | 1991) as the reference state. |
The imposed latitudinal shear was chosen to be that observed in the solar convection zone. (with Qs—]).15cos?80.15cost in units of Q,). | The imposed latitudinal shear was chosen to be that observed in the solar convection zone (with $\tilde{\Omega}_{\rm cz} = 1-0.15
\cos^2 \theta - 0.15 \cos^4 \theta$ in units of $\Omega_{\star}$ ). |
As the rotation rate O, is varied. it appears that only two parameters control the »haviour of the svstem: the Ekman number (which controls he bouncdarv-Iaver behaviour and the How velocities) and a new number which controls the cdvnamies of the bulk of the radiative interior. | As the rotation rate $ \Omega_{\star}$ is varied, it appears that only two parameters control the behaviour of the system: the Ekman number (which controls the boundary-layer behaviour and the flow velocities) and a new number which controls the dynamics of the bulk of the radiative interior. |
In the case of slow rotation. the numerical results suggest that Z and the poloidal components of the velocity wa scale with ££, and A as: where the quantities with bars are the scaled (quantities. of order of unity. | In the case of slow rotation, the numerical results suggest that $\tilde{T}$ and the poloidal components of the velocity $u_{r,\theta}$ scale with $\Enu$ and $\lambda$ as: where the quantities with bars are the scaled quantities, of order of unity. |
Lt is also found that wv, and 9 are always of order of unity. which is expected. | It is also found that $\up$ and $\tilde{\Phi}$ are always of order of unity, which is expected. |
Note that the scaling for the meridional motions is a local Ecdineton-Sweet scaling (sce Spiegel Zahn. 1992). | Note that the scaling for the meridional motions is a local Eddington-Sweet scaling (see Spiegel Zahn, 1992). |
Using this ansatz into the svsten of equations given in (1)). an expansion in powers of 1/À reveals that the angular-momoentunmr balance is dominated to zeroth order by. viscous transport through: which determines the angular velocity. profile uniquely. | Using this ansatz into the system of equations given in \ref{eq:basic1}) ), an expansion in powers of $1/\lambda$ reveals that the angular-momentum balance is dominated to zeroth order by viscous transport through: which determines the angular velocity profile uniquely. |
Using this result in the first order equations provides a relation. between the temperature and gravitational potential perturbations: which can be solved. independently for T and. Oc/06. | Using this result in the first order equations provides a relation between the temperature and gravitational potential perturbations: which can be solved independently for $\overline{T}$ and $\ptl
\tilde{\Phi}/\ptl \theta $. |
Finally. the temperature Ductuations lead to. meridional motions through the energy. advyection-dilfusion equation: In Fig. 1.. | Finally, the temperature fluctuations lead to meridional motions through the energy advection-diffusion equation: In Fig. \ref{fig:lambdabig}, |
E show the results of the numerical solutions for the angular velocity profile and the meridional motions corresponding to a slowly rotating solar-tvpe star (for which Ac 107). | I show the results of the numerical solutions for the angular velocity profile and the meridional motions corresponding to a slowly rotating solar-type star (for which $\lambda
\simeq 10^{4}$ ). |
The angular velocity profile is exactly the sane as one obtained through solving (Vou),=0. | The angular velocity profile is exactly the same as one obtained through solving $(\grad^2 \bu)_{\phi} = 0$. |
The meridional Dow. velocities decrease strongly with depth and the streamlines follow a lavered structure reminiscent. of a Lolton Low. | The meridional flow velocities decrease strongly with depth and the streamlines follow a layered structure reminiscent of a Holton flow. |
In the case of rapid rotation the correct scaling seems to be This time. L perform an asymptotic expansion in the small parameter A. | In the case of rapid rotation the correct scaling seems to be This time, I perform an asymptotic expansion in the small parameter $\lambda$. |
In this limit the temperature Huctuations are strongly damped by the rapid heat dillusion (as Aκ1 is equivalent to the small Pranclt number limit) and. the system reaches an equilibrium that is determined. by. the zeroth order equations: These equations describe a geostrophic equilibrium. and can in principle be solved. for (7, and & alone. | In this limit the temperature fluctuations are strongly damped by the rapid heat diffusion (as $\lambda \ll 1$ is equivalent to the small Prandlt number limit) and the system reaches an equilibrium that is determined by the zeroth order equations: These equations describe a geostrophic equilibrium, and can in principle be solved for $\up^2$ and $\tilde{\Phi}$ alone. |
The solutions provide. to the next order in A. an equation for the meridional How through the advection dillusion balance: and. finally. the temperature Ductuations through The results of the numerical simulations for small A (Ac10 7) are shown in Fig. 2.. | The solutions provide, to the next order in $\lambda$, an equation for the meridional flow through the advection diffusion balance: and, finally, the temperature fluctuations through The results of the numerical simulations for small $\lambda$ $\lambda
\simeq 10^{-2}$ ) are shown in Fig. \ref{fig:lambdasmall}. |
lt is interesting. to note thatthe radial shear is much larger than the latituclinal shear. which suggests to consider. οςOO00 to perform. a first. analytical approximation. | It is interesting to note thatthe radial shear is much larger than the latitudinal shear, which suggests to consider $\ptl \Omega/\ptl
\xi \gg \ptl \Omega /\ptl \theta$ to perform a first analytical approximation. |
Similarly. L suppose that if V=0b700 then OW/OS> £00. | Similarly, I suppose that if $\Psi=\ptl \tilde{\Phi}/ \ptl \theta$ then $\ptl
\Psi/\ptl \xi \gg \ptl \Psi/\ptl \theta$ . |
οσο approximations can be used into | These approximations can be used into |
coordinates: could not be tied: to the USSGZG DC.B2.8wh systend | coordinates could not be tied to the USNO B2.0 system. |
5tlfeeu attribute their low flux level to an offset tài &láfive Ao:@ovd—Eitiuu with respect to the rest of oursample. | We attribute their low flux level to an offset in relative coordinates with respect to the rest of our sample. |
A final sample of23 reliable Fremantle is= obtained.Ξ of which 15 have archival HST imag meastrenesasmenteibis. | A final sample of 23 reliable measurements is obtained, of which 15 have archival HST imaging available. |
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