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Although we do not investigate such a scenario. multiple companions may successfully. expel the envelope such that all or a subset survive. | Although we do not investigate such a scenario, multiple companions may successfully expel the envelope such that all or a subset survive. |
For the à=0.7 model. the binding energy. of the envelope is minimized when pas the peak radius on the AGB. | For the $\eta=0.7$ model, the binding energy of the envelope is minimized when past the peak radius on the AGB. |
An SA Ad) mass companion is sullicient to unbind the envelope and survive a CLE phase. | An 8.4 $M_{\rm J}$ mass companion is sufficient to unbind the envelope and survive a CE phase. |
For the y=1 model. the envelope binding energy is minimize at the peak of the RGB. | For the $\eta=1$ model, the envelope binding energy is minimized at the peak of the RGB. |
For this svstem. a 9.8 Ady companion is sullicient to survive a CIS phase. | For this system, a 9.8 $M_{\rm J}$ companion is sufficient to survive a CE phase. |
The jj=5 model does no achieve a sullicient core mass to undergo a helium Iash. | The $\eta=5$ model does not achieve a sufficient core mass to undergo a helium flash. |
As such. its maximal radial extent is less than the jj=0.7 ane ηΞd models while the envelope binding energy. ds reduced. | As such, its maximal radial extent is less than the $\eta=0.7$ and $\eta=1$ models while the envelope binding energy is reduced. |
In this case. à ~7.S Mj mass companion is sullicicnt to survive a CIE phase. | In this case, a $\sim$ 7.8 $M_{\rm J}$ mass companion is sufficient to survive a CE phase. |
For all of our models. most companions more massive than ~LO Mj will supply sullicient energy to unbine the envelope and survive the CLEP at greater orbital If a massive planctary companion or a brown cwarl is engulfed and survives the common envelope phase. it could be left in à close. short-period orbit. around the resultant WD. | For all of our models, most companions more massive than $\sim$ 10 $M_{\rm J}$ will supply sufficient energy to unbind the envelope and survive the CEP at greater orbital If a massive planetary companion or a brown dwarf is engulfed and survives the common envelope phase, it could be left in a close, short-period orbit around the resultant WD. |
In such a scenario. if the orbit were oriented. edge-on. it is conceivable that an entirely new regime of exoplanetary transiting observations would allow for detection. of the companion. | In such a scenario, if the orbit were oriented edge-on, it is conceivable that an entirely new regime of exoplanetary transiting observations would allow for detection of the companion. |
In contrast to the observations ploneered by 7 and ?exoplanets... where transiting Jupiter- planets cause dips of ~1% in the light curves of main sequence solar tvpe stars. a massiveJupiter transiting a WD could. cause a total eclipse (a dip). | In contrast to the observations pioneered by \citet{Henry:2000lr} and \citet{Charbonneau:2000fk}, where transiting Jupiter-sized planets cause dips of $\sim$ in the light curves of main sequence solar type stars, a massiveJupiter transiting a WD could cause a total eclipse (a dip!). |
Llow Likely are such detections? | How likely are such detections? |
In this context. rather than the oft- a priori probability of A.fa that an exoplanct’s orbit | In this context, rather than the oft-quoted a priori probability of $R_*/a$ that an exoplanet's orbit |
associated. cosmological pzuwanaeters are given in Table I in App. | associated cosmological parameters are given in Table I in App. |
A of paper I. We compare the predictions of our nodel with πιοΊσα. sinulatious for these six aternative cosinologics im Figs. | A of paper I. We compare the predictions of our model with numerical simulations for these six alternative cosmologies in Figs. |
5Γ and 6 for the two-point ad tiree-yout statistics at E"=l. | \ref{fig_xi_comp} and \ref{fig_kappa3_comp}
for the two-point and three-point statistics at $z_s=1$. |
To avoid overcrowding: the Heures we did not plot tlio error bars of the ποασ] simulations. | To avoid overcrowding the figures we did not plot the error bars of the numerical simulations. |
Each pair n. iiOne, and wy gives two curves hat are roughly sveric around the fiducial cosiiology result. because we consider siuall deviations of c10. | Each pair $n_s$, $\Omega_{\rm c}h^2$, and $w_0$ gives two curves that are roughly symmetric around the fiducial cosmology result, because we consider small deviations of $\pm 10\%$. |
The deviations are largest for the η case. which changes he shape of the initia power spectrum as well as the rormalization c4. | The deviations are largest for the $n_s$ case, which changes the shape of the initial power spectrum as well as the normalization $\sigma_8$. |
These six cases roughly cover the rauge ha is allowed by current data. and the ay. case ds already. somewhat bevoid the observational bounds (?).. | These six cases roughly cover the range that is allowed by current data, and the $n_s$ case is already somewhat beyond the observational bounds \citep{Komatsu2011}. |
Therefore. they provide a good check of the robustuess of our model for realistic sce111105. | Therefore, they provide a good check of the robustness of our model for realistic scenarios. |
Like for the Fouricr-space statistics studied iu paver 1. the dependence on Cosimooev of the two-point statistics is well reproduced by οιr model. | Like for the Fourier-space statistics studied in paper I, the dependence on cosmology of the two-point statistics is well reproduced by our model. |
For the threc-voit statistics it is not easv to 1nake a VOYV precise Cconipalison because the αποσα. restIts show a greater level of nolse and are seusitive to finite resolutiou aud finite size effects. | For the three-point statistics it is not easy to make a very precise comparison because the numerical results show a greater level of noise and are sensitive to finite resolution and finite size effects. |
ILlowever. where the simulaions are reliable. we also otain a good mmatch with our predictions. | However, where the simulations are reliable, we also obtain a good match with our predictions. |
We obtained similar results for τς0.6 aud τς1.5. as well as for other cosnmiologies where we vary sl. or Qa by ιο, | We obtained similar results for $z_s=0.6$ and $z_s=1.5$, as well as for other cosmologies where we vary $A_s$ or $\Omega_{\rm de}$ by $\pm 10\%$. |
This shows that our model and. more geucrallv. iiodels based on colmbinatious of perturbation theory aud halo models provide a good modeling of the matter distribution aud of weak gravitational leusiug effects aud capture their dependence on cosmology. | This shows that our model and, more generally, models based on combinations of perturbation theory and halo models provide a good modeling of the matter distribution and of weak gravitational lensing effects and capture their dependence on cosmology. |
Moreover. Figs. | Moreover, Figs. |
5 and 6 clearly show that this analytical modeling is compotitive with current ray-tracing simulations. because it provides reliable predictions over a ereater range of scales. | \ref{fig_xi_comp} and \ref{fig_kappa3_comp} clearly show that this analytical modeling is competitive with current ray-tracing simulations, because it provides reliable predictions over a greater range of scales. |
Iu particular. Figs. | In particular, Figs. |
5 aud 6 show that the accuracy of our model is sufficient to coustrain i4. Ὁμή, aud wy to better than 105. | \ref{fig_xi_comp} and \ref{fig_kappa3_comp} show that the accuracy of our model is sufficient to constrain $n_s$, $\Omega_ch^2$, and $w_0$ to better than $10\%$. |
Tn the previous sections we considered. single-scale moments. ο... associated. with onesmoothing window Wo with ao single angular radius ().. | In the previous sections we considered single-scale moments, $\lag X_s(\theta_s)^p\rag$ , associated with onesmoothing window $W_{\theta_s}^{X_s}$ with a single angular radius $\theta_s$. |
One can also use multi-poiut statistics such as NOVorNAOyOy) associated. with p windows centered on p differeut directions 9; aud with p differeut radi 0,. | One can also use multi-point statistics such as $\lag X_s(\vtheta_1;\theta_{s1}) .. X_s(\vtheta_p;\theta_{sp})\rag$ associated with $p$ windows centered on $p$ different directions $\vtheta_i$ and with $p$ different radii$\theta_{si}$ . |
In this section. we briefiv check the valicity of | In this section, we briefly check the validity of |
the preceding session. | the preceding session. |
The experiment is first carried out in the absence of the beam until the simulated atmosphere attains a quasi-steady state. | The experiment is first carried out in the absence of the beam until the simulated atmosphere attains a quasi-steady state. |
The beam heating is (hen added afterwards. | The beam heating is then added afterwards. |
Figure |. shows the snapshot structures of the stellar aimosphere for D,= 1 and 5 G denoted respectively by "small ancl "large D ancl de. and compares the results with and without the electron-beam injection. | Figure 1 shows the snapshot structures of the stellar atmosphere for $B_*=$ 1 and 5 G denoted respectively by “small" and “large $B$ and $dv$ ", and compares the results with and without the electron-beam injection. |
In the B,=LG case. the results for Fj;=10” erg 7s ! (blue curves) are almost identical to those for the no-beam case (green curves). | In the $B_*=1$ G case, the results for $F_{bi}=10^2$ erg $^{-2}$ $^{-1}$ (blue curves) are almost identical to those for the no-beam case (green curves). |
In contrast. the atmospheric structures change noticeably in (he larger D, (=5G) and <de.> (=3.6 km/s) cases. as illustrated by the red and black curves. | In contrast, the atmospheric structures change noticeably in the larger $B_*$ $=5$ G) and $<dv_\perp>$ $=3.6$ km/s) cases, as illustrated by the red and black curves. |
In the absence of the beam heating. (he stronger dissipation arising from (he larger <de> raises the chromospheric temperature anc therefore causes (he chromosphere {ο evaporate aid expancl (see (he red curve in the middle panel). | In the absence of the beam heating, the stronger dissipation arising from the larger $<dv_\perp>$ raises the chromospheric temperature and therefore causes the chromosphere to evaporate and expand (see the red curve in the middle panel). |
The local density scale height increases ancl (he density. drops more slowly with r. | The local density scale height increases and the density drops more slowly with $r$. |
Therefore. the locations of the transition region lie at higher altitucdes in the larger «απ> case. | Therefore, the locations of the transition region lie at higher altitudes in the larger $<dv_\perp>$ case. |
A hot and dense region forms in the chromosphere. which we lerm as "a warm region" (ie. > I0!k. see the bottom panel). | A hot and dense region forms in the chromosphere, which we term as “a warm region" (i.e. $T>10^4$ K, see the bottom panel). |
It then becomes difficult to [urther heat up the warm region to the coronal temperature (i.e. Z105 IN) as a result of the ellicient radiative cooling at ZT<10" IK (Landini&Monsignori-Fossi1990). | It then becomes difficult to further heat up the warm region to the coronal temperature (i.e. $\gtrsim 10^6$ K) as a result of the efficient radiative cooling at $T\lesssim 10^5$ K \citep{LM90}. |
. Llowever. when the beam energv of £j;=10 erg 7s ! is added. the chromosphere is further heated up and thus an even larger warm region can form at the location of r=1.001—1.01, (see the black curve in the bottom panel). | However, when the beam energy of $F_{bi}=10^4$ erg $^{-2}$ $^{-1}$ is added, the chromosphere is further heated up and thus an even larger warm region can form at the location of $r=1.001 - 1.01 R_*$ (see the black curve in the bottom panel). |
The black curve in the middle panel indicates that the density is on the average much higher in the beam-heated region. | The black curve in the middle panel indicates that the density is on the average much higher in the beam-heated region. |
Consequently. the heating rale per unit mass becomes smaller and the temperature of the coronal region ad rZ1.014, then becomes lower in the case of £j;=10! erg ? ! than in the case of £j;=0 (see the bottom panel). | Consequently, the heating rate per unit mass becomes smaller and the temperature of the coronal region at $r \gtrsim 1.01R_*$ then becomes lower in the case of $F_{bi}=10^4$ erg $^{-2}$ $^{-1}$ than in the case of $F_{bi}=0$ (see the bottom panel). |
The sharp transition region disappears. | The sharp transition region disappears. |
The top panel of Figure 1 shows the radial profile of the stellar wind velocity c.. | The top panel of Figure 1 shows the radial profile of the stellar wind velocity $v_r$ . |
The winds in the outer region (r= 222,) ave faster in the large DB, case. | The winds in the outer region $r\gtrsim
2R_*$ ) are faster in the large $B_*$ case. |
B, equals B,,//. which in [act specifies the flux tube properties. | $B_*$ equals $B_{ph}/f$, which in fact specifies the flux tube properties. |
The larger the [actor μεf is. the more the wave enerev dissipates in the outer region of the atmosphere. leading to faster winds 2006). | The larger the factor $B_{ph}/f$ is, the more the wave energy dissipates in the outer region of the atmosphere, leading to faster winds \citep{Kojima,Suzuki06}. |
. However. (he winds on the average are not significantly amplified by the beam heating. as indicated by the large overlap between the red and black curves in the plot. | However, the winds on the average are not significantly amplified by the beam heating, as indicated by the large overlap between the red and black curves in the plot. |
The wind speed at the planets orbit is ©300 km !. which is lower than the Alfvénn speeds ~500 and 1000 kin ! (here in our cases lor 2,=1 and 5 CG. That is. the planet lies inside the Alfvénn radius. | The wind speed at the planet's orbit is $\approx 300$ km $^{-1}$, which is lower than the Alfvénn speeds $\sim 500$ and 1000 km $^{-1}$ there in our cases for $B_*=1$ and 5 G. That is, the planet lies inside the Alfvénn radius. |
IIaving described the snapshot structures. we should note that the thermal properties of (he stellar atmosphere actually fluctuate with (me owing to wave propagation and dissipation. | Having described the snapshot structures, we should note that the thermal properties of the stellar atmosphere actually fluctuate with time owing to wave propagation and dissipation. |
As a result. the location and thermal propertiesof the warm region fluctuate with time as | As a result, the location and thermal propertiesof the warm region fluctuate with time as |
where the symbol + indicates the possibility that a line of sight intersects twice the working surface. | where the symbol $\pm$ indicates the possibility that a line of sight intersects twice the working surface. |
Therefore. the transformation equations between the two frames of reference are. The intersection conditions of the j-th working surface. formed in the approaching (z> 0) cone or the recedingσι (σ΄< 0) cone. are obtainedby comparison of equation 14 with the edges of the caps (Er,4tj]cos 64). | Therefore, the transformation equations between the two frames of reference are, The intersection conditions of the $j$ -th working surface, formed in the approaching $z'> 0$ ) cone or the receding $z'< 0$ ) cone, are obtainedby comparison of equation \ref{tracooreng} with the edges of the caps $\pm r_{ws[j]}~ \mbox{cos}~ \theta_a$ ). |
AS a consequence. the y-th working surface is intersected by a given line of sight when. or. Assuming that the observer is located at a distance D from the source. a given line of sight intersects the plane of the sky [y.z] at the point (DsinOsinb,DsinOcos Φ). where © and ᾧ are the inclination and the azimuthal angles. respectively. | As a consequence, the $j$ -th working surface is intersected by a given line of sight when, or, Assuming that the observer is located at a distance $D$ from the source, a given line of sight intersects the plane of the sky $[y,z]$ at the point $D\,\mbox{sin}\,\Theta\,\mbox{sin}\,\Phi,
D\,\mbox{sin}\,\Theta\,\mbox{cos}\,\Phi$ ), where $\Theta$ and $\Phi$ are the inclination and the azimuthal angles, respectively. |
From equations (14))-(16)). we obtain the intersection conditions in terms of these new vartables. or. where we have assumed that the observer is far enough that all points of the polar caps are located at the same distance from the observer (D>> ry]. | From equations \ref{tracooreng}) \ref{c2ceng}) ), we obtain the intersection conditions in terms of these new variables, or, where we have assumed that the observer is far enough that all points of the polar caps are located at the same distance from the observer $D\gg r_{ws[j]}$ ). |
We consider the model described in section 3.2. | We consider the model described in section 3.2. |
First. we add the optical depths of the working surfaces intersected by each line of sight to obtain the total optical depth along this line of sight. | First, we add the optical depths of the working surfaces intersected by each line of sight to obtain the total optical depth along this line of sight. |
Then. we use this optical depth to estimate the intensity emerging from this direction. | Then, we use this optical depth to estimate the intensity emerging from this direction. |
Finally. the total flux emitted by the system can be estimated by integrating this intensity over the solid angle. | Finally, the total flux emitted by the system can be estimated by integrating this intensity over the solid angle. |
Using the numerical models developed by Ghavamian & Hartigan (1998) for the free-free emission for a planar interstellar shocks. Gonzállez & Cantó (2002) estimate the average optical depth of a shock wave. | Using the numerical models developed by Ghavamian $\&$ Hartigan (1998) for the free-free emission for a planar interstellar shocks, Gonzállez $\&$ Cantó (2002) estimate the average optical depth of a shock wave. |
Assuming an average excitation temperature of 104K. these authors found that their results can be represented by tT,=fineviy7. where ne is the preshock density. uv, the shock velocity. and v is the frequency. | Assuming an average excitation temperature of $10^4$ K, these authors found that their results can be represented by $\tau_{\nu}= \beta\,
n_{0}\,\upsilon_{s}^{\gamma}\,\nu^{-2.1}$, where $n_0$ is the preshock density, $\upsilon_s$ the shock velocity, and $\nu$ is the frequency. |
The constants f and y depends on the shock speed. | The constants $\beta$ and $\gamma$ depends on the shock speed. |
We note that the optical depth of each working surface has the the contribution of the internal and external shocks. | We note that the optical depth of each working surface has the the contribution of the internal and external shocks. |
Using this representation. the optical depth of the j-th working surface Is given by. where ne, 1s the preshock density of the external shock. and Πρ 1s the preshock density of the internal shock at its time of dynamical evolution /;=!—(jDP. | Using this representation, the optical depth of the $j$ -th working surface is given by, where $n_{0,1}$ is the preshock density of the external shock, and $n_{0,2}$ is the preshock density of the internal shock at its time of dynamical evolution $t_j= t - (j-1)P$. |
At a given line of sight (specified by the angles @ and Φ). we add the contribution of the i-th intersected working surfaces to obtain the total optical depth 7,(O,Φ) along this line of sight. that is. where jj=cos4;. being 6; the angle between the line of sight and the normal vector to the j-th working surface at the intersection point. | At a given line of sight (specified by the angles $\theta$ and $\Phi$ ), we add the contribution of the $i$ -th intersected working surfaces to obtain the total optical depth $\tau_{\nu}(\Theta, \Phi)$ along this line of sight, that is, where $\mu_j= \mbox{cos}\,\theta_j$, being $\theta_j$ the angle between the line of sight and the normal vector to the $j$ -th working surface at the intersection point. |
It is easy to show that jj can be written as. Finally. the flux density at radio frequencies from. the bipolar outflow can be calculated by. where B, is the Planck function in the Rayleigh-Jeans approximation (B,=2KkT,1?/c? being Kk the Boltzmann constant. 7, the electron temperature and e the speed of light). and O,=atan(r«q1j/D). | It is easy to show that $\mu_j$ can be written as, Finally, the flux density at radio frequencies from the bipolar outflow can be calculated by, where $B_{\nu}$ is the Planck function in the Rayleigh-Jeans approximation $B_{\nu}= 2kT_e\,\nu^2/c^2$ being $k$ the Boltzmann constant, $T_e$ the electron temperature and $c$ the speed of light), and $\Theta_c= \mbox{atan}\, (r_{ws[1]}/D)$. |
In this section. we present a numerical example for the predicted radio-continuum flux at 2= ccm from a bipolar outflow with a smusoidal ejection velocity. | In this section, we present a numerical example for the predicted radio-continuum flux at $\lambda=$ cm from a bipolar outflow with a sinusoidal ejection velocity. |
The opening angle of the cones is 6,=30° and the inclination angle between the outflow axis and the sky plane is 8;= 429. | The opening angle of the cones is $\theta_a= 30^{\sf o}$ and the inclination angle between the outflow axis and the sky plane is $\theta_i= 42^{\sf o}$ . |
The outflow | The outflow |
In the absence of limb darkening, the depth ὃ is approximately the areal ratio between the sky projection of the oblate spheroid and the stellar disk, where k= is the planet-to-star radius ratio, i is the orbital inclination Req/R,with respect to the sky plane and e is theellipticity, A derivation of this expression is given in the Appendix. | In the absence of limb darkening, the depth $\delta$ is approximately the areal ratio between the sky projection of the oblate spheroid and the stellar disk, where $k \equiv R_{\rm eq}/R_\star$ is the planet-to-star radius ratio, $i$ is the orbital inclination with respect to the sky plane and $\epsilon$ is the, A derivation of this expression is given in the Appendix. |
Figure 2 shows the fractional amplitude of the transit depth variations for the case i= 90°, as a function of f and 0. | Figure \ref{fig:amps} shows the fractional amplitude of the transit depth variations for the case $i = 90^\circ$ , as a function of $f$ and $\theta$. |
For Saturn-like values of oblateness and obliquity, the depth variations would be a few percent. | For Saturn-like values of oblateness and obliquity, the depth variations would be a few percent. |
The transit duration will also vary over the precession period, due to the changing dimension of the planet's sky projection in the direction of orbital motion:R where are the quantities describing the oblateness and obliquity of the exoplanet's | The transit duration will also vary over the precession period, due to the changing dimension $R_\parallel$ of the planet's sky projection in the direction of orbital motion: where are the quantities describing the oblateness and obliquity of the projected exoplanet's shape. |
Derivations of these expressions are projectedalso given in the shape.Appendix. | Derivations of these expressions are also given in the Appendix. |
For circular orbit, the ingress/egress duration (first to second contact,a or third to fourth contact) is approximately where Po» is the orbital period, a is the orbital distance, and b=acosi/R, is the normalized impact parameter. | For a circular orbit, the ingress/egress duration (first to second contact, or third to fourth contact) is approximately where $P_{\rm orb}$ is the orbital period, $a$ is the orbital distance, and $b \equiv a \cos i/ R_\star$ is the normalized impact parameter. |
The full transit duration (first to fourth contact) is approximately (e.g., Seager Mallénn-Ornelas 2003) These approximations are valid as long as the transit is not too close to grazing. | The full transit duration (first to fourth contact) is approximately (e.g., Seager Mallénn-Ornelas 2003) These approximations are valid as long as the transit is not too close to grazing. |
The fractional amplitude of the 7 variations (T7 V) is comparable to that of depth variations (TóV). | The fractional amplitude of the $\tau$ variations $\tau$ V) is comparable to that of depth variations $\delta$ V). |
The amplitude of transit full-duration variations (TDV) depends on k and b, in addition to f and 0, and therefore cannot be summarized in a single contour plot such as Figure 2.. | The amplitude of transit full-duration variations (TDV) depends on $k$ and $b$, in addition to $f$ and $\theta$, and therefore cannot be summarized in a single contour plot such as Figure \ref{fig:amps}. |
In this section we suppose that the planet's precession is caused exclusively by the gravitational torque from the star. | In this section we suppose that the planet's precession is caused exclusively by the gravitational torque from the star. |
In order for the detection of TÓVs or TDVs to be feasible, the planet mustprecession-induced be close enough to the star for precession to produce observable effects in a human lifetime. | In order for the detection of precession-induced $\delta$ Vs or TDVs to be feasible, the planet must be close enough to the star for precession to produce observable effects in a human lifetime. |
However, if the planet is too close to the star, then tidal dissipation should slow down the planet's rotation until it is synchronized with the orbital period, and drive the obliquity to zero, which would cause the signal to be undetectable. | However, if the planet is too close to the star, then tidal dissipation should slow down the planet's rotation until it is synchronized with the orbital period, and drive the obliquity to zero, which would cause the signal to be undetectable. |
Hence, we must ask if there is a range of distances from the star that is close enough for rapid precession, and yet far enough to avoid spin-orbit synchronization. | Hence, we must ask if there is a range of distances from the star that is close enough for rapid precession, and yet far enough to avoid spin-orbit synchronization. |
The spin precession period for a planet on a fixed circular orbit is given by (Ward 1975), where P,o; is the planet's rotation period and C is its moment of inertia divided by M, pReq: | The spin precession period for a planet on a fixed circular orbit is given by (Ward 1975), where $P_{\rm rot}$ is the planet's rotation period and $\mathds{C}$ is its moment of inertia divided by $M_p R_{\rm
eq}^2$ . |
The numerical scaling of 13.5 for 6/0. is the estimated value for Saturn (Ward Hamilton 2004). | The numerical scaling of 13.5 for $\mathds{C}/J_2$ is the estimated value for Saturn (Ward Hamilton 2004). |
According to this expression, orbital periods shorter than Ps 30 days will lead to rapid enough precession to be observed over decadal timescales, depending on the planet's obliquity and internal structure. | According to this expression, orbital periods shorter than $P_{\rm orb}\sim$ 30 days will lead to rapid enough precession to be observed over decadal timescales, depending on the planet's obliquity and internal structure. |
In Fig. 3,, | In Fig. \ref{fig:times}, |
the thick solid line shows the precession period as a function of orbital distance, for an spinexoplanet with the same Po, C, ο. and 0 as Saturn. ( | the thick solid line shows the spin precession period as a function of orbital distance, for an exoplanet with the same $P_{\rm rot}$, $\mathds{C}$, $J_2$, and $\theta$ as Saturn. ( |
The thin solidlines show the more rapid precession rates produced by hypothetical planetary satellites, as discussed in 5. | The thin solidlines show the more rapid precession rates produced by hypothetical planetary satellites, as discussed in \ref{sec:disc}. |
.)The approximate timescale for tidal spin-orbit synchronization is | .)The approximate timescale for tidal spin-orbit synchronization is |
llere we will simply state the solution: for a detailed derivation see Sari(2005) or (2005). | Here we will simply state the solution; for a detailed derivation see \cite{sari05} or \cite{nakayama05}. |
. We assume the effect of the stars gravity on the shock propagation is negligible. | We assume the effect of the star's gravity on the shock propagation is negligible. |
Following Sari(2005).. we let R(/) be the solutions characteristic position. which we choose to be the position of the shock front while the shock is within the star. | Following \cite{sari05}, we let $R(t)$ be the solution's characteristic position, which we choose to be the position of the shock front while the shock is within the star. |
We take |—0 al the time the shock reaches the stars surlace (2=0). and we take Ro<0 when |«c0. | We take $t=0$ at the time the shock reaches the star's surface $R=0$ ), and we take $R<0$ when $t<0$. |
We take E(/). PO). and NO‘) to be respectively the characteristic Lorentz Ιασίου. pressure. and number density. aud we define Following Blandford&/MeIxee(1976)... we define the similarity. variable as Note that for Ro<0. c<AR ancl the relevant range in y is —o€«X<I as long as mo>—]. | We take $\Gamma(t)$, $P(t)$, and $N(t)$ to be respectively the characteristic Lorentz factor, pressure, and number density, and we define Following \cite{blandford76}, we define the similarity variable as Note that for $R<0$, $x\leq R$ and the relevant range in $\chi$ is $-\infty<\chi<1$ as long as $m>-1$. |
We define the hydrodvonamie variables(he Lorentz factor 5. the pressure p. and the number density7 as follows: llere g. f. and give the profiles of 5. p. and nm: expressions for the dependence of n on b and for g. f. h as functions of 4 make up the entire sell-similar solution. | We define the hydrodynamic variables—the Lorentz factor $\gamma$ the pressure $p$, and the number density$n$ —as follows: Here $g$, $f$ , and $h$ give the profiles of $\gamma$, $p$, and $n$; expressions for the dependence of $m$ on $k$ and for $g$, $f$, $h$ as functions of $\chi$ make up the entire self-similar solution. |
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