Tidal Disruption Flares from Stars on Marginally Bound and Unbound Orbits
TTidal disruption flares from stars on marginally bound andunbound orbits
Gwanwoo
Park and Kimitake Hayasaki [email protected] Received ; accepted Department of Astronomy and Space Science, Chungbuk National University, Cheongju361-763, Korea a r X i v : . [ a s t r o - ph . H E ] J un ABSTRACT
We study the mass fallback rate of tidally disrupted stars on marginally boundand unbound orbits around a supermassive black hole (SMBH) by performingthree-dimensional smoothed particle hydrodynamic (SPH) simulations with threekey parameters. The star is modeled by a polytrope with two different indexes( n = 1 . e = 0 .
98 to 1 .
02 and five different penetration factorsranging from β = 1 to 3, where β represents the ratio of the tidal disruptionto pericenter distance radii. We derive analytic formulae for the mass fallbackrate as a function of the stellar density profile, orbital eccentricity, and pen-etration factor. Moreover, two critical eccentricities to classify tidal disruptionevents (TDEs) into five different types: eccentric ( e < e crit , ), marginally eccentric( e crit , (cid:46) e < e = 1), marginally hyperbolic (1 < e < e crit , ),and hyperbolic ( e (cid:38) e crit , ) TDEs, are reevaluated as e crit , = 1 − q − / β k − and e crit , = 1 + 2 q − / β k − , where q is the ratio of the SMBH to stellar masses and0 < k (cid:46)
2. We find the asymptotic slope of the mass fallback rate varies withthe TDE type. The asymptotic slope approaches − / Subject headings: accretion, accretion disks – black hole physics – galaxies: nuclei- galaxies: star clusters: general – stars: kinematics and dynamics – methods:numerical
1. Introduction
There is growing evidence that supermassive black holes (SMBHs) ubiquitously resideat the center of galaxies, based on observations of stellar proper motion, stellar velocitydispersion or accretion luminosity (Kormendy & Ho 2013). Tidal disruption events (TDEs)provide a distinct opportunity to probe dormant SMBHs in inactive galaxies. Once a starapproaches a SMBH and enters inside the tidal sphere, the star is tidally disrupted by theSMBH. The stellar debris then falls back to the SMBH at a super-Eddington rate, leadingto a prominent flaring event with a luminosity exceeding the Eddington luminosity forweeks to months (Rees 1988; Phinney 1989; Evans & Kochanek 1989).Tidal disruption flares have been observed over the broad range of waveband fromoptical (van Velzen et al. 2011; Gezari et al. 2012; Holoien et al. 2016) to ultraviolet (Gezariet al. 2006, 2008; Chornock et al. 2014) to soft X-ray (Komossa & Bade 1999; Saxton et al.2012; Maksym et al. 2013; Auchettl et al. 2017) wavelengths. The TDE rates have beenestimated to be 10 − − − per year per galaxy for soft-X-ray selected TDEs (Donley et al.2002; Esquej et al. 2008) and for optical-selected TDEs (van Velzen & Farrar 2014; Holoienet al. 2016; van Velzen 2018; Hung et al. 2018). The observed rates are consistent withthe theoretically expected rates (Wang & Merritt 2004; Stone & Metzger 2016; see alsoKomossa 2015 and Stone et al. 2020 for a recent review). On the other hand, high-energyjetted TDEs have been detected through non-thermal emissions at radio (Zauderer et al.2011; Alexander et al. 2016; van Velzen et al. 2016) and/or hard X-ray (Burrows et al.2011; Brown et al. 2015) wavelengths with much lower event rate than the thermal TDEcase (Farrar & Piran 2014). Spectroscopic studies have confirmed H I and He II (Arcaviet al. 2014) as well as metal lines (Leloudas et al. 2019). Recently discovered blue-shifted(0.05c) broad absorption lines are likely to result from a high velocity outflow produced bythe candidate TDE AT2018zr (Hung et al. 2019). 5 –It still remains under debate how the standard, theoretical mass fallback rate, which isproportional to t − / (Rees 1988; Phinney 1989; Evans & Kochanek 1989), can be translatedinto the observed light curves. Dozens of X-ray TDEs have light curves shallower than t − / (Auchettl et al. 2017), while many optical/UV TDEs are well fit by t − / (e.g. Hunget al. 2017) but some deviate from t − / (Gezari et al. 2012; Chornock et al. 2014; Arcaviet al. 2014; Holoien et al. 2014). Specifically, the slope of the light curve depends on whenmeasurements are taken relative to the peak. PS1-10JH shows a light curve more consistentwith t − / at late times as shown by Gezari et al. (2015). The decay rate becomes flatterat very late times (van Velzen et al. 2019). This flattening likely reflects the evolution ofa viscously spreading disk rather than the continued evolution of the mass fallback rate(Cannizzo et al. 1990; Cannizzo & Gehrels 2009; Shen & Matzner 2014).There have been some arguments that the mass fallback rate itself can deviate fromthe t − / decay rate. Lodato et al. (2009) demonstrated that the stellar internal structuremakes the mass fallback rate deviate from the standard fallback rate in an early time.When the star is simply modeled by a polytrope, the stellar density profile is characterizedby a polytropic index. In this case, the polytrope index is a key parameter to determine themass fallback rate.The penetration factor, which is the ratio of the tidal disruption to pericenter radii, isalso an important parameter for TDEs. Guillochon & Ramirez-Ruiz (2013) showed thatresultant light curves can be steeper because of the centrally condensed core surviving afterthe partial disruption of the star. It would happen if the penetration factor is relativelylow ( β (cid:46) e < e crit , ), marginally eccentric ( e crit , (cid:46) e < e = 1), marginally hyperbolic (1 < e < e crit , ), and hyperbolic ( e (cid:38) e crit , ) TDEs,respectively, where e crit , = 1 − q − / β − and e crit , = 1 + 2 q − / β − , and q is the ratio ofthe SMBH to stellar masses. Based on this classification, they also examined the frequencyof each TDE by N-body experiments. They pointed out that stars on marginally ellipticaland hyperbolic orbits can be a main TDE source in a spherical star cluster. Therefore, it isclear that the orbital eccentricity (and semi-major axis through the penetration factor) isalso a key parameter to make the mass fallback rate deviate from the standard t − / decayrate.However, there is still little known about how the three key parameters: polytropicindex, penetration factor, and orbital eccentricity, and their combinations affect the massfallback rate. In this paper, we therefore revisit the mass fallback rate onto an SMBH or 7 –IMBH by taking account of the three key parameters. In Section 2, we give a new conditionto classify the TDEs by the stellar orbital type based on the assumption that the spreadin debris energy is proportional to the k -th power of the penetration factor, where k ispresumed to range for 0 < k < k on them. In Section 3, we describe our numerical simulation approach and comparethe semi-analytical solutions of the mass fallback rates with the simulation results. Finally,Section 4 is devoted to the conclusion of our scenario.
2. Revisit of mass fallback rates
In this section, we revisit the mass fallback rate by taking account of a stellar densityprofile (Lodato et al. 2009) and the orbital eccentricity (Hayasaki et al. 2018), including thedependence of the penetration factor, β = r t /r p , where r p is the pericenter distance, on aspread in debris specific energy. As a star approaches to a SMBH or an IMBH, it is tornapart by the tidal force of the black hole, which dominates the self-gravity of the star at thetidal disruption radius: r t = (cid:18) M bh m ∗ (cid:19) / r ∗ ≈ (cid:18) M bh M (cid:12) (cid:19) − / (cid:18) m ∗ M (cid:12) (cid:19) − / (cid:18) r ∗ R (cid:12) (cid:19) r S . (1)Here we denote the black hole mass as M bh , stellar mass and radius as m ∗ and r ∗ , and theSchwarzschild radius as r S = 2G M bh / c , where G and c are Newton’s gravitational constantand the speed of light, respectively.Following Stone et al. (2013), the tidal force produces a spread in specific energy of thestellar debris: ∆ E = β k ∆ (cid:15), (2)where k is the power-law index of the penetration factor in the leading order term of the 8 –tidal potential energy (hereafter, tidal spread energy index) and∆ (cid:15) = GM bh r ∗ r (3)is the standard spread energy (Rees 1988; Evans & Kochanek 1989). If β = 1 or k = 0,Equation (2) reduces to the standard equation. The possible range of k has been taken as0 ≤ k ≤ dMdt = dMd(cid:15) d(cid:15)dt , (4)where dM/d(cid:15) is the differential mass distribution of the stellar debris with specific energy (cid:15) . Because the thermal energy of the stellar debris is negligible compared with the debrisbinding energy, (cid:15) ≈ (cid:15) d : (cid:15) d ≡ − GM bh a d , (5)where a d is the semi-major axis of the stellar debris. Applying the Kepler’s third law toequation (5), we obtain that d(cid:15) d dt = 13 (2 πGM bh ) / t − / . (6) Lodato et al. (2009) included the effect of the stellar density profile on the differentialmass distribution of the stellar debris as dMd(cid:15) d = dMd ∆ r d ∆ rd(cid:15) d , (7)where ∆ r is the radial width of the star. In our case, the relation between the radial widthand the debris specific binding energy is given by∆ rr ∗ = | (cid:15) d | ∆ E = A c a d , (8) 9 –where A c is the critical semi-major axis: A c = a c β − k , (9)with a c ≡ ( M bh /m ∗ ) / r t /
2. If β = 1 or k = 0 is adopted here, equation (8) reduces to thatof Lodato et al. (2009). Moreover, the radial width depends on the orbital period of thestellar debris through the binding energy, i.e., ∆ r ∝ a − ∝ t − / .The internal density structure of the star is given by the radial integral of the stellardensity dMd ∆ r = 2 π (cid:90) r ∗ ∆ r ρ ( r ) r d r, (10)where ρ ( r ) is the spherically symmetric mass density of the star and the polytropes with nostellar rotation are considered. We can integrate equation (10) by solving the Lane-Emdenequation: (1 /ξ ) d ( ξ dθ/dξ ) /dξ = − θ n , where θ ( ξ ) = ρ/ρ c , ξ = r/r c , ρ c is the normalizationdensity, r c = (cid:113) ( n + 1) Kρ /n − / πG is the normalization radius, n is a polytropic index, K is a polytropic constant, respectively (Chandrasekhar 1967). Substituting equations (8)and (10) into equation (7), we obtain the differential mass distribution as dMd(cid:15) d = 32 (cid:18) ρ c ¯ ρ (cid:19) (cid:18) r c r ∗ (cid:19) (cid:16) m ∗ ∆ E (cid:17) (cid:90) r ∗ /r c ∆ r/r c θ ( ξ ) ξ d ξ, (11)where ¯ ρ = m ∗ / (4 πr ∗ /
3) is the mean density of the star. Because θ ( ξ ) is obtained by solvingthe Lane-Emden equation numerically, dM/d(cid:15) d is semi-analytically determined (see alsoFigure 2). Following the Kepler’s third law, we can estimate the orbital period of the mosttightly bound debris as t (cid:48) mtb = 2 π (cid:115) A GM bh = t mtb β − k/ , (12)where t mtb = 2 π (cid:112) a c /GM bh corresponds to the β = 1 case. Substituting equations (6) and 10 –(11) into equation (4), we obtain the mass fallback rate: dMdt = (cid:18) ρ c ¯ ρ (cid:19) (cid:18) r c r ∗ (cid:19) (cid:18) m ∗ t (cid:48) mtb (cid:19) (cid:18) tt (cid:48) mtb (cid:19) − / (cid:90) r ∗ /r c ∆ r/r c θ ( ξ ) ξ d ξ = (cid:18) ρ c ¯ ρ (cid:19) (cid:18) r c r ∗ (cid:19) (cid:18) β k (cid:19) (cid:18) m ∗ t mtb (cid:19) (cid:18) tt mtb (cid:19) − / (cid:90) r ∗ /r c ∆ r/r c θ ( ξ ) ξdξ (13)For n = 3 and ξ (cid:28)
1, the normalized density can be expanded to be θ ( ξ ) ≈ − ξ / O ( ξ ). Since we obtain from equation (8) and (12) that ∆ r/r c =( r ∗ /r c )( t/t mtb ) − / β − k , we can approximately estimate the mass fallback rate as dM/dt ≈ (1 / ρ c / ¯ ρ )(1 /β k )( m ∗ /t mtb )( t/t mtb ) − / [1 − ( t/t mtb ) − / β − k ][1 − (1 / r ∗ /r c ) (1 +( t/t mtb ) − / β − k )]. We find that the mass fallback rate depends on not only the stellardensity profile but also the penetration factor and the tidal spread energy index, whichleads to the deviation from t − / . In this section, we investigate stars that approach the SMBH on parabolic, eccentric,and hyperbolic orbits. The specific energy of the stellar debris is in the range of − ∆ E −
GM/ (2 a ) ≤ (cid:15) d ≤ ∆ E + GM/ (2 a ), where a is the orbital semi-major axis of theapproaching star.Following Hayasaki et al. (2018), the TDEs are classified by the critical eccentricitiesin terms of the orbital eccentricity of the star: ≤ e < e crit , eccentric TDEs e crit , ≤ e < e = 1 parabolic TDEs1 < e ≤ e crit , marginally hyperbolic TDEs e crit , < e hyperbolic TDEs , (14) 11 –where e crit , and e crit , are modified as e crit , = 1 − q − / β k − ,e crit , = 1 + 2 q − / β k − (15)with q ≡ M bh /m ∗ , respectively. If β = 1 or k = 0, these two terms reduce to the previouslydefined critical eccentricities (see equations (5) and (6) of Hayasaki et al. 2018). Themodified specific binding energy of the most tightly bound stellar debris for eccentric orhyperbolic stellar orbits is given by (cid:15) mtb = − ∆ E ± GM bh a = − (cid:18) ∓ A c a (cid:19) ∆ E , (16)where the negative and positive signs of the specific orbital energy of the star indicate theeccentric and hyperbolic orbit cases, respectively. The orbital period of the most tightlybound debris is also changed as τ mtb = 2 π (cid:115) A GM bh (cid:18) ∓ A c a (cid:19) − / = t mtb (cid:0) β − k/ (cid:1) (cid:18) ∓ A c a (cid:19) − / , (17)where we use equation (12) and A c /a should be smaller than unity for the upper negativesign (hyperbolic TDE) case. The differential mass distribution is thus changed fromequation (11) to dMd(cid:15) d = 32 (cid:18) ρ c ¯ ρ (cid:19) (cid:18) r c r ∗ (cid:19) (cid:16) m ∗ ∆ E (cid:17) (cid:90) r ∗ /r c ∆ r (cid:48) /r c θ ( ξ ) ξ d ξ, (18)where ∆ r (cid:48) is the newly defined radial width of the star and is given by∆ r (cid:48) r ∗ = | (cid:15) (cid:48) d | ∆ E = A c a d (cid:16) ∓ a d a (cid:17) (19)with the modification of the debris binding energy, i.e., (cid:15) (cid:48) d = − GM/ (2 a d ) ± GM / (2 a ). Notethat θ ( ξ ) = 0 if ∆ r (cid:48) /r ∗ is greater than unity because there is no stellar gas there. 12 –Substituting equations (6) and (18) into equation (4) and applying equations (16) and(17), we can obtain the modified mass fallback rate as dMdt = (cid:18) ρ c ¯ ρ (cid:19) (cid:18) r c r ∗ (cid:19) (cid:18) ∓ A c a (cid:19) (cid:18) m ∗ τ mtb (cid:19) (cid:18) tτ mtb (cid:19) − / (cid:90) r ∗ /r c ∆ r (cid:48) /r c θ ( ξ ) ξ d ξ. = (cid:18) ρ c ¯ ρ (cid:19) (cid:18) r c r ∗ (cid:19) (cid:18) β k (cid:19) (cid:18) m ∗ t mtb (cid:19) (cid:18) tt mtb (cid:19) − / (cid:90) r ∗ /r c ∆ r (cid:48) /r c θ ( ξ ) ξ d ξ. (20)Applying θ ( ξ ) (cid:39) − ξ / O ( ξ ) to equation (20) for the n = 3 and ξ (cid:28) dM/dt ≈ ( ρ c / ¯ ρ )(1 /β k )( m ∗ /t mtb )( t/t mtb ) − / [1 − ( t/t mtb ) − / β − k [1 + ( t/t mtb ) / ( a c /a )] ][1 − (1 / r ∗ /r c ) (1 + ( t/t mtb ) − / β − k [1 +( t/t mtb ) / ( a c /a )] )]. This is applied only for eccentric to parabolic orbit cases and is notvalid for the hyperbolic orbit case, because the expansion formula of θ ( ξ ) corresponds tothe density profile of the central part of the approaching star, which is unbound to theblack hole if the star is on a hyperbolic orbit. We find the mass fallback rate depends onthe stellar density profile with the orbital eccentricity (or semi-major axis), the penetrationfactor, and the tidal spread energy index, leading to the deviation from t − / . We test thishypothesis by numerical simulations and describe our results in Section 3. We evaluate how well the analytical solution matches the numerical simulations byusing a three-dimensional (3D) Smoothed Particle Hydrodynamics (SPH) code. TheSPH code is developed based on the original version of Benz (1990); Benz et al. (1990)and substantially modified as described in Bate et al. (1995) and parallelized using bothOpenMP and MPI.Two-stage simulations are performed to model a tidal interaction between a star anda black hole. In the first-stage simulation, we model the star by a polytrope with n = 1 . n = 3 for a solar-type star. We run the simulations until the polytrope is virialized. 13 –In the second-stage simulation, the star is initially located at a distance of three timesthe tidal disruption radii and approaches the SMBH following Kepler’s third law for fiveorbital eccentricities and five different penetration factors per each orbital eccentricity.In summary, we run a total 50 simulations in the second-stage. The stellar mass, stellarradius, and black hole mass are held constant throughout the simulations at m ∗ = 1 M (cid:12) , r ∗ = 1 R (cid:12) , and M bh = 10 M (cid:12) , respectively. The total number of SPH particles usedin each simulation is slightly more than 10 and the run time is measured in units ofΩ − ∗ = (cid:112) r ∗ / ( Gm ∗ ) (cid:39) . × s.Tables 1 and 2 present a summary of the SPH simulation models and the correspondingspread energy indices obtained from the simulations. For both tables, the first to thirdcolumns show the polytropic index ( n ), the penetration factor ( β ), and the orbitaleccentricity ( e ), respectively. The fourth and fifth columns show two critical eccentricities e crit , and e crit , , respectively (see equation 15). The final column presents the tidal spreadenergy index ( k ), which is estimated by fitting the simulation data (see the detail inSection 3.2). We find from Tables 1 and 2 that the e = 0 .
98 and e = 0 . e = 1 . e = 1 . e = 1 .
02 cases correspond to marginally eccentric, parabolic, marginally hyperbolic,hyperbolic TDEs, respectively. 14 –Table 1: Summary for parameters of our simulations. The first column shows the polytropeindex ( n ). The second and third columns present the penetration factor ( β ) and the orbitaleccentricity ( e ), respectively. The fourth and fifth columns show the two critical eccentricities e crit , and e crit , , respectively (see equation 15). The final column presents the specific energyindex ( k ), which is obtained by fitting the simulation data (see equation 21. n β e e crit , e crit , k − − − n β e e crit , e crit , k − − n = 3 case. n β e e crit , e crit , k1 0.980 1.02 − − − n β e e crit , e crit , k − −
3. Results
In this section, we describe the simulation results in order to compare our semi-analyticalprediction with that of the SPH simulations.
We first compare the simulated differential mass distribution of the stellar debris overthe specific energy measured at a run time of t = 4 with the Gaussian-fitted distributions;we then compare our findings with the semi-analytical solution given by equation (20). Therelevance of the fitting is also discussed.Figure 1 shows the energy distribution of the debris for n = 1 . e = 1 . e = 0 . .
99) and marginally hyperbolic ( e = 1 .
01 and e = 1 .
02) TDE cases, respectively. Inall the panels, the red, green, blue, magenta, and brown color points are the differentialmass distributions for β = 1, β = 1 . β = 2, β = 2 .
5, and β = 3, respectively. Thecorresponding fitted curves are obtained using the FindFit model provided by
Mathematica and are represented in the same color format. It can find a non-linear fitting with thegaussian function, f ( x ) = (1 / √ πσ ) exp[ − (1 / x − µ ] /σ ) ], where µ is the position of thecenter of the peak and σ is the standard deviation which is proportional to a half width athalf maximum (HWHM). We also simply evaluate the accuracy of the fitting by using theroot mean square (RMS) and its square is given by RMS = (cid:80) N d i =1 ([ y i − f ( x i )] /f ( x i )) /N d ,where N d is the number of data points, and x i and y i are the normalized specific energy atthe i -th data point and the corresponding differential mass distribution, respectively. Weevaluate the RMS using only points within 1 σ of the Gaussian fitted curve, ∆ E sim . We 17 –confirm that for parabolic TDEs, the differential mass is distributed around a specific energyof zero and also a half of the debris mass, independently of β , is bound by the SMBH. Aswe predicted in Section 2.2, the position of the peak of the differential mass is shifted inthe negative direction for eccentric TDEs and the resultant differential mass is distributedover there, while the position of the peak is positively shifted for the hyperbolic TDEcases. The deviation of the peak position corresponds to a c /a to an accuracy of less than2%. Most of the debris mass can fallback to the SMBH because of their negative bindingenergy in marginally eccentric TDEs, whereas most of the debris mass moves far awayfrom the SMBH because of their positive binding energy in marginally hyperbolic TDEcase. The debris mass becomes more widely distributed for the debris specific energy as thepenetration factor increases, particularly for the marginally eccentric and hyperbolic TDEs.In all cases, the peak of the differential mass distribution is smaller as the penetrationfactor is larger.Figure 2 is the same format as Figure 1 but for the n = 3 case. For the parabolicTDEs, a half of stellar debris is bound, whereas other half is unbound to the SMBH. Thedifferential mass distributions for β = 2, β = 2 .
5, and β = 3 are similar as those of n = 1 . β = 1 is steeper than that of n = 1 . n = 3 polytrope is one order magnitude higher than n = 1 . ∼ β = 1 .
5, whereas the maximum RMS value is less than 5% at β = 2 .
5. As shown inpanel (b), the maximum RMS value is ∼
24% at β = 2 .
0, whereas the maximum RMSvalue is less than 7% at β = 1 .
0. We also find that the RMS does not depend on the orbital 18 –eccentricity for the n = 1 . n = 3 cases. Figures 4 and 5 show the comparison betweenthe Gaussian-fitted curves with the semi-analytic solutions with n = 1 . n = 3, whichis given by equation (20), respectively. We find from the figures that the simulated curvesare good agreement with the semi-analytical solutions.As another evaluation of the Gaussian fitting model, we compare the mass of the boundpart of the stellar debris for both the simulated data and the corresponding Gaussian fittedmodel. Each bound mass is calculated by m b , Fit = (cid:90) (cid:15) d , b (cid:18) dMd(cid:15) (cid:19) Fit d(cid:15)m b , SPH = (cid:90) (cid:15) d , b (cid:18) dMd(cid:15) (cid:19) SPH d(cid:15), where (cid:15) d , b = − ( GM bh /a d + GM bh /a ) / (cid:15) d , b → − GM bh /a d because of a → ∞ for parabolic TDEs. We can then evaluate the error rate as ∆ m = | m b , SPH − m b , Fit | /m b , SPH .For all the models of the marginally eccentric and parabolic TDEs, the error rate isdistributed over 0 . (cid:46) ∆ m (cid:46) .
04. On the other hand, the mass of the bound part of thestellar debris is so tiny for the marginally hyperbolic TDEs that we instead evaluate theunbound mass of the stellar debris. For the simulated data and the corresponding fitteddata, each unbound mass is calculated by m ub , Fit = (cid:90) (cid:15) d , ub (cid:18) dMd(cid:15) (cid:19) Fit d(cid:15)m ub , SPH = (cid:90) (cid:15) d , ub (cid:18) dMd(cid:15) (cid:19) SPH d(cid:15), where (cid:15) d , ub = ( GM bh /a d + GM bh /a ) /
2. For all the models of the marginally hyperbolic TDEs,the error rate is distributed over 0 . (cid:46) ∆ m (cid:46) .
09, where ∆ m = | m ub , SPH − m ub , Fit | /m ub , SPH .We note that the largest error rate (∆ m ∼ .
09) is seen in the case of n = 1 . β = 2 . e = 1 .
02, and the error rate ranges 0 . (cid:46) ∆ m (cid:46) .
04 in the other cases. While thedeviation between the simulated data and the corresponding fitted data is less than 5% for 19 –all the models of marginally eccentric and parabolic TDEs, it is less than 10% for the casesof marginally hyperbolic TDEs. 20 – - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) ● β = ● β = ● β = ● β = ● β = - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) Fig. 1.— Simulated energy distribution of stellar debris for a n = 1 . m ∗ / ∆ (cid:15) ). Each panel shows a different orbitaleccentricity. The different colors correspond to different penetration factors ( β ). The solidlines are Gaussian fits to the simulation data. 21 – ● β = ● β = ● β = ● β = ● β = - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14) (cid:15) = (cid:16)(cid:17)(cid:18) ) - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17) (cid:12) = (cid:18)(cid:19)(cid:20)(cid:20) ) - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16) (cid:11) = (cid:17)(cid:18)(cid:19)(cid:20) ) - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20) (cid:12) = (cid:21)(cid:22)(cid:23)(cid:21) ) - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:1)(cid:4)(cid:12)(cid:13)(cid:8)(cid:6)(cid:14) (cid:15)(cid:16)(cid:17) ( (cid:7) = (cid:18)(cid:19) (cid:1) = (cid:20)(cid:21)(cid:22)(cid:23) ) Fig. 2.— The same format as Figure 1, but for the case of n = 3 22 – ● ● ● ● ●▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★◆ ◆ ◆ ◆ ◆ β R M S ( (cid:1) ) (cid:2)(cid:3)(cid:4) (cid:5)(cid:6)(cid:7)(cid:8)(cid:6)(cid:9) (cid:10) (cid:11)(cid:6)(cid:12)(cid:13)(cid:1) ( (cid:9) = (cid:10)(cid:14)(cid:15) ) ◆ e = ★ e = ■ e = ▲ e = ● e = ● ● ● ● ●▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★◆ ◆ ◆ ◆ ◆ β R M S ( (cid:1) ) (cid:2)(cid:3)(cid:4) (cid:5)(cid:6)(cid:7)(cid:8)(cid:6)(cid:9) (cid:10) (cid:11)(cid:6)(cid:12)(cid:13)(cid:14) ( (cid:9) = (cid:15) ) Fig. 3.— The root-mean square (RMS) between simulated data points and the Gaussian-fitted curves. Panels (a) and (b) panel show a n = 1 . n = 3 polytrope cases, respectively.The definition of RMS is shown in the second paragraph of Section 3.1. 23 – - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) β = β = β = β = β = - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) - ϵ / Δϵ d M d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13) ( (cid:14) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:1) = (cid:15)(cid:16)(cid:19)(cid:20) ) Fig. 4.— Comparison between the Gaussian fits to the simulation data and the correspond-ing semi-analytical distributions for an n = 1 . β = β = β = β = β = - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14) (cid:15) = (cid:16)(cid:17)(cid:18) ) - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17) (cid:12) = (cid:18)(cid:19)(cid:20)(cid:20) ) - - - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16) (cid:11) = (cid:17)(cid:18)(cid:19)(cid:20) ) - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20) (cid:12) = (cid:21)(cid:22)(cid:23)(cid:21) ) - / Δϵ d M / d ϵ [ m * Δ ϵ ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:1)(cid:4)(cid:12)(cid:13)(cid:8)(cid:6)(cid:14) (cid:15)(cid:16)(cid:17) ( (cid:7) = (cid:18)(cid:19) (cid:1) = (cid:20)(cid:21)(cid:22)(cid:23) ) Fig. 5.— The same format as Figure 2, but for the case of n = 3 25 – Assuming that the standard deviation of the Gaussian-fitted curve, ∆ E sim , correspondsto the analytical spread energy ∆ E , we can evaluate the power-law index of spread in tidalenergy from equation (2) by k = log(∆ E sim / ∆ (cid:15) )log β . (21)Figure 6 shows the dependence of the simulated spread energy on the penetrationfactor. Panels (a) and (b) show β -dependence of ∆ E sim / ∆ E for the n = 1 . n = 3cases, respectively. Panels (c) and (d) depict β -dependence of ∆ E sim / ∆ (cid:15) for the n = 1 . n = 3 cases, respectively. In each panel, the blue circles, magenta triangles, red squares,black stars and green rhombuses represent results of e = 0 .
98, 0 .
99, 1 .
0, 1 .
01 and 1 . n = 1 . n = 3 cases, although theerror of 2% is obtained for β = 1. In addition, ∆ E sim / ∆ (cid:15) slightly increases for β = 1 asthe orbital eccentricity decreases. We also find from panels (c) and (d) that the simulatedspread energy, overall, increases beyond ∆ (cid:15) as the penetration factor increases. k with different penetration factors for the n = 1 . n = 3 cases, respectively.Panels (a) to (d) show results for β = 1 . , β = 2 , β = 2 .
5, and β = 3, respectively. Thesetwo figures show that the value of k is distributed between 0 . . n = 1 .
5, whilethe value of k takes between 0 .
95 and 2 . n = 3 case. The detailed values of k canbe seen at the sixth column of Tables 1 and 2. We also find that the value of k is largerthan 2 only for the β = 1 . n = 3 case.Figure 7 depicts the dependence of the tidal spread energy index on the orbitaleccentricity with different penetration factors: β = 1 . , β = 2 , β = 2 .
5, and β = 3. Eachpanel shows a different polytrope. These two figures show that the value of k is distributed 26 –between 0 . . n = 1 .
5, while the value of k takes between 0 .
95 and 2 . n = 3 case. The detailed values of k can be seen at the sixth column of Tables 1 and 2. Itis found from panel (b) that the value of k is higher as β is smaller for a n = 3 polytrope,and this tendency is independent of the orbital eccentricity. We also find that the value of k can be larger than 2 only for the β = 1 . n = 3 case. ● ● ● ● ●▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★◆ ◆ ◆ ◆ ◆ β Δ ℰ s i m / Δ ℰ ( (cid:1) ) (cid:1) = (cid:2)(cid:3)(cid:4) ◆ e = ★ e = ■ e = ▲ e = ● e = ● ● ● ● ●▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★◆ ◆ ◆ ◆ ◆ β Δ ℰ s i m / Δ ℰ ( (cid:1) ) (cid:1) = (cid:2) ● ● ● ●▲ ▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★◆ ◆ ◆ ◆ ◆ β Δ ℰ s i m Δ ϵ ( (cid:1) ) (cid:1) = (cid:2)(cid:3)(cid:4) ● ● ● ● ●▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★◆ ◆ ◆ ◆ ◆ β Δ ℰ s i m Δ ϵ ( (cid:1) ) (cid:1) = (cid:2) Fig. 6.— Dependence of the simulated spread energy on the penetration factor. Panels (a)and (b) show β -dependence of ∆ E sim / ∆ E for the n = 1 . n = 3 cases, respectively. Panels(c) and (d) depict β -dependence of ∆ E sim / ∆ (cid:15) for the n = 1 . n = 3 cases, respectively.Note that the relation between respective normalization is given by ∆ E = ∆ (cid:15) β k (see alsoequation 2). In each panel, the blue circles, magenta triangles, red squares, black stars, andgreen rhombuses represent results for e = 0 .
98, 0 .
99, 1 .
0, 1 .
01 and 1 .
02, respectively. 27 – ● ● ● ● ●▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★ e k ( (cid:1) ) (cid:1) = (cid:2)(cid:3)(cid:4) ★ β = ■ β = ▲ β = ● β = ● ● ● ● ●▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■★ ★ ★ ★ ★ e k ( (cid:1) ) (cid:1) = (cid:2) Fig. 7.— Orbital eccentricity dependence of the tidal spread energy index. Panels (a)and (b) represent it for the n = 1 . n = 3 cases, respectively. The blue circles, greentriangles, red squares, and black stars denote the value of k for β = 1 . β = 2, β = 2 .
5, and β = 3, respectively. 28 – In this section, we evaluate the mass fallback rate of each model by using equation (4),where dM/d(cid:15) is given by the Gaussian fitted curves.Figures 8 and 9 depict the Gaussian-fitted mass fallback rates for n = 1 . n = 3polytropes, respectively. The red and blue curves represent the mass fallback rates of β = 1and β = 2 . β = 1 and β = 3 cases in Figure 9.Panels (a)-(d) depict the mass fallback rate of the parabolic TDE ( e = 1 . e = 0 .
99 and e = 0 . e = 1 . β ishigher, except for the marginally hyperbolic TDE of the n = 1 . /β k term. The reason why the β = 1 rate is overallhigher than the β = 2 . β = 1 is larger than that of β = 2 . t mtb for the case of a n = 1 . dM/dt = At s ,the slope of the mass fallback rate is given by s = t (cid:18) d M/dt dM/dt (cid:19) , (22)where the proportionality coefficient, A , is a constant value. The solid blue, magenta, red,and black lines represents the slope of e = 0 . e = 0 . e = 1 .
0, and e = 1 .
01 cases, 29 –respectively, whereas the dashed line denotes the asymptotic slope of the mass fallback ratefor the standard case, − /
3. Each panel depicts a different penetration factor. We find thatthe mass fallback rates of all types of TDEs are flatter than t − / at early times, while theyare different at very late times for respective TDEs. The mass fallback rate asymptoticallyapproaches to t − / for parabolic TDEs, is steeper than t − / for marginally eccentric TDEs,and is flatter for marginally hyperbolic TDEs. The time evolution of the slope in the n = 3case is qualitatively same as the evolution of the slope in the n = 1 . dM/d(cid:15) = Bt α ( B > s = α − , (23)where we adopt the Keplerian third law to equation (4). Equation (23) indicates thatthe mass fallback rate is flatter (steeper) than t − / if α is positive (negative). Because d M/d(cid:15) = 3 (2 πGM bh ) − / Bαt α +2 / where we used equation (6) for the derivation, α should be positive (negative) if the slope of dM/d(cid:15) about (cid:15) is positive (negative). It isclearly seen from Figure 1 that since the inclination of dM/d(cid:15) at the far negative side of (cid:15) is positive for all types of TDEs and all the given range of the penetration factor, α shouldbe positive at early times. That is why the mass fallback rate is, independently of β , flatterthan t − / at early times for all types of TDEs. On the other hand, it is clear from Figure 1that the inclination of dM/d(cid:15) of parabolic TDEs is nearly flat at zero energy so that α ≈ β , − / dM/d(cid:15) is negative at zero energy (i.e., α <
0) for all the givenrange of the penetration factor so that the asymptotic slope is steeper than − / dM/d(cid:15) is positive at zero energy 30 –(i.e., α >
0) for all the given range of the penetration factor so that the asymptotic slope isflatter than − / t / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14)(cid:15)(cid:16) (cid:17) = (cid:13)(cid:14)(cid:18) ) β = β = / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17)(cid:18)(cid:19) (cid:12) = (cid:20)(cid:17)(cid:21)(cid:21) ) - / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16)(cid:17)(cid:18) (cid:11) = (cid:19)(cid:16)(cid:20)(cid:21) ) - - / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20)(cid:21)(cid:22) (cid:12) = (cid:19)(cid:20)(cid:23)(cid:19) ) Fig. 8.— Gaussian-fitted mass fallback rates for a n=1.5 polytrope. They are normalizedby m ∗ /t tmb = 0 . M (cid:12) / yr ( M bh / M (cid:12) ) − / ( M ∗ /M (cid:12) ) ( r ∗ /R (cid:12) ) − / ( β/ k/ , where k is ob-tained from Table 1. The red and blue solid lines show the normalized mass fallback rates of β = 1 and β = 2 .
5, respectively. The dashed line shows the normalized Eddington accretionrate. Each panel shows a different orbital eccentricity. 31 – / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11) ( (cid:12) = (cid:13)(cid:14) (cid:15) = (cid:16)(cid:17)(cid:18) ) β = β = / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:11)(cid:11)(cid:12)(cid:7)(cid:13)(cid:4)(cid:6)(cid:11) (cid:14)(cid:15)(cid:10) ( (cid:7) = (cid:16)(cid:17) (cid:12) = (cid:18)(cid:19)(cid:20)(cid:20) ) / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:1)(cid:1)(cid:11)(cid:7)(cid:12)(cid:4)(cid:6)(cid:1) (cid:13)(cid:14)(cid:10) ( (cid:7) = (cid:15)(cid:16) (cid:11) = (cid:17)(cid:18)(cid:19)(cid:20) ) × / t mtb d M / d t [ m * t m t b ] ( (cid:1) ) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:8)(cid:9) (cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:8)(cid:6)(cid:15) (cid:16)(cid:17)(cid:18) ( (cid:7) = (cid:19)(cid:20) (cid:12) = (cid:21)(cid:22)(cid:23)(cid:21) ) Fig. 9.— The same format as Figure 8, but for the n = 3 case. The red and blue solid linesshow the normalized mass fallback rates of β = 1 and β = 3, respectively. 32 – - - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:2)(cid:3)(cid:6) e = = = = - - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:2)(cid:3)(cid:4) - - - - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:6)(cid:3)(cid:7) - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:6)(cid:3)(cid:4) - - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:6)(cid:3)(cid:7) Fig. 10.— The slope of the Gaussian-fitted mass fallback rate as a function of time normal-ized by t mtb for a n = 1 . e = 0 . e = 0 . e = 1 .
0, and e = 1 .
01, respectively. The dashed line represents − /
3, which is the slope of a standardTDE. 33 – - - - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:6)(cid:3)(cid:4) e = = = = - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:6)(cid:3)(cid:7) - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:6)(cid:3)(cid:4) - - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:6)(cid:3)(cid:7) - - - - - - t / t mtb s ( (cid:1) ) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) = (cid:2)(cid:3)(cid:4) Fig. 11.— The same format as Figure 10, but for the n = 3 case. 34 –
4. Discussion and Conclusions
We have revisited the mass fallback rates of marginally bound to unbound TDEsby taking account of the penetration factor ( β ), tidal spread energy index ( k ), orbitaleccentricity ( e ), and stellar density profile with a polytropic index ( n ). We have comparedthe semi-analytical solutions with 3D SPH simulation results. Our primary conclusions aresummarized as follows:1. We have analytically derived the formulations of both the differential mass distributionand corresponding mass fallback rate and obtained the semi-analytical solutions forthem. Both the differential mass distribution and corresponding mass fallback ratedepend on the penetration factor, tidal spread energy index, orbital eccentricity (orsemi-major axis), and stellar density profile (see equations 11 and 18 for differentialmass distributions and equations 13 and 20 for mass fallback rates).2. The differential mass distributions obtained by the SPH simulations show goodagreement with the Gaussian-fitted curves, with errors of ∼
5% to 18% for n = 1 . ∼
7% to 24% for n = 3. We find that the Gaussian-fitted curves are in goodagreement with the semi-analytical solutions, indicating that the analytically derivedmass fallback rates can match the simulated rates within the range of the fittingaccuracy.3. The simulated spread in debris energy is larger than ∆ (cid:15) = GM bh /r t ( r ∗ /r t ) as thepenetration factor increases for all the cases, which is consistent with our assumptionthat the spread in debris energy is proportional to β k (see also equation (4) of Stoneet al. 2013). While the tidal spread energy index is distributed over 0 . (cid:46) k (cid:46) . n = 1 . β and e , it is distributed over 0 . (cid:46) k (cid:46) .
58 for n = 3 and any value of β and e except for the β = 1 . k ranges 35 –from 2 .
08 to 2 .
17 depending on the orbital eccentricity.4. We have updated the two critical eccentricities to classify five types of TDEs (seealso Hayasaki et al. 2018), based on the spread in energy being proportional to β k ,as follows: e crit , = 1 − M bn /m ∗ ) − / β k − and e crit , = 1 + 2( M bh /m ∗ ) − / β k − (see equation 15). Again, TDEs can be classified by five different types: eccentric( e < e crit , ), marginally eccentric ( e crit , (cid:46) e < e = 1), marginallyhyperbolic (1 < e < e crit , ), and hyperbolic ( e (cid:38) e crit , ) TDEs, respectively.5. The mass fallback rates of the marginally eccentric TDEs are an order of magnitudeor more larger than those of the parabolic TDEs, while the mass fallback rates of themarginally hyperbolic TDEs are less than or comparable to the Eddington rate andabout one or a few orders of magnitude smaller than those of the parabolic TDEs.6. We find that the mass fallback rates of all the types of TDEs are flatter than t − / at early times, while they are different at late times for the respective TDEs. Themass fallback rate asymptotically approaches to t − / for the parabolic TDEs, issteeper than t − / for the marginally eccentric TDEs, and is flatter for the marginallyhyperbolic TDEs. The flatter nature at early times is because the inclination ofthe differential mass-energy distribution at the far side of negative debris energyis positive, while the steeper (flatter) nature at late times is because it is negative(positive) at zero energy. The reason why the asymptotic slope is − / . < e < e crit , (i.e., marginally hyperbolicTDEs), only a little fraction of stellar mass can fall back to the black hole, whichleads to the formation of advection dominated accretion flow (ADAF) or radiativelyinefficient accretion flows (RIAF). The marginally hyperbolic TDEs can be an origin 36 –of ADAFs (or RIAFs) around dormant SMBHs. If the orbital eccentricity is morethan e crit , (i.e., hyperbolic TDEs), no stellar debris can fall back to the black hole,which leads to a failed TDE.Given the black hole mass, stellar mass and mass, and stellar density profile with apolytrope index, we see from Figures 8 and 9 that the mass fallback rate depends on theorbital eccentricity and penetration factor, and the orbital eccentricity enhances the peakof mass fallback rate than the penetration factor. The peak of the mass fallback rate canchange by orders of magnitude over the range of 0 . ≤ e ≤ .
01 for both n = 1 . n = 3 cases. On the other hand, the mass fallback rate also strongly depends on the stellartypes (i.e., stellar mass and radius): a white dwarf disruption shows a much larger fallbackrate than the main sequence cases, whereas a red giant disruption represents a significantlylower rate (Law-Smith et al. 2017). TDEs of different stellar types are distinguishable, e.g.,through the spectral line diagnosis because these stars have different compositions. Thedegeneracy between the stellar type and the orbital eccentricity should, then, be solvableby comparison with the TDEs of the same stellar types.Partial tidal disruptions are the events where the outer layers of a star are tidallystripped by the black hole tidal forces. According to Guillochon & Ramirez-Ruiz (2013);Mainetti et al. (2017), a star can be partially disrupted if β (cid:46) . n = 1 . β (cid:46) . n = 3 polytrope. No partial tidal disruption is seen for n=1.5 because of β ≥ . n = 3 case with β = 1 . β = 1 .
5. In these cases, the mass fallback rate is thought to be asymptoticallyproportional to t − / (Guillochon & Ramirez-Ruiz 2013; Coughlin & Nixon 2019) becauseof the influence of the survived core on the stellar debris. However, the asymptotic slopeof the mass fallback rate is − / (cid:46) β (cid:46) β max , where β max = r t /r S ≈
24 ( M bh / M (cid:12) ) − / ( m ∗ /M (cid:12) ) − / ( r ∗ /R (cid:12) ) isthe maximum possible value of the penetration factor, the general relativistic (GR) effectsget significantly important. In this case, the spread in tidal energy would not follow thesimple power law of the penetration factor anymore. For example, if β = 10 and k ∼ β -dependence of spread in debris energy in tidal disruption of such a deep-plungingstar, although some existing studies show deviation from the scaling law (Evans et al. 2015;Darbha et al. 2019; Steinberg et al. 2019).Hayasaki & Yamazaki (2019) proposed that high-energy neutrinos with ∼
38 –7 . M bh / M (cid:12) ) / can be emitted from an ADAF and/or RIAF formed aftertidal disruption of a star by the decay of charged pions originated in ultra-relativisticprotons. In the standard TDE theory, the RIAF phase would start at t RIAF ∼ s for10 M (cid:12) black hole after a solar-type star disruption. For marginally unbound TDEs, theRIAF phase would start at about four orders of magnitude earlier than the standard case,i.e., t RIAF ∼ s for 10 M (cid:12) black hole and a solar-type star. Because the neutrino energygeneration rate is estimated to be L ν t RIAF R , where L ν and R are the neutrino luminosityand the TDE rate respectively, such a short timescale can significantly enhance the energygeneration rate even if the event rate of marginally hyperbolic TDEs would be subdominant. Acknowledgments
The authors thank Matthew Bate, Atsuo Okazaki, Takahiro Tanaka, and NicholasC. Stone for their helpful comments and suggestions. The authors thank the anonymousreferee for constructive comments and suggestions that helped to improve the manuscript.The authors also acknowledge the Yukawa Institute for Theoretical Physics (YITP) atKyoto University. Discussions during the YITP workshop YITP-T-19-07 on InternationalMolecule-type Workshop ”Tidal Disruption Events: General Relativistic Transients wereuseful to complete this work. This research has been supported by Basic Science ResearchProgram through the National Research Foundation of Korea (NRF) funded by theMinistry of Education (2016R1A5A1013277 and 2017R1D1A1B03028580 to K.H.), andalso supported by the National Supercomputing Center with supercomputing resourcesincluding technical support (KSC-2019-CRE-0082 to G.P. and K.H.). 39 –
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