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We must therefore inchide in the potential cucrey some estimate of the electrical poteutial enerey. | We must therefore include in the potential energy some estimate of the electrical potential energy. |
The mean uuuber density of clectrous is so woe estimate the electrical poteutial energy as where the first bracket gives the umuber of electrous iu the whole body and. is a dimensionless coustaut to be determined. | The mean number density of electrons is so we estimate the electrical potential energy as where the first bracket gives the number of electrons in the whole body and $\beta$ is a dimensionless constant to be determined. |
The gravitational potential enerev is proportional to AL? while the electrical is proportional to ALM? at fixed R. | The gravitational potential energy is proportional to $M^2$ while the electrical is proportional to $M^{4/3}$ at fixed $R$. |
Particular interest ceutres ou the mass at which these two contributions are equal. | Particular interest centres on the mass at which these two contributions are equal. |
Calling it AZ, we find with a=2 Except for a structure factor of order παν Mj is given by (ENCin?]). the ratio of the electrical to gravitational forces between two protons. raised to the 3/2 power. times ir. | Calling it $M_1$ we find with $\alpha = {\scriptstyle{3\over 5}}$ Except for a structure factor of order unity $M_1$ is given by $\left ({e^2 \over
Gm^2_p}\right )^{3/2}$, the ratio of the electrical to gravitational forces between two protons, raised to the $3/2$ power, times $m_H$. |
Notice that this mass does not depeud ou Plauck’s coustaut. | Notice that this mass does not depend on Planck's constant. |
In adapting formula (21) to rocky planets asteroids we have used p,,=3.65 aud ji,=2 | In adapting formula (21) to rocky planets asteroids we have used $\rho_o = 3.65$ and $\mu_e = 2$ . |
The total energy is now elven by The equilibrimn radius has the nininuun £ for αν 2 so equation (25) is readily solved for CAFUPR.+ aud theucefor R where | The total energy is now given by The equilibrium radius has the minimum $E$ for any $R$ so equation (25) is readily solved for $\zeta M^{1/3}R^{-1}$ and thencefor $R$ where |
The region of the 8-5; plane between r z100 and r c107 is mainly dominated by sources identified as (racio) galaxies. and this population constitutes the majority of the id-sample. | The region of the $b_J$ plane between r $\simeq 100$ and r $\simeq
10^4$ is mainly dominated by sources identified as (radio) galaxies, and this population constitutes the majority of the id-sample. |
For rZ107 and 5;<20.5 stellar identifications. most likely to be high-z QSO. dominate the sample. | For r $\simgt 10^4$ and $b_J\le 20.5$ stellar identifications, most likely to be high-z QSO, dominate the sample. |
Fainter than 6;=20.5 this effect is mainly an artefact of the data because the optical images are too faint to resolve galaxies. | Fainter than $b_J=20.5$ this effect is mainly an artefact of the data because the optical images are too faint to resolve galaxies. |
Figure 7 illustrates the [uüx-maegnitude diagram in the I band. | Figure \ref{fig:FR} illustrates the flux-magnitude diagram in the $R$ band. |
Apart from the features already. seen in Figure 6.. we note an interesting shift of the population of radio ealaxies towards brighter red. magnitudes. | Apart from the features already seen in Figure \ref{fig:FB}, we note an interesting shift of the population of radio galaxies towards brighter red magnitudes. |
This shift is especially significant for κ10 mJv and is not observed for the stellar icentifications. hiehliehtine the [act that galaxies are twpically red. earlv-tvpe objects. | This shift is especially significant for S $\simgt 10$ mJy and is not observed for the stellar identifications, highlighting the fact that radio-galaxies are typically red, early-type objects. |
This effect can be seen more clearly in Figure δι, which plots the byRR colour of sources in the ic-samiple for objects with 5; 20.5. | This effect can be seen more clearly in Figure \ref{fig:FRB}, , which plots the $b_J-R$ colour of sources in the id-sample for objects with $b_J$ $\le 20.5$. |
Phe majority of the id-sample consists of galaxies with very red colours. up to 5;I. | The majority of the id-sample consists of galaxies with very red colours, up to $b_J -R \sim 4$. |
This conclusion is actually made even stronger as in our analysis we might have missed very red sources which had 5; magnitudes too faint to be included in the original catalogue. | This conclusion is actually made even stronger as in our analysis we might have missed very red sources which had $b_J$ magnitudes too faint to be included in the original catalogue. |
The ic-sample was drawn from a catalogue selected on b; magnitude: the red. magnitudes. were added: later as explained in Section 2. | The id-sample was drawn from a catalogue selected on $b_J$ magnitude; the red magnitudes were added later as explained in Section 2. |
Note the elear distinction in Figure S between radio galaxies. which appear mostly with b;2z1 and QSO. which dominate the region 0&byRS1. | Note the clear distinction in Figure \ref{fig:FRB} between radio galaxies, which appear mostly with $b_J-R
\simgt 1$ and QSO, which dominate the region $0 \simlt b_J-R
\simlt 1$. |
This division becomes striking for radio Dluxes 8 z 10. where the two distributions take the shape of a tuning fork. | This division becomes striking for radio fluxes S $\simgt$ 10, where the two distributions take the shape of a tuning fork. |
Again we can see that star-forming galaxies showing blue (i.e. for byRS 0) colours and low ( £ 10) radio-Iuxes constitute onlv a few tens of objects in the lower part of the plot. | Again we can see that star-forming galaxies showing blue (i.e. for $b_J-R \simlt 0$ ) colours and low (S $\simlt 10$ ) radio-fluxes constitute only a few tens of objects in the lower part of the plot. |
Also when we consider the number of sources per unit of racio Lux (see Figure 9)) we see that stellar ancl non-stellar ids show cdillerent behaviours. | Also when we consider the number of sources per unit of radio flux (see Figure \ref{fig:hist_flux}) ) we see that stellar and non-stellar ids show different behaviours. |
The number of galaxies (the dashed line). shows a steeper slope than the number of QSO (the solid line). | The number of galaxies (the dashed line), shows a steeper slope than the number of QSO (the solid line). |
This dillerence rellects thefact that galaxies tend to have lower radio Uuxes. while QSO dominate the region Syice,<20 my. | This difference reflects thefact that galaxies tend to have lower radio fluxes, while QSO dominate the region $S_{1.4 {\rm
GHz}}\simgt 20$ mJy. |
catalog possibly because of the faintuess since (heir Fy.«3.0Jy. | catalog possibly because of the faintness since their $_{12}<$ 3.0Jy. |
The co-addecl MSX images neither show clear evidence of detection. | The co-added MSX images neither show clear evidence of detection. |
The photometric results in the MSX band A which is the kev band for this study are available in the MSX PSC1.2 catalog for all the associated objects except IRAS 03469+5833. | The photometric results in the MSX band A which is the key band for this study are available in the MSX PSC1.2 catalog for all the associated objects except IRAS 03469+5833. |
Since the counterpart of IRAS 034694-5333 is bright in the MSX C band with Fe—5.5 Jv close to its IRAS 12jan intensity F4»=7.3Jv which also proves the right association. the detection in band A is expected. | Since the counterpart of IRAS 03469+5833 is bright in the MSX C band with $_{\rm C}=$ 5.5 Jy close to its IRAS $\mu$ m intensity $_{12}=$ 7.3Jy which also proves the right association, the detection in band A is expected. |
Indeed. the object is clearly detected in the AISN image in band. A. The missing of bright sources in the MSX PSC V1.2 may be ascribed to the incompleteness of the catalog. | Indeed, the object is clearly detected in the MSX image in band A. The missing of bright sources in the MSX PSC V1.2 may be ascribed to the incompleteness of the catalog. |
Further discussion is confined to the objects that are actually in the M5X PSCH.2. | Further discussion is confined to the objects that are actually in the MSX PSC1.2. |
The cross-associated sample consists o£ 310 stars. from which 288 are detected with SiO maser emission at both the v—1 and v=2 J=1-0 transitions. 11 detected at only v—1 J-1-0 transition and 11 at only v—2 J=1-0 transition. and from which 309 stars are measured in the MSN A band. 305 measured in the MSX C band. | The cross-associated sample consists of 310 stars, from which 288 are detected with SiO maser emission at both the v=1 and v=2 J=1-0 transitions, 11 detected at only v=1 J=1-0 transition and 11 at only v=2 J=1-0 transition, and from which 309 stars are measured in the MSX A band, 305 measured in the MSX C band. |
There are 298 v—1 J=1-0 SiO maser stars with association in the MSN A band and 298 v—2 J=1-0 SiO maser stars wilh association in the MSN A banc. | There are 298 v=1 J=1-0 SiO maser stars with association in the MSX A band and 298 v=2 J=1-0 SiO maser stars with association in the MSX A band. |
Most of (he sources are variables by comparing their fIux intensity in (he MSX C band Fe with that at the IRAS 12;an Γιο. | Most of the sources are variables by comparing their flux intensity in the MSX C band $_{\rm C}$ with that at the IRAS $\mu$ m $_{12}$. |
Though the MSX C band is not identical to the IRAS 125a band. it is designed to be a narrower analog of the IRAS 12;mu filter and its effective wavelength is just 12.134 (Priceetal.2001). | Though the MSX C band is not identical to the IRAS $\mu$ m band, it is designed to be a narrower analog of the IRAS $\mu$ m filter and its effective wavelength is just $\mu$ m \citep{pri01}. |
. Based on the MSX photometric accuracy. the flux differences greater than is indicative of variability rather than. photometric error (Cohenetal.2000). | Based on the MSX photometric accuracy, the flux differences greater than is indicative of variability rather than photometric error \citep{coh00}. |
. The relative dillerences of Fe to Ε mostly are greater than (Fig. 2)). | The relative differences of $_{\rm C}$ to $_{12}$ mostly are greater than (Fig. \ref{fig2}) ), |
which means the dilferences between the MISN and IRAS observations are intvinsically [rom the objects. | which means the differences between the MSX and IRAS observations are intrinsically from the objects. |
Further supports to the sources being variable come Irom the independent redundant observations by MSX and IRAS to some of the objects. | Further supports to the sources being variable come from the independent redundant observations by MSX and IRAS to some of the objects. |
checked the IRAS variability indexes of the SiO maser stars in the Galactic bulge by then and found about 6vo-thirds with the indexes greater than 90. | \citet{jia95} checked the IRAS variability indexes of the SiO maser stars in the Galactic bulge by then and found about two-thirds with the indexes greater than 90. |
In the MSX PSCI.2 calalog. 167 of the 310 associated stars have (he variability indexes being 1 in at least one band which means the variation over 30 (Eganetal.1999). | In the MSX PSC1.2 catalog, 167 of the 310 associated stars have the variability indexes being 1 in at least one band which means the variation over $\sigma$ \citep{ega99}. |
. Most. if not all. the sources are variables. | Most, if not all, the sources are variables. |
The upper limit of (Fe—Fy)/Fy is about 1.5 and this further implies that the amplitude of variation at (his band is not very large. | The upper limit of ${\rm (F_C-F_{12})/F_{12}}$ is about 1.5 and this further implies that the amplitude of variation at this band is not very large. |
The property of variation is consistent with the SiO maser stus being mostly variable late-tvpe stars (Jiangetal.1995). | The property of variation is consistent with the SiO maser stars being mostly variable late-type stars \citep{jia95}. |
. First of all. the relation of SiO maser peak intensity to the infrared. radiation in the MISX A bond is diseussed. | First of all, the relation of SiO maser peak intensity to the infrared radiation in the MSX A band is discussed. |
Shown in Fig. | Shown in Fig. |
3. are the 298 SiO maser stars that are detected | \ref{fig3} are the 298 SiO maser stars that are detected |
The Edclinegton acretion rate for the embedded black hole is where ce, is the radiative clliciency of an accretion How. | The Eddington acretion rate for the embedded black hole is where $\epsilon_a$ is the radiative efficiency of an accretion flow. |
We analyze the structure of the dise by repeating the steps outlined in the previous section. and by adopting the upper limit (AlMia) for the number of the black holes embectelect in the disc. | We analyze the structure of the disc by repeating the steps outlined in the previous section, and by adopting the upper limit $(M/M_{\rm bh})^{1/3}$ for the number of the black holes embedded in the disc. |
We find and Llowever an embedded black hole will open a gap if its mass is bigger than see Eqs. ( | We find and However an embedded black hole will open a gap if its mass is bigger than see Eqs. ( |
4) of Lin and Papaloizou (1986). | 4) of Lin and Papaloizou (1986). |
Using Ίσα. (42)). | Using Eq. \ref{hreddington}) ), |
we see that The inequality above holds unless we allow the individual masses of the embedded: holes be of order of tens of thousands solar masses. | we see that The inequality above holds unless we allow the individual masses of the embedded holes be of order of tens of thousands solar masses. |
“Pherefore it is impossible to maintain a stable disc by using the feedback from Ecclington-limitec black holes: either. these holes are not massive enough to provide sullicient feedback. or they are so massive that they open gaps in and become decoupled from the disc. | Therefore it is impossible to maintain a stable disc by using the feedback from Eddington-limited black holes: either these holes are not massive enough to provide sufficient feedback, or they are so massive that they open gaps in and become decoupled from the disc. |
An unsupported. dise would. fragment. completely and the fuel supply through the dise to the central engine would be stopped. | An unsupported disc would fragment completely and the fuel supply through the disc to the central engine would be stopped. |
Lt is likely that the newlv-born black hole in the disc inspirals towards the central black hole. | It is likely that the newly-born black hole in the disc inspirals towards the central black hole. |
We imagine that the stellar-mass black hole is embedded into a massive accretion disc which forms due to continuing infall of gas from the galactic bulge. after the black hole is born. | We imagine that the stellar-mass black hole is embedded into a massive accretion disc which forms due to continuing infall of gas from the galactic bulge, after the black hole is born. |
Lo the black hole opens a eap in the disc. it will move towards the central black hole together with the dise (tvpe-HLE migration: Gould and. Rix 2000. and Armitage and Natarajan 2001). | If the black hole opens a gap in the disc, it will move towards the central black hole together with the disc (type-II migration; Gould and Rix 2000, and Armitage and Natarajan 2001). |
The timescale for such inspiral is the accretion timescale. If on the other hand. the black hole is not. massive enough to open the gap. it will migrate inwards by exciting density waves in the disc (tvpe-LE migration). | The timescale for such inspiral is the accretion timescale, If on the other hand, the black hole is not massive enough to open the gap, it will migrate inwards by exciting density waves in the disc (type-I migration). |
The speed. of this inward drift is given by ef | The speed of this inward drift is given by cf. |
Eqs. ( | Eqs. ( |
9). (B4) and (B5) of Ralikov (2002). and Ward (1986). | 9), (B4) and (B5) of Rafikov (2002), and Ward (1986). |
Llere 3 is a numerical factor of order 5 for Qδν1. and can be significantly larger for Q—1. | Here $\beta$ is a numerical factor of order $5$ for $Q\gg 1$, and can be significantly larger for $Q\sim 1$. |
For a sell-gravitating disc with Q~1. we find What determines whether the gap is open or not is whether a stellar-mass black hole has time to accrete a few housand solar masses of gas. which would put it. above he eap-opening threshold. for the tvpical cise parameters. | For a self-gravitating disc with $Q\sim 1$, we find What determines whether the gap is open or not is whether a stellar-mass black hole has time to accrete a few thousand solar masses of gas, which would put it above the gap-opening threshold for the typical disc parameters. |
The Ecldineton-limited accretion occurs on a timescale of 10° vr. much longer than the characteristic tvpe-L inspiral ime. | The Eddington-limited accretion occurs on a timescale of $\sim 10^8$ yr, much longer than the characteristic type-I inspiral time. |
Thus. in this case the black holes don't acerete much on their wav in. | Thus, in this case the black holes don't accrete much on their way in. |
On the other hand. the Boncdi-Lovle formula xediets the mass e-folding timescale ofa few hundred vears. | On the other hand, the Bondi-Hoyle formula predicts the mass e-folding timescale of a few hundred years. |
Thus. if the black hole is allowed to accrete at the Bondi- rate. its mass increases rapidly. until it opens a gap in he disc. | Thus, if the black hole is allowed to accrete at the Bondi-Hoyle rate, its mass increases rapidly until it opens a gap in the disc. |
Then. the inspiral proceeds via type-Ll migration. | Then, the inspiral proceeds via type-II migration. |
From Ies. (45)) | From Eqs. \ref{typeII}) ) |
and (47)) we see that the embedded slack hole experiencing tvpe-L or tvpe-HL migration would merge with the central black hole on the timescale 10? 10° vears. shorter than the tvpical timescale of an AGN activity. | and \ref{typeI}) ) we see that the embedded black hole experiencing type-I or type-II migration would merge with the central black hole on the timescale $10^6$ $10^7$ years, shorter than the typical timescale of an AGN activity. |
Thus it is plausible that the daughter clise-born black hole is brought towards the parent central black hole: the mass of the daughter. black hole. might grow significantly on the way in. | Thus it is plausible that the daughter disc-born black hole is brought towards the parent central black hole; the mass of the daughter black hole might grow significantly on the way in. |
Gravitational radiation will eventually become the dominant mechanism driving the inspiral. ancl the final merger will produce copious amount of gravitational waves. | Gravitational radiation will eventually become the dominant mechanism driving the inspiral, and the final merger will produce copious amount of gravitational waves. |
In the next subsection we show that these waves are detectable by LISA for a broad range of the black hole masses and the disc accretion rate. | In the next subsection we show that these waves are detectable by LISA for a broad range of the black hole masses and the disc accretion rate. |
Lt is realistic to expect that LISA would follow the last vear of the inspiral of the disc-born black hole into the central black hole. | It is realistic to expect that LISA would follow the last year of the inspiral of the disc-born black hole into the central black hole. |
Generally. one must develop a set of templates which densely span the parameter space of possible inspiral signals. | Generally, one must develop a set of templates which densely span the parameter space of possible inspiral signals. |
In order for the final inspiral to he detectable. one of the templates must. follow the signal with the ohaseshift. between the two not exceeding a fraction of a cvcle. | In order for the final inspiral to be detectable, one of the templates must follow the signal with the phaseshift between the two not exceeding a fraction of a cycle. |
Therefore. if the drag from the cise will alter the waveform by a fraction of a evele over the signal integration ime (e.e@.. 1 vear). detection of the signal with high sienal-o-noise ratio will become problematic. | Therefore, if the drag from the disc will alter the waveform by a fraction of a cycle over the signal integration time (e.g., $1$ year), detection of the signal with high signal-to-noise ratio will become problematic. |
Below we address the influence of the accretion disc on the final inpiral waveform. | Below we address the influence of the accretion disc on the final inpiral waveform. |
The issue of gas-drag influence on the LISA signal was irst addressed. by Naravan (2000): see also Chakrabarti (1996). | The issue of gas-drag influence on the LISA signal was first addressed by Narayan (2000); see also Chakrabarti (1996). |
Naravan's analysis is directlv. applicable το low-uminocitv non-racliative quasi-spherical aceretion Lows. which might exist around. supermassive black holes when he accretion rate is κO.O1 of the IExddington limit. | Narayan's analysis is directly applicable to low-luminocity non-radiative quasi-spherical accretion flows, which might exist around supermassive black holes when the accretion rate is $<0.01$ of the Eddington limit. |
aravan concluded that non-racliative Hows will not have any observable inlluence on the gravitational-wave signals seen by LISA. | Narayan concluded that non-radiative flows will not have any observable influence on the gravitational-wave signals seen by LISA. |
Below we extend Naravan's analysis to the | Below we extend Narayan's analysis to the |
results with the observed mean Dhuuinositv of the red eiut clump in the Ilipparcos sample. coucludiug for the possible need of a revision of tlic "nost updated iuput physics used iu recent evolutionarv models | results with the observed mean luminosity of the red giant clump in the Hipparcos sample, concluding for the possible need of a revision of the "most updated" input physics used in recent evolutionary models. |
The origin of differences iu the predicted dDuiumositios are finallv discussed iu section 5. by compari,ο the results of selected evolutionary codes to the lisht of the adopted plivsical ingredients. | The origin of differences in the predicted luminosities are finally discussed in section 5, by comparing the results of selected evolutionary codes to the light of the adopted physical ingredients. |
A short section of «muicluding remarks will close the paper. | A short section of concluding remarks will close the paper. |
According to Cürardi et al. ( | According to Girardi et al. ( |
1998). the Hipparcos sample of ucighboring He 1nunug ooeiauts is larecly populated by stars with masses below or iu the range of the so-called Red Ciaut Brauch phase trausitio1 (RGD-pt). | 1998), the Hipparcos sample of neighboring He burning giants is largely populated by stars with masses below or in the range of the so-called Red Giant Branch phase transition (RGB-pt). |
Since the pionecring paper by Sweigart ct al. ( | Since the pioneering paper by Sweigart et al. ( |
1990) this ranec of stellar masses has been the suject of several careful evolutiowry investigations. | 1990) this range of stellar masses has been the subject of several careful evolutionary investigations. |
As we known. as we go from stars wdh M41 M. to higher i.assos, we progressively fiud II-«ο buiie stars with a kΜΟΥ degree [9] electron degeneracy in their cores. | As well known, as we go from stars with $\sim$ 1 $_{\odot}$ to higher masses, we progressively find H-shell burning stars with a lower degree of electron degeneracy in their cores. |
Eveutuavy. Πο is quietly igied in the ceuter of the structure. | Eventually, He is quietly ignited in the center of the structure. |
As a COnsequeuce. the nass of the IIe core at he Te ignition xogressivelv ¢CCYCHSCR. reaching a iiini at a star mass which depends ou he original chemical composition. | As a consequence, the mass of the He core at the He ignition progressively decreases, reaching a minimum at a star mass which depends on the original chemical composition. |
After this ΙΙ. he Ile core grows again with mass. fowing the iucreasing size of the central convective core in the Main Seqince structures. | After this minimum, the He core grows again with mass, following the increasing size of the central convective core in the Main Sequence structures. |
Fig. | Fig. |
1 gives selected quantities cOLnceriuge the behavior of Te burning models across the RGB-pt as computed for Z=0.02 x the two alternative asstnipions Y=0.27 or 0.23. | 1 gives selected quantities concerning the behavior of He burning models across the RGB-pt as computed for Z=0.02 and the two alternative assumptions Y=0.27 or 0.23. |
All models have been conipued according to the theoretical scenario already preseuted i1 C99. which Incorporates al the most recent evaluations of the iuput plivsies. | All models have been computed according to the theoretical scenario already presented in C99, which incorporates all the most recent evaluations of the input physics. |
As evervwhere iu the folowing. cuantities eiveu lu Fis.l refer to tje first 1iodoel whic1. after ieuitiug central Te. has already reached the IIR. di:wrau location where it will speud the uajor phase of ceural He τιrniug. | As everywhere in the following, quantities given in Fig.1 refer to the first model which, after igniting central He, has already reached the HR diagram location where it will spend the major phase of central He burning. |
Let us here notice that the luminosity oft1ese He burning models is not related oulv to the mass ο| the Πο core. | Let us here notice that the luminosity of these He burning models is not related only to the mass of the He core. |
DIuitiallv this luminosity increases du spite of the decreasing He COKE. suce he increased. efficiency of the II burniie shell overcomes the decreased outpi to: We-burning reactions. | Initially this luminosity increases in spite of the decreasing He core, since the increased efficiency of the H burning shell overcomes the decreased output of He-burning reactions. |
However. eventually the decrease o ‘the He core doniuates alu the luminosity reaches 1s uum. whereas the litenue in the central helium Iwhinge phase imcereases following the decrease of the officiuev of the Te nruuue reactions. | However, eventually the decrease of the He core dominates and the luminosity reaches its minimum, whereas the lifetime in the central helium burning phase increases following the decrease of the efficiency of the He burning reactions. |
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