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The source with the lowest value of absorbing column density is source 38 for which N(H)=0.05£0.06x107? em. consistent with all the YSOs for which spectra and light curves have been studied here being part of the Serpens complex (rather than being foreground objects). | The source with the lowest value of absorbing column density is source 38 for which $N({\rm H}) = 0.05 \pm
0.06 \times 10^{22}$ $^{-2}$, consistent with all the YSOs for which spectra and light curves have been studied here being part of the Serpens complex (rather than being foreground objects). |
t reffie:spe,ableshowsthevaluesoftheabsorbingcolumndensitiesvs.the plas psfit..exceptforthesourcesundergoingastrong flare(G6. 79and − The d | \\ref{fig:spe_table} shows the values of the absorbing column densities vs. the plasma temperatures of the sources in \\ref{tab:psfit}, except for the sources undergoing a strong flare (66, 79 and 44) or unlikely to be stars (6, 13 and 70), which have been omitted. |
iagram in | A trend of decreasing absorbing column densities from Class I to Class III is clearly present. |
reffig:spe,a | There is a lack of low-plasma-temperature high-absorption sources, which is expected, as with low plasma temperature higher luminosities are necessary in order to see a source through higher column densities. |
bleissimilartotheonede | There is also, possibly, a lack of sources in the lower-right region of the diagram (where sources have a high plasma temperature and small absorption, and are thus easy to detect), hinting at an evolutionary effect from the Class I (high X-ray temperature, high extinction) to the Class III stage (lower X-ray temperature and extinction). |
rivedbyOzawaet al | \\ref{tab:medians} lists the median values of $\nh$ and $kT$ for the different classes from which the trend of decreasing absorbing column density from Class I to Class III is clear and a trend of decreasing plasma temperature is also somewhat present (although less clear). |
,(2005) fort | Note also that, since these median values are based on detected sources only (we did not include upper limits, because of the lack of a parent catalogue), the trend in $kT$ could be due to a selection effect (i.e. Class I/II soft sources, being very absorbed, are not detected). |
hep Oph cloud core. | The diagram in \\ref{fig:spe_table} is similar to the one derived by \citet{ogm05} for the $\rho$ Oph cloud core. |
They also note a lack of sources in the high- small-absorption region of the diagram and interpret this as evidence of an evolutionary effect. | They also note a lack of sources in the high-plasma-temperature small-absorption region of the diagram and interpret this as evidence of an evolutionary effect. |
A similar effect is noted by Flaccomioetal.(2006) in their study of the 22264 star-forming complex. | A similar effect is noted by \cite{fms2006} in their study of the 2264 star-forming complex. |
A quantity which. under the assumption of isotropy. fully describes the ECRB is the differential photou intensity Je (photous per unit area-time-cucrey-solid angle). | A quantity which, under the assumption of isotropy, fully describes the EGRB is the differential photon intensity $I_E$ (photons per unit area-time-energy-solid angle). |
In this section. we calculate Zzg for a population of "resolved. sources; | In this section, we calculate $I_E$ for a population of unresolved sources. |
We make the following assunptious for the population of unresolved eanuna-ray sources We consider. | We make the following assumptions for the population of unresolved gamma-ray sources we consider. |
The differential flux of cach object of this population. ΕΤ. isvelated to Fy through provided à>1. | The differential flux of each object of this population, $F_E$, is related to $F_8$ through provided $\alpha>1$. |
Each unresolved source with a flux Fx at energies >Ly. has a contribution fp.(Fy.a) to the diffuse enission which is eiven by where a is the spectral index of the source. and Ly=100MeV in the case of 3EC fluxes. | Each unresolved source with a flux $F_8$ at energies $>E_0$, has a contribution $I_{E,1}(F_8, \alpha)$ to the diffuse emission which is given by where $\alpha$ is the spectral index of the source, and $E_0 = 100
{\rm \, MeV}$ in the case of 3EG fluxes. |
The 1li normalization factor comes from assuming an isotropic distribution of sources. the collective cuuission of which is uniformly distributed over the celestial sphere. | The $1/4\pi$ normalization factor comes from assuming an isotropic distribution of sources, the collective emission of which is uniformly distributed over the celestial sphere. |
The differeutial flux distribution of the population is The collective diffuse euission due to this population of uuresolved sources with fluxes between Εκ uid Fax will be: where fy=LOTon2sjwGeV(pm). | The differential flux distribution of the population is The collective diffuse emission due to this population of unresolved sources with fluxes between $F_{\rm 8,min}$ and $F_{\rm 8,
max}$ will be: where $ I_0 = 10^{-7} \rm cm^{-2} s^{-1} sr^{-1} GeV ^{-1} / (4\pi)$. |
Physically. the wpper-lanit fux Fija represeuts the scusitivity lait of the telescope (objects with tux higher than this are resolved and do not contribute to the background). while the lower-linüt fux Fi, represents the flux below which the approximation of ((2)) for the cumulative flux distribution breaks down. | Physically, the upper-limit flux $F_{\rm 8,max}$ represents the sensitivity limit of the telescope (objects with flux higher than this are resolved and do not contribute to the background), while the lower-limit flux $F_{\rm 8,min}$ represents the flux below which the approximation of \ref{fit1}) ) for the cumulative flux distribution breaks down. |
The features of the spectral shape of this emission iu the simple case when pla) is à Caussian are discussed in Appendix C.. | The features of the spectral shape of this emission in the simple case when $p(\alpha)$ is a Gaussian are discussed in Appendix \ref{shape}. |
Equation (6)) can provide a first simple cstimate for the possible contribution of unresolved sources of the same class as extragalactic unidentified ECRET sources to the isotropic diffuse ewmuna-rav backeround. | Equation \ref{contr}) ) can provide a first simple estimate for the possible contribution of unresolved sources of the same class as extragalactic unidentified EGRET sources to the isotropic diffuse gamma-ray background. |
The required iuputs are the cumulative flux function of ((2)). and the spectral iudex distribution of the source class. pla}. | The required inputs are the cumulative flux function of \ref{fit1}) ), and the spectral index distribution of the source class, $p(\alpha)$. |
Iu this section. we derive these inputs from the three source samples discussed in 8?7.. | In this section, we derive these inputs from the three source samples discussed in \ref{samples}. |
Possible source variability has not been accounted for iu this analvsis. | Possible source variability has not been accounted for in this analysis. |
A discussion of possible effects of source variability. as well as our reasoning for not considering thei for the purposes of our study. are presented iu Appendix D.. | A discussion of possible effects of source variability, as well as our reasoning for not considering them for the purposes of our study, are presented in Appendix \ref{Var}. |
We use data from the resolved objects in our three suuples to coustruct the cumulative flux distribution of ECRET uuideuti&ed sources. | We use data from the resolved objects in our three samples to construct the cumulative flux distribution of EGRET unidentified sources. |
The cumulative fiux distributions of the three samples are shown iu ((2)). | The cumulative flux distributions of the three samples are shown in \ref{lognlogs}) ). |
Due to nou-tniformities iu EGRET exposure. the lowest resolvable flux is not a hard-set number. but rather the cficiency in resolving low-fHux objects | Due to non-uniformities in EGRET exposure, the lowest resolvable flux is not a hard-set number, but rather the efficiency in resolving low-flux objects |
MMR, as shown by Pipino Matteucci (2006) (see, e.g., the models discussed in their Sec. | MMR, as shown by Pipino Matteucci (2006) (see, e.g., the models discussed in their Sec. |
3.3). | 3.3). |
On the other hand, we can have the extreme case in which the galaxy is created by several progenitors of the kindB. | On the other hand, we can have the extreme case in which the galaxy is created by several progenitors of the kind. |
. The results of this analysis show that this model can reproduce the MMAR but it predicts an underabundance of a-elements relative to Fe, at variance with observations. | The results of this analysis show that this model can reproduce the MMR but it predicts an underabundance of $\alpha$ -elements relative to Fe, at variance with observations. |
We conclude that we cannot form massive spheroids from a sequence of several dry-mergers between building blocks of the same kind (similar mass and chemical properties), even if the progenitors are chosen to have the correct a-enhancement. | We conclude that we cannot form massive spheroids from a sequence of several dry-mergers between building blocks of the same kind (similar mass and chemical properties), even if the progenitors are chosen to have the correct $\alpha$ -enhancement. |
For the same reason, present-day low mass ellipticals which satisfy both the MMR and the MFMR cannot be the building blocks of massive ones. | For the same reason, present-day low mass ellipticals which satisfy both the MMR and the MFMR cannot be the building blocks of massive ones. |
This conclusion can be extended to the CMR, since the colour differences are mainly driven by metallicity. | This conclusion can be extended to the CMR, since the colour differences are mainly driven by metallicity. |
Remarkably, similar conclusions have been obtained by Ciotti et al. ( | Remarkably, similar conclusions have been obtained by Ciotti et al. ( |
2007) by studying the dynamical properties of ellipticals. | 2007) by studying the dynamical properties of ellipticals. |
Therefore, pure dissipationless merger of similar stellar systems Now we relax the extreme assumption of the previous section and allow for two or more kind of progenitors for our massive elliptical. | Therefore, pure dissipationless merger of similar stellar systems Now we relax the extreme assumption of the previous section and allow for two or more kind of progenitors for our massive elliptical. |
If, for simplicity sake, we have a fraction f=50% of the final mass coming from progenitors of the type A and 1-f=50% from progenitors B, the final stellar metallicity distribution (shaded hystogram in Fig. 3)) | If, for simplicity sake, we have a fraction $f=$ of the final mass coming from progenitors of the type A and $f=$ from progenitors B, the final stellar metallicity distribution (shaded hystogram in Fig. \ref{prog_AB}) ) |
will be closer to the one expected for a normal elliptical (solid line in Fig. 3)) | will be closer to the one expected for a normal elliptical (solid line in Fig. \ref{prog_AB}) ) |
and the outcome will match the CMR and the MMR, being its final [<Fe/H >] = 0.66 dex, but the predicted [<Mg/Fe>] = 0.06 dex is still too low. | and the outcome will match the CMR and the MMR, being its final $<Fe/H>$ ] = 0.66 dex, but the predicted $<Mg/Fe>$ ] = 0.06 dex is still too low. |
Moreover, this scenario cannot represent a solution for the still too low SFR per unit mass. | Moreover, this scenario cannot represent a solution for the still too low SFR per unit mass. |
If we repeat the same exercise with model and B, these latter quantities get in a better agreement with the values inferred from the observations, and we also notice an improvement for the [<Mg/Fe >] which now amounts to 0.1 dex. | If we repeat the same exercise with model and B, these latter quantities get in a better agreement with the values inferred from the observations, and we also notice an improvement for the $<Mg/Fe>$ ] which now amounts to 0.1 dex. |
We admit that the portion of the parameter space that we are investigating is quite small - although the choice of the models is sensible - the main aim of this investigation being the study of a few clear and extreme cases. | We admit that the portion of the parameter space that we are investigating is quite small - although the choice of the models is sensible - the main aim of this investigation being the study of a few clear and extreme cases. |
Such examples serve to probe to which extent the random nature of the merger process models can be accomodated within the observational uncertainties. | Such examples serve to probe to which extent the random nature of the merger process models can be accomodated within the observational uncertainties. |
A more comprehensive analysis featuring a proper merger history based on the hierachical growth of structure and a self-consistent chemical evolution is in preparation (Pipino et al, 2008). | A more comprehensive analysis featuring a proper merger history based on the hierachical growth of structure and a self-consistent chemical evolution is in preparation (Pipino et al, 2008). |
However, even in the case in which either two more suitable progenitors can be found, or a different mixture of several progenitors can predict the right final chemical properties for a given final galactic mass, several questions arise: i) why are only dSph (i.e. progenitor A) still observable in the local universe? ( | However, even in the case in which either two more suitable progenitors can be found, or a different mixture of several progenitors can predict the right final chemical properties for a given final galactic mass, several questions arise: i) why are only dSph (i.e. progenitor A) still observable in the local universe? ( |
but see Robertson et al. | but see Robertson et al. |
2005). | 2005). |
ii) why is the fraction f such that none of the two classes is predominant? | ii) why is the fraction $f$ such that none of the two classes is predominant? |
iii) since the [<Mg/Fe >] correlates with the final galactic mass, one should expect progenitors with different initial (i.e. pre-merger) properties - which scale accordingly to the final mass of the object - to live in the early universe. | iii) since the $<Mg/Fe>$ ] correlates with the final galactic mass, one should expect progenitors with different initial (i.e. pre-merger) properties - which scale accordingly to the final mass of the object - to live in the early universe. |
How it is possible that they know in advance what they are about to build later on? | How it is possible that they know in advance what they are about to build later on? |
Finally, even if a selection mechanism is at work and it leads to an agreement between model and observed chemical properties, | Finally, even if a selection mechanism is at work and it leads to an agreement between model and observed chemical properties, |
lugher than the mass derived from the maui Bolocam data. | higher than the mass derived from the mm Bolocam data. |
Four possibilities for explaining these ciscrepaucies are 1) uncertainties m the dust properties 2) variations in the temperature within the cores. 3) errors im the measurement of the nua and/or μαι fluxes. aud 1)iun the case of Perbol5. we are likely to have resolved out some of the Sinmun continua enissiou. | Four possibilities for explaining these discrepancies are 1) uncertainties in the dust properties 2) variations in the temperature within the cores, 3) errors in the measurement of the mm and/or mm fluxes, and 4) in the case of Perbo45, we are likely to have resolved out some of the mm continuum emission. |
Note that the decouvolved size of PerbolS iu the Bolocam surveys is rouglily53".. so the amouut offlax resolved out by our CARMA observatious is likely to be substautial. | Note that the deconvolved size of Perbo45 in the Bolocam surveys is roughly, so the amount of flux resolved out by our CARMA observations is likely to be substantial. |
The masses of Perbods derived from the 1.112111 aud Sinmun data could be brought iuto agreement if the enüssvitv spectral iudex were given bv ο20.51 instead of ο)=2. as we assumed. | The masses of Perbo58 derived from the mm and mm data could be brought into agreement if the emissivity spectral index were given by $\beta \simeq 0.5-1$ instead of $\beta = 2$, as we assumed. |
Although the two best measurements of the enissivitv spectral iudex im starless cores sugeest that (2/22 djs a much better fit than 9<1 (Shirleyetal.2005:Schuecect2010)... it is possible that erain growth leads to a shallower cuuissivity spectral iudex at the high deusitics likely to be found at the centers of prestellar cores. | Although the two best measurements of the emissivity spectral index in starless cores suggest that $\beta \geq 2$ is a much better fit than $\beta
\leq 1$ \citep{Shirley05, Schnee10}, it is possible that grain growth leads to a shallower emissivity spectral index at the high densities likely to be found at the centers of prestellar cores. |
For stance. the cuussivity spectral index ina sample of nearby protostars has Όσσα ueasured to be in the range 0.25—1.5 (Arce&Sareent 2006).. and values in this range nueht be plausible Or starless cores given the recent evidence for erain erowthn in starless cores (Steinackeretal.2009). | For instance, the emissivity spectral index in a sample of nearby protostars has been measured to be in the range $0.25-1.5$ \citep{Arce06}, and values in this range might be plausible for starless cores given the recent evidence for grain growth in starless cores \citep{Steinacker09}. |
. Because 1ο nasses in Table 30 are derived from the 1.212nuu opacity (Ossenkopf&Denning1991) extrapolated out ο Suu. au incorrect assumption about the emissivity spectral index would lead to masses off by a factor of a ow. | Because the masses in Table \ref{MMFITSTAB} are derived from the mm opacity \citep{Ossenkopf94} extrapolated out to mm, an incorrect assumption about the emissivity spectral index would lead to masses off by a factor of a few. |
If the cmussivity spectral iudex in starless cores is closer to 9=1 than 3=2. then the expected fluxes at τὰ as extrapolated from the πα Bolocam observations would be a factor of ~3 larger than we have assumed. | If the emissivity spectral index in starless cores is closer to $\beta = 1$ than $\beta = 2$, then the expected fluxes at mm as extrapolated from the mm Bolocam observations would be a factor of $\sim$ 3 larger than we have assumed. |
Iu this case the nou-detection of maui contimmiun enission towards nine of the eleven cores iu our sample becomes au even more siguificaut result (see Section 3.2)). | In this case the non-detection of mm continuum emission towards nine of the eleven cores in our sample becomes an even more significant result (see Section \ref{RATE}) ). |
Alternatively. if PerboSs were siguificauth wariucr at the ceuter than its average temperature. the higher resolution 2muuu observations would be more biased towards Biel masses aud deusities than the low resolution l.luunuu data. | Alternatively, if Perbo58 were significantly warmer at the center than its average temperature, the higher resolution mm observations would be more biased towards high masses and densities than the low resolution mm data. |
It is generally observed that starless cores are colder at them centers due to sclfshicldine (c.g.Crapsictal.2007:Schuce2007). | It is generally observed that starless cores are colder at their centers due to self-shielding \citep[e.g.][]{Crapsi07, Schnee07}. |
. [eating due to contraction can result in a teniperatures a few degrees higher iu the ceuter of a starless core (νου&Caselli 2009).. but this on its own caunot account for the large differeuce im masses derived from the nau aud ια coutinuun data. | Heating due to contraction can result in a temperatures a few degrees higher in the center of a starless core \citep{Keto09}, but this on its own cannot account for the large difference in masses derived from the mm and mm continuum data. |
If there were a weak internal Dhuunositv source in PerhoSs (as discussed iu Section 2.13). then one would expect some central heating that could bring the nua ΛΣ mto agreciuent. | If there were a weak internal luminosity source in Perbo58 (as discussed in Section \ref{SELECTION}) ), then one would expect some central heating that could bring the mm and mm-derived masses into agreement. |
A third possible cause for the disagreement. betweeEE he Llama aud 212nuu derived masses is the uncertai absolute calibration of the emission maps. | A third possible cause for the disagreement between the mm and mm derived masses is the uncertain absolute calibration of the emission maps. |
The two core:PO, detected by CARAIA/SZA are both in the dense iux xieht cluster NGC13232. which results iu artifacts 1 he bolometer maps. | The two cores detected by CARMA/SZA are both in the dense and bright cluster NGC1333, which results in artifacts in the bolometer maps. |
However. detailed testing of the accuracy of the recovery of ua. fux from Bolocam das shown that the quoted fluxes are accurate to within (Enochetal.2006). | However, detailed testing of the accuracy of the recovery of mm flux from Bolocam has shown that the quoted fluxes are accurate to within \citep{Enoch06}. |
. The absolute calibration of he 2111111 continuun maps is accurate to within20%. | The absolute calibration of the mm continuum maps is accurate to within. |
. Therefore. uncertainties in the absolute calibration of the enission maps alone cannot account for the factor of a few difference between the 1ànun aud 2121ua derived πο | Therefore, uncertainties in the absolute calibration of the emission maps alone cannot account for the factor of a few difference between the mm and mm derived masses. |
Due to uncertiüntios in the dust emissivity ancl temperature profiles of PerbolS aud Perbobs. we will defer a iore detailed analysis of their masses wutil more data are available aud ouly note that the 3212114 fluxes of PerbolS aud Perbo58 are not what we expected based ou their wan fiuxes as mapped by Dolocau. | Due to uncertainties in the dust emissivity and temperature profiles of Perbo45 and Perbo58, we will defer a more detailed analysis of their masses until more data are available and only note that the mm fluxes of Perbo45 and Perbo58 are not what we expected based on their mm fluxes as mapped by Bolocam. |
We calculate the density of the detected starless cores from the derived mass and effective radius using the equation: where rag is the effective radius. jj=2.33 is the mean molecular weight per particle and ο is the mass of hydrogen. | We calculate the density of the detected starless cores from the derived mass and effective radius using the equation: where $r_{\rm eff}$ is the effective radius, $\mu = 2.33$ is the mean molecular weight per particle and $m_H$ is the mass of hydrogen. |
The effective radius is the ecometric nean of the decouvolved semüanajor and senianinor axes. derived froma Caussian fit to the flux distribution. | The effective radius is the geometric mean of the deconvolved semi-major and semi-minor axes, derived from a Gaussian fit to the flux distribution. |
The density for each detected core is presented i1 Table 3.. | The density for each detected core is presented in Table \ref{MMFITSTAB}. |
The deusities of Perbol5 axl Perbo 58 «erived from. the πια coutiumua data iu this survey arelarger than the values derived from the nuu Bolocaudata. | The densities of Perbo45 and Perbo 58 derived from the mm continuum data in this survey are larger than the values derived from the mm Bolocam data. |
The densities derived from the ια data (see Table 3)) are roughly a few 410910% cm7. while the mean deusity ina 10 AAU diameter aperature derived from the Lum data are on the order of a few <LO? emi? | The densities derived from the mm data (see Table \ref{MMFITSTAB}) ) are roughly a few $\times 10^6 - 10^7$ $^{-3}$, while the mean density in a $10^4$ AU diameter aperature derived from the mm data are on the order of a few $\times 10^5$ $^{-3}$. |
Since the 311112 observations are higher resolutiou than the l.luuuni observations. if is uot surprising that the 2iunuu-derived densitics are larger. | Since the mm observations are higher resolution than the mm observations, it is not surprising that the mm-derived densities are larger. |
Though high. deusitiesa. in the 10010* ? range have been reported in LISLL (Crapsictal.2007) and iu several starless cores in Orion (Nutter&Ward-Thompsou2007). | Though high, densities in the $10^6 - 10^7$ $^{-3}$ range have been reported in L1544 \citep{Crapsi07} and in several starless cores in Orion \citep{Nutter07}. |
. The densitiesa. iu Table 3. would be lower if the assumed teirperature were higher. as might be expected if the detected cores harbor low-luuinosity protostars. | The densities in Table \ref{MMFITSTAB} would be lower if the assumed temperature were higher, as might be expected if the detected cores harbor low-luminosity protostars. |
A simple model of the density profile of a starless core C1 be given by: where sg. ry aud a are constants. | A simple model of the density profile of a starless core can be given by: where $n_0$, $r_0$ and $\alpha$ are constants. |
This formulation with a=2.5 is a good approximation for the deusitv profile of a Dounor-Ebert sphere (Tafallaetal.2001) and provides a qualitatively good fit to the observed density profiles of starless cores. which are fairly flat at small radi and are steeper at larger radii. | This formulation with $\alpha=2.5$ is a good approximation for the density profile of a Bonnor-Ebert sphere \citep{Tafalla04} and provides a qualitatively good fit to the observed density profiles of starless cores, which are fairly flat at small radii and are steeper at larger radii. |
We investigate whether or uot the density profiles of the cores in Perbolb aud PerboSs are consistent witli Equation 3.. | We investigate whether or not the density profiles of the cores in Perbo45 and Perbo58 are consistent with Equation \ref{MODELEQ}. |
First we measure the fiux profile iu anuuli centered on the positions of Perbo15 and Perbohs given in Table 3.. | First we measure the flux profile in annuli centered on the positions of Perbo45 and Perbo58 given in Table \ref{MMFITSTAB}. |
We then calculate the dus profile that would be observed for a spherical core with the density profile given by Equation 3. assunuiug that the cores are isothermal aud have coustaut dust enmuüssivities. | We then calculate the flux profile that would be observed for a spherical core with the density profile given by Equation \ref{MODELEQ} assuming that the cores are isothermal and have constant dust emissivities. |
We normalize the observed aud model fluxes such that the flux at the center of the core is equal to unity. so the Do term in Equation 3. is not meanineful. | We normalize the observed and model fluxes such that the flux at the center of the core is equal to unity, so the $n_0$ term in Equation \ref{MODELEQ} is not meaningful. |
We show iu Figure 3 that under the stated assuniptious of the dust properties aud geoimoetnies that a deusity profile like that | We show in Figure \ref{DENSITYMAPS}
that under the stated assumptions of the dust properties and geometries that a density profile like that |
The fragmentation of the shell. ie. its fully developed transverse collapse. occurs when Writing ηεἰς lor the gradient ancl using σας. | The fragmentation of the shell, i.e. its fully developed transverse collapse, occurs when Writing $- i \eta /R_s$ for the gradient and using Eqs. |
3. ane 4.. Eqs. | \ref{g1_sigma1} and \ref{sigma_0}, , Eqs. |
1 and 2. successively become: Seven parameters intervene in the ability of the shell to undergo a transverse collapse: In what follows. the influence of each of the parameters involved in the fragmentation process is investigated. the final aim being to check whether some reasonable sets of conclitions can lead to the shell fragmentation. | \ref{eq:per_cont} and \ref{eq:per_motion} successively become: Seven parameters intervene in the ability of the shell to undergo a transverse collapse: In what follows, the influence of each of the parameters involved in the fragmentation process is investigated, the final aim being to check whether some reasonable sets of conditions can lead to the shell fragmentation. |
The amplituce of the perturbed surface density at a time {. σι). results from its initial value σι) ancl from the transverse Llows which redistribute the shell mass accumulated while the shell was propagating throughout the progenitor cloud. | The amplitude of the perturbed surface density at a time $t$, $\tilde \sigma_1(t)$, results from its initial value $\tilde \sigma_1(t_{em})$ and from the transverse flows which redistribute the shell mass accumulated while the shell was propagating throughout the progenitor cloud. |
The fragmentation is therefore favoured if these transverse Hows converge towards the clumps initially present within the shell. | The fragmentation is therefore favoured if these transverse flows converge towards the clumps initially present within the shell. |
Such à situation corresponds. to a phase dillerence AG = between the two perturbed cuantities a(/.6) ancl e(f.6). that is. the transverse velocity exhibits a phase celay of a quarter of a wavelength with respect to the perturbecl surface density (see Fig. 3)). | Such a situation corresponds to a phase difference $\Delta \phi$ = $-\frac{\pi}{2}$ between the two perturbed quantities $\sigma _1(t,\phi)$ and $v(t,\phi)$, that is, the transverse velocity exhibits a phase delay of a quarter of a wavelength with respect to the perturbed surface density (see Fig. \ref{fig:fragshell}) ). |
Replacing ;Xó by this value. Eqs. | Replacing $\Delta \phi$ by this value, Eqs. |
15 and 10 respectively become: In what follows. the roles of the other. parameters is studied. assuming that AO= w/2. ie. through the numerical integration of I2qs. | \ref{eq:per_cont_num} and \ref{eq:per_motion_num} respectively become: In what follows, the roles of the other parameters is studied assuming that $\Delta \phi = - \pi $ /2, i.e. through the numerical integration of Eqs. |
17. - I8.. | \ref{eq:per_cont_num_phi} - \ref{eq:per_motion_num_phi}. |
In order to assess whether the shell fragments or not during its propagation through the hot protogalactic background. we compare in Fig. | In order to assess whether the shell fragments or not during its propagation through the hot protogalactic background, we compare in Fig. |
4. the evolutions with time of the perturbed and unperturbed surface densities. namely σι(1) ancl σου). For different values of the external pressure 1%, 10U apnd P"ddeneem.2 7) and numbers A of SNcLL (100 ancl 200). | \ref{fig:disc_sig1init} the evolutions with time of the perturbed and unperturbed surface densities, namely $\tilde{\sigma} _1(t)$ and $\sigma _0(t)$, for different values of the external pressure $P_h$ $\times 10^{-10}$ and $^{-10}$ $^{-2}$ ) and numbers $N$ of SNeII (100 and 200). |
The corresponding metallicity is indicated in each panel. | The corresponding metallicity is indicated in each panel. |
Phe other parameters are kept the same in every ease: the sound speed e; of the shell material is + the perturbed velocity οἐν) is Vs). the number of forming clumps η is LO and the initial perturbed surface density is assumed to be of order one per cent of the unperturbecl value. namely σιένο)=0.01Tullo). | The other parameters are kept the same in every case: the sound speed $c_s$ of the shell material is $^{-1}$, the perturbed velocity $v(t_{em})$ is $V_s(t_{em})$, the number of forming clumps $\eta$ is 10 and the initial perturbed surface density is assumed to be of order one per cent of the unperturbed value, namely $\sigma _1(t_{em}) = 0.01 \times \sigma _0(t_{em})$. |
This choice may seem rather arbitrary but the next. paragraph will show that the initial amplitude of σι does not alfect the results significantly. | This choice may seem rather arbitrary but the next paragraph will show that the initial amplitude of $\sigma _1$ does not affect the results significantly. |
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