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We uote that the term vield iu this paper connotes the total ejected mass in individual clements or in metals. and uot the uct vield after accounting for the original metal composition of the star. | We note that the term yield in this paper connotes the total ejected mass in individual elements or in metals, and not the net yield after accounting for the original metal composition of the star. |
We do not consider main-sequence mass loss in aly of our cases below: this is uulikelv to be important for low-Z, stellar populations (see Tuuliusonetal.200: ou this poiut). | We do not consider main-sequence mass loss in any of our cases below; this is unlikely to be important for $Z_\star$ stellar populations (see \citealt{tsv} on this point). |
We assume the stellar TIME. to lave the foin o(M)=ωμλ". and the IME slope. a. to be the Salpeter value of 2.35. since small slope variatious have been shown to male little difference (Cüroux&Shapiro 2003).. | We assume the stellar IMF to have the form, $\phi(M) = \phi_0 M^{-\alpha}$, and the IMF slope, $\alpha$ , to be the Salpeter value of 2.35, since small slope variations have been shown to make little difference \citep{gs96,madshull,sch02,sch03}. . |
AIL the cases here are normalized over their respective mass ranges as, {dJALALO(AL)=1. | All the cases here are normalized over their respective mass ranges as, $ \int dM M
\phi(M) = 1 $. |
For the ionizing spectra. we use Dronuuct aud Schaerer(2003) for a top-heavy IAIF. Tiuliusouetal.(2003). for d100 ALL. Z,=O0 stars. aud Leithererotal.(1995) for non-zero Z, cases. | For the ionizing spectra, we use \citet{bromm} and \citet{sch03} for a top-heavy IMF, \citet{tsv} for 1–100 $_\odot$, $Z_\star=0$ stars, and \citet{leitherer} for non-zero $Z_\star$ cases. |
We note that theintegrated. ionizing photon uunber from VALSs happens to be roughly the same as that from Z,=0 stars in an IAIF over 1100. AZ... despite the ereatly boosted iouizinerate froii VMSS. owing to their brief litetimes. | We note that the ionizing photon number from VMSs happens to be roughly the same as that from $Z_\star=0$ stars in an IMF over 1–100 $M_\odot$, despite the greatly boosted ionizing from VMSs, owing to their brief lifetimes. |
The stellar cluission rate of iouizine photous drops to less thaw of its initial value after 30 Myr for a prescut-day IME (Venkatesanetal.2003).. but takes only about 3.5 Myr for a top-heavy IME (Schaerer2003). | The stellar emission rate of ionizing photons drops to less than of its initial value after 30 Myr for a present-day IMF \citep{vts}, but takes only about 3.5 Myr for a top-heavy IMF \citep{sch03}. |
. The cause is directly related to the lifetimes of the lougest-lived (motal-free) star that is relevant for reionization in each of the IMEs: a 10 AL. star (~ 2 «107 vr) versus a LOO AL. star (~ 3 <10° vr). | The cause is directly related to the lifetimes of the longest-lived (metal-free) star that is relevant for reionization in each of the IMFs: a 10 $M_\odot$ star $\sim$ 2 $\times 10^7$ yr) versus a 100 $M_\odot$ star $\sim$ 3 $\times 10^6$ yr). |
The net difference iu the time-uteerated ionizing radiation isa factor of only ~ 1.1 (Bromunetal.2001:Schaerer2002. 2005). | The net difference in the time-integrated ionizing radiation is a factor of only $\sim$ 1.4 \citep{bromm, sch02, sch03}. |
. Lastly. for the conversion from gps Το NiveNi. we assunie that Zap~ οσο, with the conespoudiuelv scaled levels of carbon aud oxvecn with respect to their solar (Shull1993.. and. references. therein). | Lastly, for the conversion from $\eta_{\rm Lyc}$ to $N_{\rm Lyc}/N_{\rm
b}$, we assume that $Z_{\rm IGM} \sim$ $^{-2.5} Z_\odot$, with the correspondingly scaled levels of carbon and oxygen with respect to their solar \citealt{shull93}, and references therein). |
We take (Ei) to be 21 eV for Z,=Z.. aud Z,=ü.001 stars. 27 eV for Z,=O stars ina 1100 AL. IMF. aud 30 eV for Z,=0 stars in a top-heavy IAIF (Schaerer2003). | We take $\langle E_{\rm Lyc} \rangle$ to be 21 eV for $Z_\star =
Z_\odot$ and $Z_\star = 0.001$ stars, 27 eV for $Z_\star = 0$ stars in a 1–100 $M_\odot$ IMF, and 30 eV for $Z_\star = 0$ stars in a top-heavy IMF \citep{sch03}. |
. We first show that carbon aud oxvecn are mostly the products of IMSs aud massive stars respectively. aud that IMSs are consequently sieuificaut coutributors to cosmolocical mctal eeucration. | We first show that carbon and oxygen are mostly the products of IMSs and massive stars respectively, and that IMSs are consequently significant contributors to cosmological metal generation. |
Figure 1 displays the total IME-weiehlted vields for metals. £C aud tO as a fiction of stellar niass for VMSs and for three values of Z, for a present-day IME. | Figure 1 displays the total IMF-weighted yields for metals, $^{12}$ C and $^{16}$ O as a function of stellar mass for VMSs and for three values of $Z_\star$ for a present-day IMF. |
Using the total ejected mass. A;. iu cach clement or metals as a function of stellar mass M. the y-axis iu the figure is calculated by weieliting each AL; with the IAIF as AZ;o(AS). | Using the total ejected mass, $M_i$ , in each element or metals as a function of stellar mass $M$, the $y$ -axis in the figure is calculated by weighting each $M_i$ with the IMF as $M_i \phi(M)$. |
Using the vields aud ionizing spectra as described in 82. we calculate pve auc NoveLN, in Table 1 for several cases: (a) as a function of Z,. (b) as defined with respect to total metal. °C. aud 1*O vield. and (0) with metal vields imeluded from three mass ranges (1100 AFL. 8100 AZ... and 100-1000 37.23. | Using the yields and ionizing spectra as described in 2, we calculate $\eta_{\rm Lyc}$ and $N_{\rm Lyc}/N_{\rm b}$ in Table 1 for several cases: (a) as a function of $Z_\star$, (b) as defined with respect to total metal, $^{12}$ C, and $^{16}$ O yield, and (c) with metal yields included from three mass ranges (1–100 $M_\odot$, 8–100 $M_\odot$, and 100-1000 $M_\odot$ ). |
Tu the second case. the conversion from gps to NiveNj involves two factors that will cause νοLN to vary within each col iu the table: νο aud Zyeap. which will both vary at fixed Z, depending on whether we are considering JL and Nive/N}, with respect to total metals. 12€. or 160. The last case is intended to evaluate the impact of the metal contribution fom various regions of the stellar IAIF. particularly from IMSs. on py iud NyyNi. | In the second case, the conversion from $\eta_{\rm Lyc}$ to $N_{\rm Lyc}/N_{\rm
b}$ involves two factors that will cause $N_{\rm Lyc}/N_{\rm b}$ to vary within each column in the table: $\eta_{\rm Lyc}$ and $Z_{\rm IGM}$, which will both vary at fixed $Z_\star$ depending on whether we are considering $\eta_{\rm Lyc}$ and $N_{\rm Lyc}/N_{\rm b}$ with respect to total metals, $^{12}$ C, or $^{16}$ O. The last case is intended to evaluate the impact of the metal contribution from various regions of the stellar IMF, particularly from IMSs, on $\eta_{\rm Lyc}$ and $N_{\rm Lyc}/N_{\rm b}$. |
The results in Table 1. although subject to the uncertainties in stellar modelling. reveal several treuds of interest to Ligh-: IGM studies. | The results in Table 1, although subject to the uncertainties in stellar modelling, reveal several trends of interest to $z$ IGM studies. |
These may be sununuiuized as follows. | These may be summarized as follows. |
First. νο can vary siguificautly with Z, aud with the clement with respect to which it is defined. | First, $\eta_{\rm Lyc}$ can vary significantly with $Z_\star$ and with the element with respect to which it is defined. |
As anticipated for the reasous outlined iu 82. there is a stroug eain in pec aud NeveiM, as Z, decreases. with Z=0 stars in a present-day IME being up to 10-20 times more cfiicicut at ecnerating ioniziug radiation per unit metal vield thau the Z=Z. stars in Madau&Shull(1996). | As anticipated for the reasons outlined in 2, there is a strong gain in $\eta_{\rm Lyc}$ and $N_{\rm
Lyc}/N_{\rm b}$ as $Z_\star$ decreases, with $Z=0$ stars in a present-day IMF being up to 10-20 times more efficient at generating ionizing radiation per unit metal yield than the $Z=Z_\odot$ stars in \citet{madshull}. |
. Second. although the values of pve approximately match those of. e.g. Schaerer(2002).. when the vields from IMSs are excluded. we see that IMSs are iu fact a substantial source of metals. particularly carbon. aud should be included in such calculations. | Second, although the values of $\eta_{\rm Lyc}$ approximately match those of, e.g., \cite{sch02}, when the yields from IMSs are excluded, we see that IMSs are in fact a substantial source of metals, particularly carbon, and should be included in such calculations. |
One possible exception might be at very high redshifts. when IMSs may uot have had the time vet to eject their uncleosvuthetic products. | One possible exception might be at very high redshifts, when IMSs may not have had the time yet to eject their nucleosynthetic products. |
Inthis case. the original definition (Aladau&Shull1996) of νο which does not iuclude IMS. inctal vields would be valid. aud pve may be stated to increase with decreasing Z, and/or mereasiug redshift. | Inthis case, the original definition \citep{madshull} of $\eta_{\rm Lyc}$ which does not include IMS metal yields would be valid, and $\eta_{\rm Lyc}$ may be stated to increase with decreasing $Z_\star$ and/or increasing redshift. |
Third. gno is the most sensitive to Z,. increasing strouely from solar-Z, to metal-free stars. for a prescut-day IME. | Third, $\eta_{\rm Lyc, O}$ is the most sensitive to $Z_\star$, increasing strongly from $Z_\star$ to metal-free stars, for a present-day IMF. |
This directly results from the following three treuds: (1) ο aud ionizing radiation are produced donuuantlv by the IME's iuassve stars. (2) with decreasing Z,. the production of 1*O decreases while that of ioniziug photons increases. aud therefore. (3) their ratio {νοο. I comparison Το pecz OY Hive. shows the strongest increase with dechning Z, for stars iu a prescut- IME. | This directly results from the following three trends: (1) $^{16}$ O and ionizing radiation are produced dominantly by the IMF's massive stars, (2) with decreasing $Z_\star$, the production of $^{16}$ O decreases while that of ionizing photons increases, and therefore, (3) their ratio $\eta_{\rm Lyc,
O}$, in comparison to $\eta_{\rm Lyc, Z}$ or $\eta_{\rm Lyc, C}$, shows the strongest increase with declining $Z_\star$ for stars in a present-day IMF. |
Fourth. the value of gps directly impacts the fraction of barvous required to be converted iuto a first stars »»pulation in order to influence reionization (c.g... Drouun 1997)). | Fourth, the value of $\eta_{\rm Lyc}$ directly impacts the fraction of baryons required to be converted into a first stars population in order to influence reionization (e.g., \citealt{bromm, mirees97}) ). |
Oue possible implication of the first result above isthat a factor of 10.20 ewer barvous need be part of such a population for them. ο be relevaut for reionization alone (which requires oulv NiveiM,~ 10). i£ such stars are not required. to account OY Zr Iu | One possible implication of the first result above isthat a factor of 10–20 fewer baryons need be part of such a population for them to be relevant for reionization alone (which requires only $N_{\rm
Lyc}/N_{\rm b} \sim$ 10), if such stars are not required to account for $Z_{\rm IGM}$ . |
this case. recionizatiou bv Pop IIIstars in a xeseut-«dav. ΠΑΤΕ occurs late (since they should generate ewer joniziue photous thanindicated by Table 1). and such stars would therefore make a neeligible contribution | In this case, reionization by Pop IIIstars in a present-day IMF occurs late (since they should generate fewer ionizing photons thanindicated by Table 1), and such stars would therefore make a negligible contribution |
It is well known that along the Hubble sequence the ealaxv color becomes bluer moving from carly to late types aud that this corresponds to a sequence in SER which is virtually zero in ellipticals aud maxima in late spirals. | It is well known that along the Hubble sequence the galaxy color becomes bluer moving from early to late types and that this corresponds to a sequence in SFR which is virtually zero in ellipticals and maximum in late spirals. |
However. especially iu spirals. there is a siguificaut dispersion in the average color from galaxy to galaxy. indicating that SER cau vary significantly even for a οἼνοιι IIubble type. | However, especially in spirals, there is a significant dispersion in the average color from galaxy to galaxy, indicating that SFR can vary significantly even for a given Hubble type. |
Convenieutlv. (£5Wyo and (G7By colors. corrected for galactic aud iuterual extinction are listed in the RC3 catalog for a fair percentage of the galaxies of our sample and respectively). | Conveniently, $(B-V)_T^0$ and $(U-B)_T^0$ colors, corrected for galactic and internal extinction are listed in the RC3 catalog for a fair percentage of the galaxies of our sample and respectively). |
Frou. these we derived also (Το colors which. allowing for the extended wavelength baseline. are nore sensitive SFR indicators. | From these we derived also $(U-V)_T^0$ colors which, allowing for the extended wavelength baseline, are more sensitive SFR indicators. |
For cach biu of galaxy morphological types we divided the galaxies iuto subsamples, coutaimiug galaxies blicr aud redder than the elobal average. | For each bin of galaxy morphological types we divided the galaxies into subsamples, containing galaxies bluer and redder than the global average. |
We then computed separately tle SN rates for cach of these subsamples. | We then computed separately the SN rates for each of these subsamples. |
The results are reported iu Table 5.. where the galaxy types are in col 1. the average colors for the galaxies of the specific subset are in cols 2 and 5. the SN rates in SNu for SN Ta and for core-collapse SNII|Ib/c iu cols 3-1 aud 6-7. | The results are reported in Table \ref{color}, where the galaxy types are in col 1, the average colors for the galaxies of the specific subset are in cols 2 and 5, the SN rates in SNu for SN Ia and for core-collapse SNII+Ib/c in cols 3-4 and 6-7. |
As expected. the rate of core collapse SNe (ITIΠοο) is lieher iu the bluer spirals. | As expected, the rate of core collapse SNe (II+Ib/c) is higher in the bluer spirals. |
By using B-V color lus effect is seen ouly for late spirals (the rate is higher by a factor of 1.7 for Sle-Scj. but becomes clear for all spirals when using U-V color (over a factor of 2). | By using B-V color this effect is seen only for late spirals (the rate is higher by a factor of 1.7 for Sbc-Sd), but becomes clear for all spirals when using U-V color (over a factor of 2). |
Iustead the rate of SN Ta is. within the uncertainties. iudependent on galaxy colors. | Instead the rate of SN Ia is, within the uncertainties, independent on galaxy colors. |
With regards to the rows labeled “AM” (which includes galaxies of all types) we should note that dividing galaxies iuto bluer and redder colors to a large extent corresponds to separating them into carly aud late type galaxies. | With regards to the rows labeled “All” (which includes galaxies of all types) we should note that dividing galaxies into bluer and redder colors to a large extent corresponds to separating them into early and late type galaxies. |
Therefore the great difference in the core collapse SN rates in bluer aud redder galaxies simply reflects the fact that core collapse SNe are not fouud in early type galaxies. | Therefore the great difference in the core collapse SN rates in bluer and redder galaxies simply reflects the fact that core collapse SNe are not found in early type galaxies. |
We cal conmare the observed SN rates with the predicted SFR in galaxies of differeut colors. | We can compare the observed SN rates with the predicted SFR in galaxies of different colors. |
This is done in Fie.l where the dots represent the SN rates iu SNu (left-hand scale) in galaxies of different 0V inteerated colors aud the line is the SFR per unit of blue luinesitv (right-hand scale) taken from the evolutionary svuthesis models of I&cnuicutt (1998)). | This is done in \ref{sfr} where the dots represent the SN rates in SNu (left-hand scale) in galaxies of different $U-V$ integrated colors and the line is the SFR per unit of blue luminosity (right-hand scale) taken from the evolutionary synthesis models of Kennicutt \cite{kenni}) ). |
Tn general. for a galaxy of huuinositv Lp. because of the short life of progenitor evolution. the number of core collapse SNe per century corresponds to the nuniber of new born stars within the appropriate mass range. namely: where "n is the mass fraction of stars which are born with mass in the range A; to Ap. the lower aud upper lit for core-collapse SN progenitors. aud <Afsy> is the average mass of SN progenitors. | In general, for a galaxy of luminosity $_B$, because of the short life of progenitor evolution, the number of core collapse SNe per century corresponds to the number of new born stars within the appropriate mass range, namely: where $f_{M_L}^{M_U}$ is the mass fraction of stars which are born with mass in the range $M_L$ to $M_U$ , the lower and upper limit for core-collapse SN progenitors, and $<M_{SN}>$ is the average mass of SN progenitors. |
According to the standard scenarios. Mg~SAL. aud Adj~LOAL... | According to the standard scenarios, $M_L \simeq 8 M_\odot$ and $M_U \simeq 40 M_\odot$. |
Adopting a Salpeter mass function. xLO aud «Magy>>LOM. which compensate thefa factor LOO which accounts for the difference in the time scale. | Adopting a Salpeter mass function, $f_{M_L}^{M_U} \simeq 10^{-1}$ and $<M_{SN}> \simeq 10 M_\odot$ which compensate the factor 100 which accounts for the difference in the time scale. |
Iu conclusion. even if the exact comcideuce of the two scales in our figure is to some deerce fortuitous. the nice agreement of the SER measured through core collapse SN rates and that deduced dy svuthesis modeling for "averages ↴∖↴↻↕↥⋅⋜↧↕∶↴⋁⋜↧↕⋜⋯↕↸∖↴∖↴∙↕↸∖∐≼↧↴∖↴↴∖↴∏∏⋯↥⋅↑↑∪↑∐↸∖∶↴∙⊾↸∖∐↸∖↥⋅⋜↧↕ scenario for stellar population evolution. | In conclusion, even if the exact coincidence of the two scales in our figure is to some degree fortuitous, the nice agreement of the SFR measured through core collapse SN rates and that deduced by synthesis modeling for “average” spiral galaxies, lends support to the general scenario for stellar population evolution. |
Conversely. the fact that the rate of SN Ta shows no dependence ou the galaxy. U-V color requires a siguificaut delay between the SFR episodes aud the onset of SN Ta events. | Conversely, the fact that the rate of SN Ia shows no dependence on the galaxy U-V color requires a significant delay between the SFR episodes and the onset of SN Ia events. |
The relation between core collapse SN rates and colors provides a useful tool for the comparison of local and high-: SN rates. | The relation between core collapse SN rates and colors provides a useful tool for the comparison of local and $z$ SN rates. |
Dudeed. for galaxies at ;omteerated colors can be measured relatively easily. whereas morphological types. requiring superb laine. are not generally available. | Indeed, for galaxies at $z$ integrated colors can be measured relatively easily, whereas morphological types, requiring superb imaging, are not generally available. |
Conversely. it is clear that reporting the average SN rates for uncharacterized galaxy saluples may turn out to be pointless for coustrainiug ealaxy evolution nmocols. | Conversely, it is clear that reporting the average SN rates for uncharacterized galaxy samples may turn out to be pointless for constraining galaxy evolution models. |
The interest in deriving the SN rate in units ofthe FIR huninosities was stressed by Jorgensen (1990)) who | The interest in deriving the SN rate in units ofthe FIR luminosities was stressed by rgensen \cite{jorg90}) ) who |
The event has a total of 265 points in dy; and 352in D. | The event has a total of 265 points in $R_M$ and 352in $B_M$. |
I eliminate 3 outliers (all αἱ baseline) and find that the remaining points have a A?/dof=0.65 in each filler separately. | I eliminate 3 outliers (all at baseline) and find that the remaining points have a $\chi^2/{\rm dof}=0.65$ in each filter separately. |
This indicates that the errors have been overestimated. | This indicates that the errors have been overestimated. |
In principle. one might under these circumstances renormalize the errors bv a [actor ν0.00=0.81. | In principle, one might under these circumstances renormalize the errors by a factor $\sqrt{0.65}=0.81$. |
However. the great majority of (he points are al baseline where the event is extremely faint. whereas most of the information ol immediate interest comes from the highly maenilied portions of the event. where (he error corrections are not likely to be (he same as for (he baseline points. | However, the great majority of the points are at baseline where the event is extremely faint, whereas most of the information of immediate interest comes from the highly magnified portions of the event, where the error corrections are not likely to be the same as for the baseline points. |
Hence I do not renormalize. | Hence I do not renormalize. |
As mentioned in the Introduction. the fit to a standard Paezviiski(1986). curve. with 7 parameters. /y. ug. Fi. fige foa Foe. ond foa. where [u(1)]?=ugτμ shows clear asymmetric residuals. | As mentioned in the Introduction, the fit to a standard \citet{pac86} curve, with 7 parameters, $t_0$, $u_0$, $t_\e$, $f_{s,R}$, $f_{b,R}$, $f_{s,B}$ , and $f_{b,B}$, where $[u(t)]^2 = u_0^2 + (t-t_0)^2/t_\e^2$, shows clear asymmetric residuals. |
See Figure L.. | See Figure \ref{fig:lc}. |
I then add two additional parameters. zi. and zz. which enter through equations (7)) and (3)). | I then add two additional parameters, $\pi_{\e,N}$ and $\pi_{\e,E}$, which enter $[u(t)]^2=[\tau(t)]^2 + [\beta(t)]^2$ through equations \ref{eqn:tauoft}) ) and \ref{eqn:betaoft}) ). |
At first. E use the no-parallax solution as my seed. | At first, I use the no-parallax solution as my seed. |
The eode converges to a solution that is inconsistent with the results of Aleoekοἱal.(2001). | The code converges to a solution that is inconsistent with the results of \citet{alcock01}. |
. see Table 1. | See Table 1. |
I therefore explore a densely samplecl grid over Che rectangle —2<zx6. —2Xmpg<A. | I therefore explore a densely sampled grid over the rectangle $-2\leq\pi_{\e,N}\leq 6$, $-2\leq\pi_{\e,E}\leq 4$. |
The likelihood contours of this search are shown in Figure 3.. | The likelihood contours of this search are shown in Figure \ref{fig:piecontours}. |
Figure 3. has a number of notable features. | Figure \ref{fig:piecontours} has a number of notable features. |
First. of course. it has two solutions. | First, of course, it has two solutions. |
The second solution (to the northwest) is the same as the one found by Aleockοἱal.(2001). | The second solution (to the northwest) is the same as the one found by \citet{alcock01}. |
. As shown in Table L. the (vo solutions differ in 47 by less than 0.1. | As shown in Table 1, the two solutions differ in $\chi^2$ by less than 0.1. |
Hence. thev are truly degenerate. | Hence, they are truly degenerate. |
Second. the high 4? contours to the southwest of the two solutions tend toward continuous straight lines with a position angle of about 149° (North through East). | Second, the high $\chi^2$ contours to the southwest of the two solutions tend toward continuous straight lines with a position angle of about $149^\circ$ (North through East). |
This is almost exactly perpendicular to the acceleration of the Earth (projected onto the plane of the sky. which has an amplitude of 0.52kms!dav.| and a position angle of 23873. | This is almost exactly perpendicular to the acceleration of the Earth (projected onto the plane of the sky, which has an amplitude of $0.52\,\kms\,\rm day^{-1}$ and a position angle of $238.\hskip-2pt^\circ 3$. |
That is. (hese contours derive Irom the parallax asymmetry that is due (ο (he acceleration of the Earth along the direction of lens motion and which is clearly visible in Figure 1.. | That is, these contours derive from the parallax asymmetry that is due to the acceleration of the Earth along the direction of lens motion and which is clearly visible in Figure \ref{fig:lc}. |
For events with weak parallax. one obtains only this one-dimensional information about the parallax (Gould.Miralda-Escudé&Baheall1994). | For events with weak parallax, one obtains only this one-dimensional information about the parallax \citep*{gmb}. |
. Evidentlv. NLACIIO-LMCT-5 is relatively close to (his situation (as one would expect [rom its brevitv). but Figure 3. shows that this event lies in a region of the πι: diagram that is beyond (his continuous degeneracy. | Evidently, MACHO-LMC-5 is relatively close to this situation (as one would expect from its brevity), but Figure \ref{fig:piecontours} shows that this event lies in a region of the $\bpi_\e$ diagram that is this continuous degeneracy. |
In fact. there are not just two solutions. but four. | In fact, there are not just two solutions, but four. |
The other two solutions are obtained from the firsttwo bv first substituting wy—αμ ancl then making very slight adjustments to | The other two solutions are obtained from the firsttwo by first substituting $u_0\rightarrow -u_0$ and then making very slight adjustments to |
the outflow point A. see Fig. 1)) | the outflow point $N$, see Fig. \ref{fig-layer}) ) |
is roughly Che same as the density nr al the center of the laver. we get Πρ κ, nudcL Aud. Next. we consider the set of equations describing the magnetic field: Ohms law. Faradays law. and Ampere’s laa. | is roughly the same as the density $n$ at the center of the layer, we get n_0 L n u L A u. Next, we consider the set of equations describing the magnetic field: Ohm's law, Faraday's law, and Ampere's law. |
Here. our treatment is quite standard. | Here, our treatment is quite standard. |
From Faradays law in a steady state in (wo dimensions it follows that the out-ol-plane electric field E. is uniform across the domain: im the ideal-MIID region just outside (he laver (point M). this field can be written as E.—leeBye. whereas al the center of the laver point (O). where both B and v vanish. resistive MIID Ohm's law vields E.=jj. where i is the plasma resistivity (related to the magnetic diffusivity ) via 1= j/0?/4x). | From Faraday's law in a steady state in two dimensions it follows that the out-of-plane electric field $E_z$ is uniform across the domain; in the ideal-MHD region just outside the layer (point M), this field can be written as $E_z = - v_{\rm rec} B_0/c$, whereas at the center of the layer point (O), where both ${\bf B}$ and ${\bf v} $ vanish, resistive MHD Ohm's law yields $E_z = \eta' j_z$, where $\eta'$ is the plasma resistivity (related to the magnetic diffusivity $\eta$ via $\eta = \eta' c^2 / 4\pi$ ). |
Estimating Jj. using Ampere’s law as Jj. —6D,/40. we eel the following important relationship between Όρος and 9: | Estimating $j_z$ using Ampere's law as $j_z \simeq - c B_0/4\pi\delta$, we get the following important relationship between $v_{\rm rec}$ and $\delta$:. |
which is the same as in the Sweet.Parker model. | which is the same as in the Sweet–Parker model. |
Furthermore. a quick comparison of the electric field at the inflow point and the oullow point N=(xL.y0). allows us to determine the characteristic reconnected magnetic field By=D,(L.0) at the end of the reconnection laver: ος—lyeD,=—uD,. ( | Furthermore, a quick comparison of the electric field at the inflow point and the ouflow point $N=(x=L, y=0)$, allows us to determine the characteristic reconnected magnetic field $B_1\equiv B_y(L,0)$ at the end of the reconnection layer: $cE_z = - v_{\rm rec} B_0 = - u B_1$. ( |
This is basically magnetic flux conservation: (he amount of the flux that enters the laver per unit time equals to the amount of [lux that leaves the laver: inother words. magnetic flux just reconnects but does not get destroved). | This is basically magnetic flux conservation: the amount of the flux that enters the layer per unit time equals to the amount of flux that leaves the layer; inother words, magnetic flux just reconnects but does not get destroyed). |
Combining this with the mass conservation equation (2.2)). we gel an estimate By = B,(L.0) = By wel. | Combining this with the mass conservation equation \ref{eq-mass-conserv}) ), we get an estimate B_1 = B_y(L,0) = B_0 A . |
is casily computed. leacing to These solutious represent à uno singular Universe which goes asviuptotically over to the correspoucding flat classical iiodel for dust (a= 0) dominated epoch (22-25)Y(Fig. 1)) | is easily computed, leading to These solutions represent a no singular Universe which goes asymptotically over to the corresponding flat classical model for dust $w=0$ ) dominated epoch \ref{class1}- \ref{class4}) )(Fig. \ref{fig1}) ) |
Iu the case &=1 aud w=0 the tinc-iudepeudeut Wheecler-DeWitt equation (30)) reduces to Defining uew variable «=126/—E! we find Equation (51)) is similar to the time-independent Schróddinger equation for à sniple harmonic oscillator with unit mass and energy A where PA=ΓΙ aud w= 1/12. | In the case $k=1$ and $w=0$ the time-independent Wheeler-DeWitt equation \ref{sle2}) ) reduces to Defining new variable $x=12a - E'$ we find Equation \ref{k1}) ) is similar to the time-independent Schröddinger equation for a simple harmonic oscillator with unit mass and energy $\lambda$ where $2\lambda = E'^{2}/144$ and $w=1/12$ . |
Therefore. the allowed values of A are (oi|1/2) aud the possible values of E? are therefore. the stationary solutions are where and ZZ, are Termite polyunonials. | Therefore, the allowed values of $\lambda$ are $w(n+1/2)$ and the possible values of $E'$ are therefore, the stationary solutions are where and $H_n$ are Hermite polynomials. |
The wave fictions (57)) are similar to the stationary quantum wormholes as defined in (| «C | The wave functions \ref{k1-final}) ) are similar to the stationary quantum wormholes as defined in \cite{Hawking}. . |
Dowever. neither of the boundary conditions (28)) can be satisfied by the thesewave functions. | However, neither of the boundary conditions \ref{boundary}) ) can be satisfied by the thesewave functions. |
Tnb= laud we case. equation (30)) reduces to | In$k=-1$ and $w=0$ case, equation \ref{sle2}) ) reduces to |
(Lilly et al. | (Lilly et al. |
2007), and the 2SLAQ-LRG (Cannon et al. | 2007), and the 2SLAQ-LRG (Cannon et al. |
2006) survey. | 2006) survey. |
In Fig. | In Fig. |
1, we show the colour-colour plots of the H-ATLAS sources. | 1, we show the colour-colour plots of the H-ATLAS sources. |
We divide these plots in terms of colours based on either SPIRE only or SPIRE and PACS data. | We divide these plots in terms of colours based on either SPIRE only or SPIRE and PACS data. |
Our flux selection results in selecting 1686, 402, and 158 sources, from top to bottom of Fig. | Our flux selection results in selecting 1686, 402, and 158 sources, from top to bottom of Fig. |
1. | 1. |
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