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Recently there Las been a considerable observational aud theoretical activity to determine the spectrau of the COSuic backeround radiatiojio and carving out ealaxy counts in the far infrared‘submillimeter range (Puget et al. | Recently there has been a considerable observational and theoretical activity to determine the spectrum of the cosmic background radiation and carrying out galaxy counts in the far infrared/submillimeter range (Puget et al. |
1996. Schlegel et al. | 1996, Schlegel et al. |
1997. Ctuxeroni et al. | 1997, Guideroni et al. |
1997). | 1997). |
Bocase the wdrogen recolubiration chawes the CBAL specruni in he same spectral rauge a detailed exact recalculation of the frequency distribujon of the recoubiation phoous Is Muportait. | Because the hydrogen recombination changes the CBM spectrum in the same spectral range a detailed exact recalculation of the frequency distribution of the recombination photons is important. |
In this paper we calculate he recombination CYOSS SOClonis exaclv bv using a coitinnous. plysical cut off for tie hiehlv excited states o| hydrogen. tzse into account he induced recombination ale explain. why the tine¢epeudenuce of the recoubination process 1s 80 haradly effected x different techlues aux by different effective uzatio) curves. | In this paper we calculate the recombination cross sections exactly by using a continuous, physical cut off for the highly excited states of hydrogen, take into account the induced recombination and explain, why the time dependence of the recombination process is so hardly effected by different techniques and by different effective ionization curves. |
The outliue of the paper ds as follows. | The outline of the paper is as follows. |
In 82 we cisctss the recolubinaion process and ¢erive the τοςΟΙüuatiou equation. | In 2 we discuss the recombination process and derive the recombination equation. |
We eive a new paranuetrization of the effective recolubinalon coefficients. taking iuto account the induced recolbination as well. | We give a new parametrization of the effective recombination coefficients, taking into account the induced recombination as well. |
Iu the third section we solve the recolbination equation. | In the third section we solve the recombination equation. |
The spectrum) of the recombination photons aud its dependence on the cosmological parameters is even πι the fourth8. and in the fifth closing section we discuss our results. | The spectrum of the recombination photons and its dependence on the cosmological parameters is given in the fourth, and in the fifth closing section we discuss our results. |
The umber of recombinations in unit tie can be calculated from the recombination equation as a functiou of the density of free electrous Εν teuuperature T. aud cosinological. and atomic coustants. | The number of recombinations in unit time can be calculated from the recombination equation as a function of the density of free electrons $n_e\,$, temperature $T\,$, and cosmological, and atomic constants. |
Iu this section we derive this equation following Peebles (1965). | In this section we derive this equation following Peebles (1968). |
At the beeinnine of the Lwdrogen recombination the helium is already: completely recombined. | At the beginning of the hydrogen recombination the helium is already completely recombined. |
The mass fraction of the helium is 25 of he total baryonic mass. | The mass fraction of the helium is 25 of the total baryonic mass. |
As was remarked by Novikov aud Ze‘dovich (1967) the direct recombinations to the eround state are inhibited. while the new born enerectic phoons ionize again almost nunediatelv when there are alreacv sone hydrogen atoms. | As was remarked by Novikov and Zel'dovich (1967) the direct recombinations to the ground state are inhibited, while the new born energetic photons ionize again almost immediately when there are already some hydrogen atoms. |
In the following we neglect coupletelv the direct recombinations to the ground sta5 | In the following we neglect completely the direct recombinations to the ground state. |
The states with principal quautuu number n2 plav a kev role in the recombination Xrocesses, | The states with principal quantum number $n = 2 \,$ play a key role in the recombination processes. |
First we calculate the raeof recombinations tov 2 for given free electron umber deusitv ο. temperature T aud ος. | First we calculate the rate of recombinations to $ n \geq 2\,$ for given free electron number density $n_e$, temperature T and $n_{2s}$. |
Tere Dar 19 the cleIs]vof atoms in the state with the principal quant number 3) ancl aueular moment quanti munber /. | Here $n_{nl}$ is the density of atoms in the state with the principal quantum number $n$ and angular momentum quantum number $l$. |
Second. we determine thenet umuber of 5>1 transitions in unit tine bv given no, and1e | Second, we determine thenet number of $2 \rightarrow 1$ transitions in unit time by given $n_{2s} $ and$n_{1s}$. |
The two rates nmt be equal. so we can eliminate 1o. from the calculations. | The two rates must be equal, so we can eliminate $n_{2s}$ from the calculations. |
The loincΠιο cherey of the η=2 state is By=Byflw=X1ceV. | The binding energy of the $n = 2 \,$ state is $_2 = {\rm B}_1/4 \approx
3.4\, eV$ . |
During the recombination process there are a large nuniber of photons with euergv less than D». therefore the excited states of the hivdrogeu atomis are in thermodynamical equilibrium above the second level. ic. Iu the case of gaseous uebulae the mean free path of low energetic photons is larger than the cdimensious of the ionized region. the svsteu is far from being in equilibrium above the η=2 level in contrast to the recombining Universe. | During the recombination process there are a large number of photons with energy less than $B_{2}$, therefore the excited states of the hydrogen atoms are in thermodynamical equilibrium above the second level, i.e. In the case of gaseous nebulae the mean free path of low energetic photons is larger than the dimensions of the ionized region, the system is far from being in equilibrium above the $ n = 2\,$ level in contrast to the recombining Universe. |
The partiion. stun $5,,7,|4Lip(Boltzmann 1s divereeut. | The partition sum $\sum_{nl} (2 l + 1) n_{nl}^{({\rm Boltzmann})} \,$ is divergent. |
As analysed w Πιο aud \Ghalas (1988) à uber of effects μπιτ the rauge of sununation. | As analysed by Hummer and Mihalas (1988) a number of effects limit the range of summation. |
In the considered telpcrature and deusitv range the action of free protous turus out to be the most important. the action of neutral atoms are nch sinaller. | In the considered temperature and density range the action of free protons turns out to be the most important, the action of neutral atoms are much smaller. |
The free protons «estrov the state n witha proability 1w,, where w,,=exp|{mylis2}|t with wv.=1075.9πο.15 (i, oerem ο). | The free protons destroy the state $n\,$ with a probability $ 1 - w_n\,$, where $w_n = \exp[\,-({n \over n_*})^{15/2}\,]\,$, with $n_* = 1075.9 \,{\rm cm}^{-2/5}\, n^{-2/15}_e\,$ $_e$ in $^{-3}$ ). |
This means that the recombined electrons become uubouud with 1te, probability. before they would beein to move towards the state. corresponding to tfrerimal equilibria. | This means that the recombined electrons become unbound with $1- w_n$ probability, before they would begin to move towards the state, corresponding to thermal equilibrium. |
The highly excited (7 larger than ~ ni) states are practically completely destroved. | The highly excited $n$ larger than $\sim n_*$ ) states are practically completely destroyed. |
The occuation nmuubers of the excited states in thermal equilibrium are: Owing to lis fact. the uuuber of electrons in bouud states is finite. axd thermia equilibrium between the »z2 bound states aud the contiuuua is possible. | The occupation numbers of the excited states in thermal equilibrium are: Owing to this fact, the number of electrons in bound states is finite, and thermal equilibrium between the $n \geq 2$ bound states and the continuum is possible. |
This approximation is good down to z z 300. | This approximation is good down to z $\approx \,$ 300. |
The »=2 levels freeze out between fre redshifts 300 aud 250. | The $n = 2 \,$ levels freeze out between the redshifts 300 and 250. |
The ground state of the lydrogen atom ds about 10.2 eV deeper thai the»)=2 level. | The ground state of the hydrogen atom is about 10.2 eV deeper than the $n = 2$ level. |
Cousequeutly. it’s occupatio ids ereater than the occupation of the excited states together:[ mi,29awitoHg. | Consequently, it's occupation is greater than the occupation of the excited states together: $ n_{1s} \gg \sum_{n \geq 2} n_n$. |
This meaus. that the density o: free. electrons plus the density of lydrogen atonis in ground state can be taken as equal to the total proton umber sa.|ys p. | This means, that the density of free electrons plus the density of hydrogen atoms in ground state can be taken as equal to the total proton number : $ n_e + n_{1s} = p\,$ . |
The totalbarvon number (determined by 05) is the sum of p aud the nuuuber of barvous iu the hela nuclei. | The totalbaryon number (determined by $\Omega_b$ ) is the sum of p and the number of baryons in the helium nuclei. |
of this part of work we really look for severe. departure from nonnormality and keep our discussion brief. | of this part of work we really look for severe departure from non–normality and keep our discussion brief. |
The assembly maps appear to have slightly more than expected points beyond the confidence region. | The assembly maps appear to have slightly more than expected points beyond the confidence region. |
Moreover. the two noncosmological [requeney maps show clear signs of normality. | Moreover, the two non–cosmological frequency maps show clear signs of non--normality. |
The two CALBonly maps also show signs of nonnormalitv- the POLL map result appearing to have departed furthest [rom normality. | The two CMB–only maps also show signs of non--normality- the TOH map result appearing to have departed furthest from normality. |
Our bivariate analysis results are shown in Figures 9--14.. | Our bivariate analysis results are shown in Figures \ref{fig:multi_skew1}- \ref{fig:multi_trans2}. |
The bivariate skewness statistics (0615/6) calculated. from he 16 maps are shown in Figures 9 ane 10.. | The bivariate skewness statistics $nb_{12}/6$ ) calculated from the 16 maps are shown in Figures \ref{fig:multi_skew1} and \ref{fig:multi_skew2}. |
The statistic appears to be abnormal for three of the assembly. maps- (QI. M2 and WM 4. | The statistic appears to be abnormal for three of the assembly maps- $Q$ 1, $W$ 2 and $W$ 4. |
Some of the other assembly maps have wo or three points outside the confidence region but visually the results do not look too unusual. | Some of the other assembly maps have two or three points outside the confidence region but visually the results do not look too unusual. |
Once again. the wo noncosmological frequencies have results that strongly indicate nonnormality. | Once again, the two non–cosmological frequencies have results that strongly indicate non–normality. |
Phis non.normality is still evident inthe Q and V band. | This non–normality is still evident in the $Q$ and $V$ band. |
The two CALB maps also have a higher han expected number of points above the confidence region. | The two CMB maps also have a higher than expected number of points above the confidence region. |
It. would appear that the bivariate skewness results ΠΟ their univariate counterparts. | It would appear that the bivariate skewness results mimic their univariate counterparts. |
The bivariate kurtosis results are cisplaved in Figures 1l and 12.. | The bivariate kurtosis results are displayed in Figures \ref{fig:multi_kurtosis1} and \ref{fig:multi_kurtosis2}. |
2s with the skewness results. bivariate kurtosis seem similar to their univariate equivalent. | As with the skewness results, bivariate kurtosis seem similar to their univariate equivalent. |
Nevertheless. in the case of the assembly maps. the shift. away from the expected value is even greater than for the univariate results. | Nevertheless, in the case of the assembly maps, the shift away from the expected value is even greater than for the univariate results. |
As before. the Galactic frequency maps are clearly found to be nonnormal. | As before, the Galactic frequency maps are clearly found to be non–normal. |
The higher value than expected. value of the bivariate kurtosis persists in the foreground:cleaned CMD maps The bivariate power transformation is shown in Figures 13. and l14.. | The higher value than expected value of the bivariate kurtosis persists in the foreground–cleaned CMB maps The bivariate power transformation is shown in Figures \ref{fig:multi_trans1} and \ref{fig:multi_trans2}. |
The assembly results do not look entirely consistent with being drawn [rom a X3 distribution. | The assembly results do not look entirely consistent with being drawn from a $\chi^2_2$ distribution. |
Five of the assembly maps produce. results with 5 or more points outside the confidence region. | Five of the assembly maps produce results with 5 or more points outside the confidence region. |
Saving that. our Gaussian MC map has four points outside this region. which makes it hard to draw definite conclusions. | Saying that, our Gaussian MC map has four points outside this region, which makes it hard to draw definite conclusions. |
This is certainly. not true for the two Galactic frequency bands that are clearly inconsistent. with normality. | This is certainly not true for the two Galactic frequency bands that are clearly inconsistent with normality. |
The two CAIBonly maps also appear to have an extremely high number. of volnts bevond the confidence region. | The two CMB–only maps also appear to have an extremely high number of points beyond the confidence region. |
o Lastly. in this subsection. we assess the linearity of the data. | Lastly, in this subsection, we assess the linearity of the data. |
We tried. adding separately three nonlinear terms to our linear model of the data (as described in 5)). | We tried adding separately three non–linear terms to our linear model of the data (as described in \ref{sec:implementation}) ). |
However. all | However, all |
satisfies equation (16)) with €z0.911. | satisfies equation \ref{pwnexp}) ) with $\bar C\simeq 0.911$. |
This derivation based on (vai) pressure balance at the immer aud outer edees of the pulsar wind nebula coufiiiis our earlier result obtained from overall energy. couservation. | This derivation based on (ram) pressure balance at the inner and outer edges of the pulsar wind nebula confirms our earlier result obtained from overall energy conservation. |
The conustaut wind Lhuuinositv assuniptfion is nof very realistic bv the time the effects of the reverse shock iud its associated reverberations have vanished. | The constant wind luminosity assumption is not very realistic by the time the effects of the reverse shock and its associated reverberations have vanished. |
The spin-down huninosity of the pulsar is more realistically described by the huninosity evolution frou a rotating magnetic dipole model: | The spin-down luminosity of the pulsar is more realistically described by the luminosity evolution from a rotating magnetic dipole model:. |
Therefore we now consider the more realistic case of a timce-dependeut huinositv given by (31)). | Therefore we now consider the more realistic case of a time-dependent luminosity given by \ref{pulsarlum}) ). |
The energy balance equation for the PWN reads: We solve this equation uunercallhv using a fourth-order Ruusc-Iutta method (e.g. Press et al. | The energy balance equation for the PWN reads: We solve this equation numerically using a fourth-order Runge-Kutta method (e.g. Press et al., |
1992). | 1992). |
As an initial condition we take the radius of the PWN equal to zero at the start of the evolution. neglecting the initial stage when the PWN is expanding supersonicallv. | As an initial condition we take the radius of the PWN equal to zero at the start of the evolution, neglecting the initial stage when the PWN is expanding supersonically. |
For the pressure f. we use the pressure at the center of the Sedov SNR (23)). | For the pressure $P_{\rm i}$, we use the pressure at the center of the Sedov SNR \ref{Shurel}) ). |
We fiud that the solution for Riya converges to Tax$85 on a time scale unich larger than the tvpical time scale for the reverse shock to hit the edee of the PWN. | We find that the solution for $R_{\rm pwn}$ converges to $R_{\rm pwn}\propto t^{0.3}$ on a time scale much larger than the typical time scale for the reverse shock to hit the edge of the PWN. |
Figure 9 shows his seuni-analvtical result together with results from lyvdrodvuamical simulations. | Figure 9 shows this semi-analytical result together with results from hydrodynamical simulations. |
For the senu-analytical equation woe Use 54=5/3. because the lvdrodvuanues code also uses this value (see section L1 below). | For the semi-analytical equation we use $\gamma_{\rm pwn} =5/3$, because the hydrodynamics code also uses this value (see section 4.1 below). |
Our simulations were performed using the Versatile Advection Code (VAC. Tótth 1996) which can inteerate the equations of gas dynaudes in a conservative fori in 1l. 2 or 3 climeusions. | Our simulations were performed using the Versatile Advection Code (VAC, Tótth 1996) which can integrate the equations of gas dynamics in a conservative form in 1, 2 or 3 dimensions. |
We used the TVD-MUSCL scheme with a Roc-tvpe approximate Riemiaun solver from the nunerical aleorithins available in VAC (Totth aud Odstréil. 1996): a discussion of this aud other schemes for merical wdrodvuamucs can be found in LeVeque (1998). | We used the TVD-MUSCL scheme with a Roe-type approximate Riemann solver from the numerical algorithms available in VAC (Tótth and Odstrčiil, 1996); a discussion of this and other schemes for numerical hydrodynamics can be found in LeVeque (1998). |
Iu this paper our caleulatiouns are limited to spherically sviunietric flows. | In this paper our calculations are limited to spherically symmetric flows. |
We use a uniform grid with a erid spacing chosen sufficiently fine to resolve both the shocks inside the PWN aud the larecr-scale shocks associated with the SNR. | We use a uniform grid with a grid spacing chosen sufficiently fine to resolve both the shocks inside the PWN and the larger-scale shocks associated with the SNR. |
Table 1 gives the plivsical scale associated with the erid size for the simulations prescutec here. | Table 1 gives the physical scale associated with the grid size for the simulations presented here. |
An expanding SNR is created by impulsively releasing the mechanical enerev of he SN explosion in the first few erid cells. | An expanding SNR is created by impulsively releasing the mechanical energy of the SN explosion in the first few grid cells. |
The thermal energy aud mass deposited. there lead to freely expanding ejecta with a nearly uniforii density. aud a huear velocity profile as a function of radius. | The thermal energy and mass deposited there lead to freely expanding ejecta with a nearly uniform density, and a linear velocity profile as a function of radius. |
single integer index i that labels the specific splitting (i=l,-,N.= 14). | single integer index $i$ that labels the specific splitting $i= 1,
\cdots, N_{\rm s}\equiv 14$ ). |
Each term of the sum is weighted with the inverse square of the standard uncertainty (c;) of the observed splittings, which are derived from the uncertainties in the frequencies given in Fu et al. ( | Each term of the sum is weighted with the inverse square of the standard uncertainty $\sigma_i$ ) of the observed splittings, which are derived from the uncertainties in the frequencies given in Fu et al. ( |
2007) and are shown in the last column of Table 2.. | 2007) and are shown in the last column of Table \ref{table2}. |
This is at variance with the preliminar study of Córrsico Althaus (2010), in which the fits of the rotational splitting were made without weighting the terms of the sum, and so, the impact of the different uncertainties of the observational data on the final result was neglected. | This is at variance with the preliminar study of Córrsico Althaus (2010), in which the fits of the rotational splitting were made without weighting the terms of the sum, and so, the impact of the different uncertainties of the observational data on the final result was neglected. |
The lower the value of x?, the better the match between the theoretical and the observed frequency splittings. | The lower the value of $\chi^2$, the better the match between the theoretical and the observed frequency splittings. |
'The theoretical rotational splittings are computed using the expressions resulting from the perturbative theory to first order in €) (the rotation rate) that assumes that the pulsating star rotates with a period (P= 1/Q) much longer than any of its pulsation periods (Unno et al. | The theoretical rotational splittings are computed using the expressions resulting from the perturbative theory to first order in $\Omega$ (the rotation rate) that assumes that the pulsating star rotates with a period $P \equiv 1/\Omega$ ) much longer than any of its pulsation periods (Unno et al. |
1989). | 1989). |
Under the assumption of rigid rotation (Q constant), the theoretical frequency splittings are given by: with m=0,+1,...,+¢, and Cy, being coefficients that depend on the eigenfunctions of the pulsation mode obtained in the non-rotating case. | Under the assumption of rigid rotation $\Omega$ constant), the theoretical frequency splittings are given by: with $m= 0, \pm 1, \ldots, \pm \ell$, and $C_{k \ell}$ being coefficients that depend on the eigenfunctions of the pulsation mode obtained in the non-rotating case. |
Such coefficients are computed as (Unno et al. | Such coefficients are computed as (Unno et al. |
1989): where £, and & are the unperturbed radial and tangential eigenfunctions, respectively. | 1989): where $\xi_r$ and $\xi_t$ are the unperturbed radial and tangential eigenfunctions, respectively. |
In the case of g-modes, when k is large then £,.<&, in such a way that Cy;>1/£(£+1) (Brickhill 1975). | In the case of $g$ -modes, when $k$ is large then $\xi_r \ll \xi_t$, in such a way that $C_{k
\ell} \rightarrow 1 / \ell(\ell+1)$ (Brickhill 1975). |
If the condition of rigid body rotation is relaxed and (spherically symmetric) differential rotation is assumed, 2= Q(r), the frequency splittings are given by (Unno et al. | If the condition of rigid body rotation is relaxed and (spherically symmetric) differential rotation is assumed, $\Omega= \Omega(r)$ , the frequency splittings are given by (Unno et al. |
1989): Γκι) being the first-order rotation kernels computed from the rotationally unperturbed eigenfunctions as (Unno et al. | 1989): $K_{k\ell}(r)$ being the first-order rotation kernels computed from the rotationally unperturbed eigenfunctions as (Unno et al. |
1989): From Eq. (4)) | 1989): From Eq. \ref{rota-diff}) ) |
it is clear that the frequency splitting for a given mode is just a weighted average of the rotation rate Q(r) throughout the star, being the rotation kernel Kxe¢(r) precisely the weighting function. | it is clear that the frequency splitting for a given mode is just a weighted average of the rotation rate $\Omega(r)$ throughout the star, being the rotation kernel $K_{k
\ell}(r)$ precisely the weighting function. |
Note that the perturbative theory to first order in Ω predicts symmetric separations of the m40 components within each multiplet with respect to the central one 0) (see Eqs. (2)) | Note that the perturbative theory to first order in $\Omega$ predicts symmetric separations of the $m \neq 0$ components within each multiplet with respect to the central one $m= 0$ ) (see Eqs. \ref{rota-rigid}) ) |
and (4))). | and \ref{rota-diff}) )). |
Therefore, in this work we are neglecting the departures from symmetric frequency splitting within the triplets centered at II~560 s and II~610 s exhibited by | Therefore, in this work we are neglecting the departures from symmetric frequency splitting within the triplets centered at $\Pi \sim 560$ s and $\Pi
\sim 610$ s exhibited by |
operated by these agencies in co-operation with ESA and NSC (Norway). | operated by these agencies in co-operation with ESA and NSC (Norway). |
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