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In this case we can detect the electron flux from Vela pulsar with such a sharp spectral cutoff | In this case we can detect the electron flux from Vela pulsar with such a sharp spectral cutoff |
is >107!M... the spectrum cannot be distinguished from a pure hydrogen atmosphere in the Chandra band. | is $>10^{-11}$, the spectrum cannot be distinguished from a pure hydrogen atmosphere in the Chandra band. |
However in the case of an intermediate. thickness hydrogen layer of ~107 tthere is a significant effect in the Chandra band (Fig. 2)). | However in the case of an intermediate thickness hydrogen layer of $\sim 10^{-12}$ there is a significant effect in the Chandra band (Fig. \ref{fig:hlaag}) ). |
For the lowest allowed hydrogen column (2x107M... see below). there is a 70 flux reduction around 55A.. diminishing rapidly for larger columns. while at shorter wavelengths the flux is higher. peaking for 4x107 aand then diminishing rapidly in the pure hydrogen limit. | For the lowest allowed hydrogen column $2\times 10^{-13}$, see below), there is a $\sim$ 70 flux reduction around 55, diminishing rapidly for larger columns, while at shorter wavelengths the flux is higher, peaking for $4\times 10^{-13}$ and then diminishing rapidly in the pure hydrogen limit. |
For this reason. we carefully reconsider the constraints to the hydrogen layer. | For this reason, we carefully reconsider the constraints to the hydrogen layer. |
From the EUVE spectrum published by (their Fig. | From the EUVE spectrum published by \citet{barstow1995} (their Fig. |
2). we estimated the equivalent widths of the Lye. B and y absorption lines (Table 4)). | 2), we estimated the equivalent widths of the $\alpha$, $\beta$ and $\gamma$ absorption lines (Table \ref{tab:hzhelines}) ). |
We compared these equivalent widths with our model calculations. | We compared these equivalent widths with our model calculations. |
None of them can be regarded as a detection. | None of them can be regarded as a detection. |
Combining all three lines. we obtain a best fit hydrogen mass M=2-4I07M... | Combining all three lines, we obtain a best fit hydrogen mass $M=2-4\times 10^{-13}$. |
This mass predicts equivalent widths for the Lye. B and y lines of 0.02. 0.04 and 0.02Α.. respectively. | This mass predicts equivalent widths for the $\alpha$, $\beta$ and $\gamma$ lines of 0.02, 0.04 and 0.02, respectively. |
The 99 confidence lower limit is L4bx107?M... | The 99 confidence lower limit is $\times 10^{-13}$. |
While the lower limit is rather strict. we cannot fully exclude that the hydrogen layer is very thick (which would make the model effectively a pure hydrogen model): the pure hydrogen limit i5 at the 90 confidence upper limit. | While the lower limit is rather strict, we cannot fully exclude that the hydrogen layer is very thick (which would make the model effectively a pure hydrogen model): the pure hydrogen limit is at the 90 confidence upper limit. |
Another constraint on the thickness of the hydrogen layer is obtained from the Lyman limit edge. | Another constraint on the thickness of the hydrogen layer is obtained from the Lyman limit edge. |
Effectively. the edge is slightly shifted due to blending with the higher Lyman series lines and occurs near 230A. | Effectively, the edge is slightly shifted due to blending with the higher Lyman series lines and occurs near 230. |
. We find from the EUVE spectrum that the edge is invisible. corresponding to an optical depth of less than 2%. | We find from the EUVE spectrum that the edge is invisible, corresponding to an optical depth of less than 2. |
. This corresponds to a lower limit to the hydrogen mass of 2.2x107M... | This corresponds to a lower limit to the hydrogen mass of $2.2\times 10^{-13}$. |
We conclude that the thickness of the hydrogen layer in HZ 43 is most likely between 2—4x107M... with lower values excluded but with no solid upper limit to the hydrogen mass. | We conclude that the thickness of the hydrogen layer in HZ 43 is most likely between $\times 10^{-13}$, with lower values excluded but with no solid upper limit to the hydrogen mass. |
Given all this. we consider only stratified models for HZ 43A with hydrogen mass »2x107? as argued before. a pure hydrogen atmosphere is a limiting case of this set of models. | Given all this, we consider only stratified models for HZ 43A with hydrogen mass $>2\times 10^{-13}$; as argued before, a pure hydrogen atmosphere is a limiting case of this set of models. |
Our best fit model | (short cut-off) has a y of 56.17(52.50). our best fit model 2 (long cut-off) has y=49.41(46.33). | Our best fit model 1 (short cut-off) has a $\chi^2$ of $56.17\,(52.50)$, our best fit model 2 (long cut-off) has $\chi^2 = 49.41\,(46.33)$. |
The numbers in brackets denote the contribution of the Chandra data only. | The numbers in brackets denote the contribution of the Chandra data only. |
With 25 data points and 12 adjustable parameters the number of degrees of freedom would be 13. and hence the value of y is slightly enhanced with respect to purely statistical noise. | With 25 data points and 12 adjustable parameters the number of degrees of freedom would be 13, and hence the value of $\chi^2$ is slightly enhanced with respect to purely statistical noise. |
However. the actual number of degrees of freedom ts higher. as several parameters are strongly correlated and the best fit is rather insensitive to others (such as the helium abundance in Sirius B. and in general the interstellar absorption column densities). | However, the actual number of degrees of freedom is higher, as several parameters are strongly correlated and the best fit is rather insensitive to others (such as the helium abundance in Sirius B, and in general the interstellar absorption column densities). |
Moreover. our additional constraints. also effectively limit the number of degrees of freedom. | Moreover, our additional constraints also effectively limit the number of degrees of freedom. |
Although hard to estimate exactly. the true number of degrees of freedom may be of the order of 20. | Although hard to estimate exactly, the true number of degrees of freedom may be of the order of 20. |
Fig. | Fig. |
| shows the observed ratio of the LETGS spectra of Sirius B to HZ 43A. together with the best fit models 1 and 2. | \ref{fig:ratcomp} shows the observed ratio of the LETGS spectra of Sirius B to HZ 43A, together with the best fit models 1 and 2. |
From this figure it is clear that there 1s some additional systematic scatter in the data points (as the models in the 50-170 wwavelength range are. as expected. rather smooth). | From this figure it is clear that there is some additional systematic scatter in the data points (as the models in the $50-170$ wavelength range are, as expected, rather smooth). |
For instance, the data point at 160 ddeviates by 43.00 or 43.7 %. | For instance, the data point at 160 deviates by $+3.0\sigma$ or $+3.7$ . |
. In this case. some of the systematic effect may be due to the fact that this wavelength is close to the edge of the spectrum in the —1 spectral order (the physical edge of the detector). | In this case, some of the systematic effect may be due to the fact that this wavelength is close to the edge of the spectrum in the $-1$ spectral order (the physical edge of the detector). |
For other data points. the relative deviations are smaller or less significant. | For other data points, the relative deviations are smaller or less significant. |
By adding a systematic uncertainty of only ] or 2 to our ratios. the v for the best fit model 2 would reduce from 49.41 to 29 or 16. respectively. ie. in the acceptable range given the - 20 degrees of freedom. | By adding a systematic uncertainty of only 1 or 2 to our ratios, the $\chi^2$ for the best fit model 2 would reduce from 49.41 to 29 or 16, respectively, i.e. in the acceptable range given the $\sim$ 20 degrees of freedom. |
This reduction by a factor of ~2 in y then suggests that we should use Ay=2.0 instead of Ay=1.0 for the original fits without systematic uncertainties. in order to determine the Ic confidence limits on the parameters. | This reduction by a factor of $\sim$ 2 in $\chi^2$ then suggests that we should use $\Delta\chi^2=2.0$ instead of $\Delta\chi^2=1.0$ for the original fits without systematic uncertainties, in order to determine the $1\sigma$ confidence limits on the parameters. |
We list the best-fit parameters in Table 5.. and the spectrum at a few selected wavelength in Table 6.. | We list the best-fit parameters in Table \ref{tab:bestpar}, and the spectrum at a few selected wavelength in Table \ref{tab:modelfluxes}. |
The absorbed spectrum of HZ 43 is represented with an accuracy of better than 0.5 over the full 43-180 wwavelength range by for model 1 and for model 2 with an accuracy better than 0.7 by Note that (5))-(6)) should not be used outside this range. | The absorbed spectrum of HZ 43 is represented with an accuracy of better than 0.5 over the full 43–180 wavelength range by for model 1 and for model 2 with an accuracy better than 0.7 by Note that \ref{eqn:hz43unconstrained1}) $-$ \ref{eqn:hz43unconstrained2}) ) should not be used outside this range. |
Our lower limit to the hydrogen mass of 2.5 to 3x107 iis an order of magnitude higher than the lower limit derived by ? based on the EUVE continuum. | Our lower limit to the hydrogen mass of $2.5$ to $3\times 10^{-12}$ is an order of magnitude higher than the lower limit derived by \citet{barstow1995}
based on the EUVE continuum. |
We could derive this tighter limit because the LETGS covers also shorter wavelengths. for which the continuum is very sensitive to the hydrogen thickness (Fig. 2)). | We could derive this tighter limit because the LETGS covers also shorter wavelengths, for which the continuum is very sensitive to the hydrogen thickness (Fig. \ref{fig:hlaag}) ). |
Our best-fit model is indistinguishable from a pure hydrogen model. and evenfor our lower limit hydrogen mass. above 50 tthe differences with a pure hydrogen model are less than a few percent. | Our best-fit model is indistinguishable from a pure hydrogen model, and evenfor our lower limit hydrogen mass, above 50 the differences with a pure hydrogen model are less than a few percent. |
The direct. estimation of στ) breaks down when the low-frequeney variance in the timing noise is dominant. | The direct estimation of $c(\tau)$ breaks down when the low-frequency variance in the timing noise is dominant. |
In this case the residuals show a slow variation that substantially. exceeds the error bars. as shown in Figures l(a) and 4(a). | In this case the residuals show a slow variation that substantially exceeds the error bars, as shown in Figures 1(a) and 4(a). |
In most such cases the power spectrum has the form PCf)=Af“ where azc 2. | In most such cases the power spectrum has the form $P(f) = A f^{-\alpha}$ where $\alpha > 2$ . |
lt is hard to estimate e(7) because there are [ον degrees of freedom. (e. 702 Lins). | It is hard to estimate $c(\tau)$ because there are few degrees of freedom (i.e., $\tau_0 \approx T_{obs}$ ). |
However it is usually possible to make a power-Iaw. model of the spectrum and to specify the aniplitucle of that power law with reasonable accuracy. | However it is usually possible to make a power-law model of the spectrum and to specify the amplitude of that power law with reasonable accuracy. |
This is because cach spectral estimate for which PCf) is greater than the white noise. can provide two degree of freedom. | This is because each spectral estimate for which $P(f)$ is greater than the white noise, can provide two degree of freedom. |
As noted earlier. one must use a spectral estimator which provides independent estimates of P(f). | As noted earlier, one must use a spectral estimator which provides independent estimates of $P(f)$. |
Steep red. processes require whitening. but we do not vet have the covariance matrix so we have to obtain a spectral estimate iteratively. | Steep red processes require whitening, but we do not yet have the covariance matrix so we have to obtain a spectral estimate iteratively. |
We start by low-pass filtering the residuals to separate the red and white components. | We start by low-pass filtering the residuals to separate the red and white components. |
The resulting red component can be interpolated on to a regular erid without much distortion because it is quite smooth after the low-pass filtering. | The resulting red component can be interpolated on to a regular grid without much distortion because it is quite smooth after the low-pass filtering. |
We can then pre-whiten it with a first dilferenee process. compute the ΠΡΙ and post-darken the result. | We can then pre-whiten it with a first difference process, compute the $|{\rm DFT}|^2$, and post-darken the result. |
This generally gives an adequateΕΙ. "first guess" at the power spectrum of the red component. | This generally gives an adequate “first guess” at the power spectrum of the red component. |
We estimate the white component by subtracting the red. component [rom the original residuals. | We estimate the white component by subtracting the red component from the original residuals. |
We find the speetrum of the white component with the Z-Ix weighted. least squares estinate. | We find the spectrum of the white component with the Z-K weighted least squares estimate. |
The next step is to fit a power-lav model of the form WO)=ALOfefPYF to the rec spectral estimate [or the frequency range below the frequency at which the red and white spectra cross over. | The next step is to fit a power-law model of the form $P_m(f) = A/(1 + (f/f_c)^2)^{\alpha/2}$ to the red spectral estimate for the frequency range below the frequency at which the red and white spectra cross over. |
We find that the "corner [requencev f; should not he fi.«1/2755; because even if the red noise is à pure power law. fitting v and £ will llatten the spectrum. of the resicluals below this [requeney. | We find that the “corner frequency” $f_c$ should not be $f_c < 1/T_{obs}$ because even if the red noise is a pure power law, fitting $\nu$ and $\dot{\nu}$ will flatten the spectrum of the residuals below this frequency. |
. We then compute e(7) by Fourier transformation of ο) and finally obtain the covariance matrix as cliscussec earlier. | We then compute $c(\tau)$ by Fourier transformation of $P_m(f)$ and finally obtain the covariance matrix as discussed earlier. |
This covariance matrix is used to re-estimate the power spectrum using the Cholesky least squares procedure. | This covariance matrix is used to re-estimate the power spectrum using the Cholesky least squares procedure. |
This spectrum is used to revise the model P,,(f). à new e(z) is found. a new covariance matrix. and a new Cholesky estimate of the power spectrum. | This spectrum is used to revise the model $P_m(f)$, a new $c(\tau)$ is found, a new covariance matrix, and a new Cholesky estimate of the power spectrum. |
At this point. the power-law mocel. the covariance matrix and the power spectral estimate are self-consistent. | At this point, the power-law model, the covariance matrix and the power spectral estimate are self-consistent. |
We then check to see that the whitened residuals look white. by computing their power spectrum using an OLS procedure (because they should. be both white and normalized). | We then check to see that the whitened residuals look white, by computing their power spectrum using an OLS procedure (because they should be both white and normalized). |
These steps are illustrated in Figure 4 for the pulsar J1539 5626. | These steps are illustrated in Figure 4 for the pulsar $-$ 5626. |
"Phe residuals obtained from the Parkes analogue filterbank (Manchester ct al. | The residuals obtained from the Parkes analogue filterbank (Manchester et al. |
are shown in the top panel (a). | are shown in the top panel (a). |
In the second. panel (b) weshow the [DET power spectrum of the red. component. as a jagged solid. line obtained. by first. cillerence pre-whitening ancl post-darkening. | In the second panel (b) weshow the $|DFT|^2$ power spectrum of the red component as a jagged solid line obtained by first difference pre-whitening and post-darkening. |
The power-law model £f) is shown as à smooth solid line. and the WLS spectrum of the white component is shown dotted. | The power-law model $P_m(f)$ is shown as a smooth solid line, and the WLS spectrum of the white component is shown dotted. |
In the third panel (ο) we show the the Cholesky spectrum as a jagged solid line and the final model ο) as a smooth solid line. | In the third panel (c) we show the the Cholesky spectrum as a jagged solid line and the final model $P_m(f)$ as a smooth solid line. |
Phe original model is shown as a dashed line. and the WLS spectrum of the white component is shown exactly as in panel (b) for comparison. | The original model is shown as a dashed line, and the WLS spectrum of the white component is shown exactly as in panel (b) for comparison. |
One expects to see the Cholesky spectrum merge into the spectrum of the white component and this does in fact occur. | One expects to see the Cholesky spectrum merge into the spectrum of the white component and this does in fact occur. |
Finally in the lowest. panel (d) we show the OLS spectrum of Ry. the €‘holesky-transformed residuals. | Finally in the lowest panel (d) we show the OLS spectrum of $R_{W}$, the Cholesky-transformed residuals. |
The mean ancl confidence limits for a unit. variance white spectrum are shown as horizontal dashed lines. | The mean and confidence limits for a unit variance white spectrum are shown as horizontal dashed lines. |
One can see that the transformed residuals are in fact quite consistent with white nolse. | One can see that the transformed residuals are in fact quite consistent with white noise. |
We have provided options in to perform this iteration with no cdillerencing. first-order dilflerencing or second-order dillerencing. | We have provided options in to perform this iteration with no differencing, first-order differencing or second-order differencing. |
Although the clilferencing is only used to get a first-guess of the spectral model. we prefer to use the least order of dilferencing if the spectra are similar with two different orders. | Although the differencing is only used to get a first-guess of the spectral model, we prefer to use the least order of differencing if the spectra are similar with two different orders. |
For example PSR. 5626 can be analyzecl either with no differencing or first-order differencing. the results are similar. | For example PSR $-$ 5626 can be analyzed either with no differencing or first-order differencing, the results are similar. |
By comparison the analysis of PSR | 2134 shown in Figure L. requires first or second-orcder dillerencing. | By comparison the analysis of PSR $+$ 2134, shown in Figure 1, requires first or second-order differencing. |
Lf the exponent of the spectral Ποο) was steeper by more than a few tenths with higher order dillerencing. we would choose the higher order. | If the exponent of the spectral model was steeper by more than a few tenths with a higher order differencing, we would choose the higher order. |
However it is always necessary low-pass filter the residuals and interpolate then onto a regular grid. | However it is always necessary to low-pass filter the residuals and interpolate them onto a regular grid. |
We perform the low-pass filter by convolving the residuals with a weighted exponential smoothing function of the form expt|[/7.])with a une constant z,zc20 days. | We perform the low-pass filter by convolving the residuals with a weighted exponential smoothing function of the form $\exp(-|t/\tau_s|)$with a time constant $\tau_s \approx 20$ days. |
The low pass filter response isd0.25 at f=ni(2z7,) | The low pass filter response is 0.25 at $f = (2\pi\tau_s)^{-1}$ . |
The smoothing time can be changed should adusted.. so that. the | The smoothing time can be changed and should be adjusted so that the |
will find a few background stars within that distance. | will find a few background stars within that distance. |
This limit is also comparable to the scale size of major nebular structures seen in the NIR images. | This limit is also comparable to the scale size of major nebular structures seen in the NIR images. |
Each sample of 20 (or less) background stars is averaged using an iterative sigma-clipping routine, deriving a robust estimate of both the mean local extinction Áy and standard deviation σαν. | Each sample of 20 (or less) background stars is averaged using an iterative sigma-clipping routine, deriving a robust estimate of both the mean local extinction $\bar{A_V}$ and standard deviation $\sigma_{\bar{A_V}}$. |
This would represent the solution to our problem, except that we want to make sure that the observed surface density and brightness distribution of the background sample is still compatible with the model. | This would represent the solution to our problem, except that we want to make sure that the observed surface density and brightness distribution of the background sample is still compatible with the model. |
To this purpose, we apply the Ay extinction to the synthetic galactic model and compare the local density of the reddened model population to the observed one. | To this purpose, we apply the $\bar{A_V}$ extinction to the synthetic galactic model and compare the local density of the reddened model population to the observed one. |
The predicted density of the reddened galactic population has to be adjusted for the completeness of the survey, which statistically accounts for the success/failure of detection based on the magnitude of both stars and local background. | The predicted density of the reddened galactic population has to be adjusted for the completeness of the survey, which statistically accounts for the success/failure of detection based on the magnitude of both stars and local background. |
In R10 we computed the completeness levels in three concentric regions centered on 6!Ori-C, neglecting any variation inside each region. | In R10 we computed the completeness levels in three concentric regions centered on $\theta^1$ Ori-C, neglecting any variation inside each region. |
For this study we improve the completeness estimates using a locally-computed completeness level, as described in Appendix??:: for each grid point in the extinction map, our new approach allows us to compute the appropriate completeness adjustment. | For this study we improve the completeness estimates using a locally-computed completeness level, as described in Appendix: for each grid point in the extinction map, our new approach allows us to compute the appropriate completeness adjustment. |
It turns out that, particularly within a few arcminutes from the Trapezium cluster, the number of predicted background stars is lower than the number of observed ones. | It turns out that, particularly within a few arcminutes from the Trapezium cluster, the number of predicted background stars is lower than the number of observed ones. |
The excess sources are, most probably, ONC members wrongly identified as background stars, having photometry matching our background selection criteria. | The excess sources are, most probably, ONC members wrongly identified as background stars, having photometry matching our background selection criteria. |
To statistically reject these extra sources, for each group of 20 stars we discard the background star with the lowest extinction Ay;,;;, assuming it is a cluster member, and replace it with a new, further background star with Ay>Aymin, if existing (we are still constrained by the 1,000 pixels maximum distance). | To statistically reject these extra sources, for each group of 20 stars we discard the background star with the lowest extinction $A_{V,min}$, assuming it is a cluster member, and replace it with a new, further background star with $A_V>A_{V,min}$, if existing (we are still constrained by the 1,000 pixels maximum distance). |
By decreasing the density of background sources, we obtain an extinction map which matches both the typical reddening and source density of the galactic component, averaged over5’,, or less. | By decreasing the density of background sources, we obtain an extinction map which matches both the typical reddening and source density of the galactic component, averaged over, or less. |
The result is shown in Fig.4, | The result is shown in Fig., |
where we have interpolated the discrete gridding of the map using the IDL routine. | where we have interpolated the discrete gridding of the map using the IDL routine. |
The angular resolution ranges between aand5', depending on the density of background sources (Fig. 2)). | The angular resolution ranges between and, depending on the density of background sources (Fig. ). |
In correspondence of the highest extinction values the map reaches its worst resolution, as expected. | In correspondence of the highest extinction values the map reaches its worst resolution, as expected. |
The error map shown in Fig. | The error map shown in Fig. |
shows a typical uncertainty Ay xl. | shows a typical uncertainty $A_V\lesssim$ 1. |
The regularity in this map is due to the fact that for each grid point, apart the Trapezium region, the algorithm collects the requested 20 stars within a distance shorter than the limit of5’. | The regularity in this map is due to the fact that for each grid point, apart the Trapezium region, the algorithm collects the requested 20 stars within a distance shorter than the limit of. |
. It follows that the number of stars in the analyzed subsample (corresponding to the given grid point) does not depend on the stellar density, but on the combination of the density and the angular resolution of the extinction map. | It follows that the number of stars in the analyzed subsample (corresponding to the given grid point) does not depend on the stellar density, but on the combination of the density and the angular resolution of the extinction map. |
In deriving our extinction map we have assumed that the foreground contamination is closely modeled by the Besangoon synthetic galactic population. | In deriving our extinction map we have assumed that the foreground contamination is closely modeled by the Besançoon synthetic galactic population. |
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