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Using our simplified geometry. depicted in I. the momentum imparted on each state is where B,=B,(v.z) (note that dt=dz in the adopted units). | Using our simplified geometry, depicted in figure 1, the momentum imparted on each state is where $B_x=B_x(y,z)$ (note that $dt=dz$ in the adopted units). |
Clearly. each of the beams will be affected in a similar way while gaining opposite momenta so that the total momentum 1s zero and the classical wave packet travels in a straight line (along the z-axis). | Clearly, each of the beams will be affected in a similar way while gaining opposite momenta so that the total momentum is zero and the classical wave packet travels in a straight line (along the $z$ -axis). |
(seeThis effect is analogous to the Stern-Gerlach experiment Fig. | This effect is analogous to the Stern-Gerlach experiment (see Fig. |
1). | 1). |
Beam splitting effects. arising due to photon-particle (polariton) mixing. have been measured in the laboratory (Karpa Weitz 2006) and that an analogy exists between this case and scalar QED (see appendix). | Beam splitting effects, arising due to photon-particle (polariton) mixing, have been measured in the laboratory (Karpa Weitz 2006) and that an analogy exists between this case and scalar QED (see appendix). |
In the limit 2,—I. the separation angle between the beams is where p is the beam momentum along the propagation direction. 1.e.. the z-axis. | In the limit $n_\pm \simeq 1$, the separation angle between the beams is where $p$ is the beam momentum along the propagation direction, i.e., the $z$ -axis. |
This expression holds for small splitting angles and assumes relativistic axions. | This expression holds for small splitting angles and assumes relativistic axions. |
We also approximated. [dz(0B,/Ov)=fGB| where fG(N) is a geometrical factor depending on the magnetic field geometry the inclination of our line-of-sight through the magnetized region (e.g.. for pulsars and magnetars the magnetic field is predominantly dipolar and fc«1: see 833). and on the photon polarization. | We also approximated $\int dz (\partial B_x /\partial y)= f_G B_\|$ where $f_G(N)$ is a geometrical factor depending on the magnetic field geometry, the inclination of our line-of-sight through the magnetized region (e.g., for pulsars and magnetars the magnetic field is predominantly dipolar and $f_G <1$; see 3), and on the photon polarization. |
An implicit assumption in the above expression is that the field is monotonically increasing or decreasing. | An implicit assumption in the above expression is that the field is monotonically increasing or decreasing. |
In situations where the magnetic field is stochastic. Op, can no longer be evaluated according to equation 6 which ts linear with distance (or time) and a better treatment is that of a random walk nature whereby the average 0. (0)e Vr; such cases are beyond the scope of this paper and are likely to be less relevant to compact astrophysical objects whose magnetic fields are thought to be relatively ordered. | In situations where the magnetic field is stochastic, $\delta p_y$ can no longer be evaluated according to equation \ref{dpy} which is linear with distance (or time) and a better treatment is that of a random walk nature whereby the average $\theta$, $\left < \theta \right > \propto \sqrt{t}$ ; such cases are beyond the scope of this paper and are likely to be less relevant to compact astrophysical objects whose magnetic fields are thought to be relatively ordered. |
Assuming splitting angles 0~10770. rrad are detectable at radio wavelengths (see $33). then the 5minimum coupling constant which can be probed. provided equation 4 holds. It is important to establish that the above condition on the photon and axion mass can be materialized under realistic conditions. | Assuming splitting angles $\theta \sim 10^{-2}\theta_{-2}$ rad are detectable at radio wavelengths (see 3), then the minimum coupling constant which can be probed, provided equation \ref{cond2} holds, is It is important to establish that the above condition on the photon and axion mass can be materialized under realistic conditions. |
Consider the non-resonant case: as the mass of ALPs is unknown. there is no reason to suspect such a condition is irrelevant. | Consider the non-resonant case: as the mass of ALPs is unknown, there is no reason to suspect such a condition is irrelevant. |
As for the photon mass. there are two contributions to the refractiveindex. of the photon in magnetized plasma: vacuum birefringence and a plasma term (e.g.. C09). | As for the photon mass, there are two contributions to the refractiveindex of the photon in magnetized plasma: vacuum birefringence and a plasma term (e.g., C09). |
The condition for the plasma term to be negligible is The fact we observe a signal at photon energies Q. means that that@met>> | The condition for the plasma term to be negligible is The fact that we observe a signal at photon energies $\omega$, means that $\omega \gg \omega_p$. |
Taking fg0.1 and 0—107? trad. this condition Is if (9,.o0,>5 (see 833). | Taking $f_G\sim 0.1$ and $\theta\sim 10^{-2}$ rad, this condition is met if $\omega/\omega_p > 5$ (see 3). |
We shall therefore neglect plasma effects in our analysis. | We shall therefore neglect plasma effects in our analysis. |
A more restrictive condition is associated with the vacuum birefringence term which was calculated by Adler (1971) and used in the context of photon-particle oscillations by C09.In particular. if the refractive index rn=I|n then we demand that For 0=10-7 rad. fc;=0.1. and G4EP—0.05 (C09) we get that the above ratio is of of order unity. | A more restrictive condition is associated with the vacuum birefringence term which was calculated by Adler (1971) and used in the context of photon-particle oscillations by C09.In particular, if the refractive index $n=1+\delta n_\|^{\rm QED}$ then we demand that For $\theta=10^{-2}$ rad, $f_G=0.1$, and $\delta n^{\rm QED}\sim 0.05$ (C09) we get that the above ratio is of of order unity. |
While this | While this |
exceed of the speed of light. in which case Newtonian fluid dynamies ts still a good approximation. | exceed of the speed of light, in which case Newtonian fluid dynamics is still a good approximation. |
A kicked neutron star will move away from the grid center but will remain the center of the neutrino-driven wind. | A kicked neutron star will move away from the grid center but will remain the center of the neutrino-driven wind. |
The neutron star motion may thus lead to a distortion of the wind- interaction and may change the location of the reverse shock in the wind and thus its influece on the wind properties. | The neutron star motion may thus lead to a distortion of the wind-ejecta interaction and may change the location of the reverse shock in the wind and thus its influence on the wind properties. |
Such influence on the detailed evolution of a particular model was indeed seen by Schecketal.(2006).. who performed runs (for fixed progenitor. boundary. coditions. and. perturbation seeds) without and with applying a Galilei transformation on the flow around a neutron star at the grid center by giving the surrounding fluid à coherent motior with the negative sign of the neutron star kick velocity. | Such influence on the detailed evolution of a particular model was indeed seen by \cite{Scheck.Kifonidis.Janka.Mueller:2006}, who performed runs (for fixed progenitor, boundary conditions, and perturbation seeds) without and with applying a Galilei transformation on the flow around a neutron star at the grid center by giving the surrounding fluid a coherent motion with the negative sign of the neutron star kick velocity. |
This procedure was assumed to capture the main effects of the neutron star movement. | This procedure was assumed to capture the main effects of the neutron star movement. |
Although such effects can be relevant for determining the detailed evolution and ejecta properties of a particular model star (which was not the goal of this work) they are unlikely to change the basic and general conclusions that we have drawn from our simulations. | Although such effects can be relevant for determining the detailed evolution and ejecta properties of a particular model star (which was not the goal of this work) they are unlikely to change the basic and general conclusions that we have drawn from our simulations. |
Since the neutron-star velocity vy. is typically small compared to the sound speed and the terminal wind velocity. vy,«ος and vy,«yr>Κιν). at is hard to imagine that allowing the neutron star to move (instead of anchoring it at the grid center as in our models) will affect the overall picture that we have obtained for the antsotropie wind-ejecta interaction. | Since the neutron-star velocity $v_\mathrm{ns}$ is typically small compared to the sound speed and the terminal wind velocity, $v_\mathrm{ns} \ll c_\mathrm{s}$ and $v_\mathrm{ns} \ll u_\mathrm{w}(r \gg R_\mathrm{ns})$, it is hard to imagine that allowing the neutron star to move (instead of anchoring it at the grid center as in our models) will affect the overall picture that we have obtained for the anisotropic wind-ejecta interaction. |
Performing simulations in two dimensions imposes the artifical constraint that all structures are axially symmetric. | Performing simulations in two dimensions imposes the artifical constraint that all structures are axially symmetric. |
Of course. this naturally raises the question what differences one might expect in three dimensions. | Of course, this naturally raises the question what differences one might expect in three dimensions. |
Ignoring the important question whether 2D/3D differences have any influence on the success of the neutrino-heating mechanism for supernova explosions (a very first statement in this context was recently published by Nordhausetal. (2010))). and relying on the viability of this mechanism (which is a fundamental assumption in our study). basic features of the 2D asymmetries observed inour calculations seem to be confirmed by recent 3D simulations: Wongwathanaratetal.(2010) followed the evolution of a small set of 3D models over a similarly long post-bounce period (but with considerably less spatial resolution than used in our 2D calculations). | Ignoring the important question whether 2D/3D differences have any influence on the success of the neutrino-heating mechanism for supernova explosions (a very first statement in this context was recently published by \cite{Nordhaus.Burrows.etal:2010}) ), and relying on the viability of this mechanism (which is a fundamental assumption in our study), basic features of the 2D asymmetries observed inour calculations seem to be confirmed by recent 3D simulations: \cite{Wongwathanarat.Janka.Mueller:2010} followed the evolution of a small set of 3D models over a similarly long post-bounce period (but with considerably less spatial resolution than used in our 2D calculations). |
The 3D slice of figure 3. right panel. in the latter publicatio exhibits the same overall behavior and structural features for the neutrino-driven wind phase 1 2D and 3D nodels. namely an essentially spherical wind. an asymmetric wind-terminatio shock. and an inhomogeneous and strongly anisotropic eject:£2 shell behind the forward shock. | The 3D slice of figure 3, right panel, in the latter publication exhibits the same overall behavior and structural features for the neutrino-driven wind phase in 2D and 3D models, namely an essentially spherical wind, an asymmetric wind-termination shock, and an inhomogeneous and strongly anisotropic ejecta shell behind the forward shock. |
Also ti 3D the reverse shock is highly deformed and its effect on the winc properties and nucleosynthesis conditions depeds on the outflow direction. | Also in 3D the reverse shock is highly deformed and its effect on the wind properties and nucleosynthesis conditions depends on the outflow direction. |
We are therefore confident that our matt findings for the interaction are valid not only in the considered 2D situation, | We are therefore confident that our main findings for the interaction are valid not only in the considered 2D situation. |
Of course. the present parameterized explosion models. which do not yield any information about the explosion properties (energy. timescale) of a progenitor star. cannot be conclusive on the consequences of these asymmetries for the explosive nucleosynthesis of individual stars. nor does our present knowledge of the explosion mechanism allow for any statements In a statistical sense. | Of course, the present parameterized explosion models, which do not yield any information about the explosion properties (energy, timescale) of a progenitor star, cannot be conclusive on the consequences of these asymmetries for the explosive nucleosynthesis of individual stars, nor does our present knowledge of the explosion mechanism allow for any statements in a statistical sense. |
Rotation. of the nascent neutron star is an additional degree of freedom. which we ignored in the models of this paper. | Rotation of the nascent neutron star is an additional degree of freedom, which we ignored in the models of this paper. |
It will cause a global pole-equator asymmetry of the wind and of the wind-ejecta interaction with potentially interesting implications. | It will cause a global pole-equator asymmetry of the wind and of the wind-ejecta interaction with potentially interesting implications. |
This should be studied in future work by systematic variations of the proto-neutron star spin. | This should be studied in future work by systematic variations of the proto-neutron star spin. |
We have shown the impact of multidimensional effects on the dynamical evolution of the neutrino-driven wind. reverse shock and supernova ejecta. | We have shown the impact of multidimensional effects on the dynamical evolution of the neutrino-driven wind, reverse shock and supernova ejecta. |
In this section. we want to briefly address the possible implications of our results for the nucleosynthesis processes occurring In supernova outflows: charged-particle reactions. alpha process (Woosley&Hoffman.1992;etal.. 1994).. vp-process (Frohlichetal..2006:Pruet2006:Wanajo. 2006).. and occasionally r-process etal..2007.fora review).. | In this section, we want to briefly address the possible implications of our results for the nucleosynthesis processes occurring in supernova outflows: charged-particle reactions, alpha process \cite[]{Woosley.Hoffman:1992, Witti.Janka.Takahashi:1994}, $\nu$ p-process \cite[]{Froehlich.Martinez-Pinedo.ea:2006,
Pruet.Hoffman.ea:2006, Wanajo:2006}, and occasionally r-process \cite[][for a review]{arnould.goriely.takahashi:2007}. |
Since the works of Cameron(1957) and Burbidgeetal. (1957).. core-collapse supernova outflows have been the best studied candidate for the production of heavy elements. | Since the works of \cite{Cameron:1957} and \cite{Burbidge.Burbidge.ea:1957}, core-collapse supernova outflows have been the best studied candidate for the production of heavy elements. |
However. this environment is facing more and more problems to fulfill the requirements (high entropy. low electron fraction and fast expansion) for the production of heavy r-process elements (A> 90). | However, this environment is facing more and more problems to fulfill the requirements (high entropy, low electron fraction and fast expansion) for the production of heavy r-process elements $>90$ ). |
The conditions found to be necessary for a robust and strong r-process (e.g..Hoffmanetal..1997) are not achieved by recent long-time supernova simulations (PaperI.Hüdepohletal..2010:Fischer 2010).. | The conditions found to be necessary for a robust and strong r-process \cite[e.g.,][]{hoffman.woosley.qian:1997}
are not achieved by recent long-time supernova simulations \cite[Paper~I,][]{Huedepohl.etal:2010, Fischer.etal:2010}. . |
This is also the case for our 2D simulations. where the wind entropies are too low to get the high neutron-to-seed ratios necessary for the r-process. | This is also the case for our 2D simulations, where the wind entropies are too low to get the high neutron-to-seed ratios necessary for the r-process. |
Yet galactic chemical evolution models (seee.g..Ishimaruetal.2004:Qian&Wasserburg.2008) suggest that at least à subset of core-collapse supernovae could be responsible of the origin of half of the heavy r-process elements. | Yet galactic chemical evolution models \cite[see
e.g.,][]{Ishimaru.etal:2004, Qian.Wasserburg:2008} suggest that at least a subset of core-collapse supernovae could be responsible of the origin of half of the heavy r-process elements. |
Therefore. one may speculate that the r-process could take place in neutrino-driven winds because of some still unknown aspect of physies that might cure the problems revealed by the present hydrodynamical models. | Therefore, one may speculate that the r-process could take place in neutrino-driven winds because of some still unknown aspect of physics that might cure the problems revealed by the present hydrodynamical models. |
In this case the reverse shock could have important consequences (Wanajoetal.2002). | In this case the reverse shock could have important consequences \cite[]{Wanajo.Itoh.ea:2002}. |
Depending on the temperature at the reverse shock the r-process path is different. | Depending on the temperature at the reverse shock the r-process path is different. |
When the reverse-shock temperature is low (7,€0.5 GK). neutron-capture and beta-decay timescales are similar (Blake&Schramm.1976) and shorter than (y.1) timescales. | When the reverse-shock temperature is low $T_{\mathrm{rs}} \lesssim 0.5$ GK), neutron-capture and beta-decay timescales are similar \cite[]{Blake.Schramm:1976} and shorter than $(\gamma,n)$ timescales. |
This is alsoknown as "cold r-process" (Wanajo.2007;Panov&Janka. 2009).. | This is alsoknown as “cold r-process” \cite[]{Wanajo:2007, Panov.Janka:2009}. . |
In contrast. when the reverse shock is at high temperatures. there is an | In contrast, when the reverse shock is at high temperatures, there is an |
Therefore. all objects present extended continu over several are sec (up to 15 are sec in SMM. | Therefore, all objects present extended continuum over several arc sec (up to $\sim$ 15 arc sec in SMM J02399-0136). |
The coutimuun in MBRC2025-218. MIBC1555- and SMAL J02399-0136 is dominated bv a bright component which is rather compact. | The continuum in MRC2025-218, MRC1558-003 and SMM J02399-0136 is dominated by a bright component which is rather compact. |
It is probably uuresolved in SMIAL JO2399-013¢G and mareiually resolved in MRC?2025-218. | It is probably unresolved in SMM J02399-0136 and marginally resolved in MRC2025-218. |
All objects show extended emission lies which present rather different spatial profiles compared to the continuum (compare. for instance. fιο Lya aud continu profiles in SAINT J02399-01:X) Ue XIRC2025-215]. | All objects show extended emission lines which present rather different spatial profiles compared to the continuum (compare, for instance, the $\alpha$ and continuum profiles in SMM J02399-0136 and MRC2025-218). |
Both lines aud continu reveal the preseuce of several spatially distinct regions. | Both lines and continuum reveal the presence of several spatially distinct regions. |
The 2-D spectrum of MBRC2025-218 shows a clear absortion feature blueshifted. with respect the CTW enissjon (see Fie. 1l) | The 2-D spectrum of MRC2025-218 shows a clear absorption feature blueshifted with respect the CIV emission (see Fig. \ref{Fig4}) ). |
Iu order to search for other absorion features; we have extracted a 1-D spectra from he continua emitting region (8 pixels or 2.2 arc sec aperure). | In order to search for other absorption features, we have extracted a 1-D spectrum from the continuum emitting region (8 pixels or 2.2 arc sec aperture). |
We fitted the profiles of all possible absorption cleteclolis. | We fitted the profiles of all possible absorption detections. |
We asstuned Gaussian profiles (although it does rot necessarily have to be the case). | We assumed Gaussian profiles (although it does not necessarily have to be the case). |
Some absorption featires are detected. | Some absorption features are detected. |
We show in Fig. | We show in Fig. |
5 (bottoni) the specrn in the range I1sNO-1700A. with the expected position of some absorption features commonly fouxd iu nearby starburst galaxies. | \ref{Fig5} (bottom) the spectrum in the range 1180-1700, with the expected position of some absorption features commonly found in nearby starburst galaxies. |
We preseut for coniparisoi the | We present for comparison the |
We may easily restrict the caleulations to T>10! K and i«&10M à so that P>>1. and resistivity will be ignored. | We may easily restrict the calculations to $T\gtrsim 10^4$ K and $n \ll 10^{10}$ $^{-3}$ , so that ${\cal P} \gg 1$, and resistivity will be ignored. |
Since these questions can be subtle however. we return to this point in 83.5. | Since these questions can be subtle however, we return to this point in 3.5. |
The theory of viscous (transport in magnetizedplasmas is presented by (1965). | The theory of viscous transport in magnetizedplasmas is presented by \citet{b65}. |
. The usual isotropic collisional viscous stress tensor can be written where 5 is the dynamical viscosity coefficient. and This form applies to a set of Cartesian axes. (.jf) being an even permutation of (X.Y.Z). | The usual isotropic collisional viscous stress tensor can be written where $\eta$ is the dynamical viscosity coefficient, and This form applies to a set of Cartesian axes, $(i, j, k)$ being an even permutation of $(X, Y, Z)$. |
As usual. 9;; denotes the Kronecker delta finction. | As usual, $\delta_{ij}$ denotes the Kronecker delta function. |
We note that the stress is traceless. | We note that the stress is traceless. |
In the paper. we work exclusively in the Doussinesq limit. aud shall sel Veo=0 in (he above. | In the paper, we work exclusively in the Boussinesq limit, and shall set $\del\bcdot\bb{v}=0$ in the above. |
In 83.4. it is shown that the Doussinesq limit is justified when the Revnolds number is large. | In 3.4, it is shown that the Boussinesq limit is justified when the Reynolds number is large. |
In the presence of a restricting magnetic field. the onlv component of σε; that remains unaffected is the momentiun fIux along the magnetic line of force due to the gradient. along the field line. | In the presence of a restricting magnetic field, the only component of $\sigma_{ij}$ that remains unaffected is the momentum flux along the magnetic line of force due to the gradient along the field line. |
Define a local Cartesian coordinate svstem (the. "field frame”) (ΑιYs.Za). chosen with the magnetic field lving along the Zi axis. | Define a local Cartesian coordinate system (the “field frame”) $(\xb, \yb, \zb)$, chosen with the magnetic field lying along the $\zb$ axis. |
Then LP bibis, where (he b; are components of the unit magnetic [field vector in an arbitrary. Cartesian frame. | Then = b_i b_j where the $b_i$ are components of the unit magnetic field vector in an arbitrary Cartesian frame. |
Draginski(1965) shows that all other components of the magnetized viscous stress tensor are smaller than oz,z, by a factor of order ε or €?. and are therefore ignored in (his calculation. | \citet{b65} shows that all other components of the magnetized viscous stress tensor are smaller than $\sigma_{\zb \zb}$ by a factor of order $\epsilon$ or $\epsilon^2$, and are therefore ignored in this calculation. |
The important exception are (the two other diagonal stress components. which to leading order in ε are identicalwith one another. | The important exception are the two other diagonal stress components, which to leading order in $\epsilon$ are identicalwith one another. |
Since the stress must always be traceless. we have | Since the stress must always be traceless, we have = = . |
We wish to estimate how well the ratio of the strengths of the broad and narrow lines can be measured. | We wish to estimate how well the ratio of the strengths of the broad and narrow lines can be measured. |
To do this we will mocel the [lux in the “th resolution of the spectrum as consisting of four contributions: the broad Lincs GoWPCA). the narrow lines 7 C/"MGN). the continuum emission (c6) and instrumental noise (N;). | To do this we will model the flux in the $i$ th resolution of the spectrum as consisting of four contributions: the broad lines $\omega' W^b_i(\Delta_b)$ ), the narrow lines $f^n W^n_i(\Delta_n)$ ), the continuum emission $c' C_i$ ) and instrumental noise $N_i$ ). |
The functions Wwe. We and C; are normalised so that their sum over all resolution elements is one and their widths (and perhaps some other shape parameters) are A, and A,,. | The functions $W^b_i$, $W^n_i$ and $C_i$ are normalised so that their sum over all resolution elements is one and their widths (and perhaps some other shape parameters) are $\Delta_b$ and $\Delta_n$. |
We are interested in the ratio wSa”ff". | We are interested in the ratio $\omega\equiv \omega'/f^n$. |
The flux is then where 4; is the extinction and e= cf". | The flux is then where $I_i$ is the extinction and $c=c'/f^n$ . |
We will rewrite the extinction curve as Z;f/"=ah; with the normalisation »»AAW?=1 so that e is the total lux in the narrow lines after extinction correction. | We will rewrite the extinction curve as $I_i f^n=a h_i$ with the normalisation $\sum_i h_i
W^n_i =1$ so that $a$ is the total flux in the narrow lines after extinction correction. |
H£ we assume Gaussian. uncorrelated noise the log of the likelihood function will be For our present. purposes it is convenient to replace the sum over resolution elements in (A2)) with an integral over wavelength by making the substitution »08A+|αλ where 9A is the width in wavelength of the resolution elements. | If we assume Gaussian, uncorrelated noise the log of the likelihood function will be For our present purposes it is convenient to replace the sum over resolution elements in \ref{likelihood}) ) with an integral over wavelength by making the substitution $\sum_i\rightarrow
\delta\lambda^{-1}\int d\lambda$ where $\delta\lambda$ is the width in wavelength of the resolution elements. |
We will assume that the noise is constant over the wavelength range considered and given hy 0. | We will assume that the noise is constant over the wavelength range considered and given by $\sigma_n$. |
The Fisher information matrix is defined as where p, and po are two parameters that are to be constrained. by the measurements. | The Fisher information matrix is defined as where $p_1$ and $p_2$ are two parameters that are to be constrained by the measurements. |
The variance of the minimum variance unbiased estimator of a parameter p is given by the Cramoer-Iao bound ‘To caleulate this we must identify the parameters in (A2)) that are to be fit to the spectrum. | The variance of the minimum variance unbiased estimator of a parameter $p$ is given by the Cramer-Rao bound To calculate this we must identify the parameters in \ref{likelihood}) ) that are to be fit to the spectrum. |
The region of the spectrum we are interested in. shown in Figure S.. is relatively small compared to the complete spectrum and it is in region where the continuum has a shallow minimum in template QSO spectra(VandenBerketal.2001). | The region of the spectrum we are interested in, shown in Figure \ref{fig:pqso}, is relatively small compared to the complete spectrum and it is in region where the continuum has a shallow minimum in template QSO \citep{vandenberk:01}. |
. For simplicity we will approximate the extinction. h(À). and continuum. C'(À). as constant over the region of interest. but we will allow their normalisations to vary. | For simplicity we will approximate the extinction, $h(\lambda)$ , and continuum, $C(\lambda)$, as constant over the region of interest, but we will allow their normalisations to vary. |
Formally. this is a valid approximation if dInC'CAJ/dA.dinh(A)/dAX«1/2NÀ where AA is the range of wavelength considered. | Formally, this is a valid approximation if $d\ln
C(\lambda)/d\lambda, d\ln h(\lambda)/d\lambda \ll 1/\Delta\lambda$ where $\Delta\lambda$ is the range of wavelength considered. |
We will also assume that the redshift of the QSO is well established so that the positions of all lines are known. | We will also assume that the redshift of the QSO is well established so that the positions of all lines are known. |
This leaves five parameters to be determined. = a. c. αν A, and «δρ. | This leaves five parameters to be determined – $\omega$, $c$, $a$, $\Delta_n$ and $\Delta_b$. |
There are then 15 independent elements in the Fisher matrix which we caleulate. | There are then 15 independent elements in the Fisher matrix which we calculate. |
In general the profiles of the lines are functions of multiple parameters which could be added to the list. but for simplicity we will leave these out here and take the lines to be Gaussian. | In general the profiles of the lines are functions of multiple parameters which could be added to the list, but for simplicity we will leave these out here and take the lines to be Gaussian. |
Once these matrix elements are calculated the matrix can be inverted. numerically. | Once these matrix elements are calculated the matrix can be inverted numerically. |
Lt is convenient to express error estimates in terms of the signal to noise per resolution clement with which the continuum can be measured. | It is convenient to express error estimates in terms of the signal to noise per resolution element with which the continuum can be measured. |
In our parameterization this is A noise level of σι~0.01 may be reasonable. | In our parameterization this is A noise level of $\sigma_o\sim 0.01$ may be reasonable. |
Table 1: shows the estimated errors for several choices of fiducial parameters. | Table \ref{table} shows the estimated errors for several choices of fiducial parameters. |
The ratio of the continuum Ilux within the range AA to the narrow line Εαν. e. is estimated from the height of the narrow lines relative to the continuum in composite spectra (VandenBerketal.2001). | The ratio of the continuum flux within the range $\Delta\lambda$ to the narrow line flux, $c$, is estimated from the height of the narrow lines relative to the continuum in composite spectra \citep{vandenberk:01}. |
.. We use only and. for the estimates and assume that no lines overlap except the BL and NL112. | We use only and for the estimates and assume that no lines overlap except the BL and NL. |
. The source is at ος=2. | The source is at $z_s=2$. |
In this analvsis we have ignored the noise associated: with subtracting the sky and the absolute (lux calibration. | In this analysis we have ignored the noise associated with subtracting the sky and the absolute flux calibration. |
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