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In this paper we present an analysis of the quasar luminosity function. and. its evolution based on new observational data. ancl argue that knowledge of 10 luminosity function alone is insullicient to allow us to unelerstancl the physical causes of quasar evolution. | In this paper we present an analysis of the quasar luminosity function and its evolution based on new observational data, and argue that knowledge of the luminosity function alone is insufficient to allow us to understand the physical causes of quasar evolution. |
The most wiclely-quotecd study of the Iuminosity 'unction to date has been that of Bovle et al. ( | The most widely-quoted study of the luminosity function to date has been that of Boyle et al. ( |
1988. hereafter BSP). who used the ANT sample of faint UVX. quasars ogether with brighter samples such as the Palomar-Green survey (Schmidt Green. 1983) to determine the luminosity Cfunetion in our redshift slices from 2=0.3 to 2=2.2. | 1988, hereafter BSP), who used the AAT sample of faint UVX quasars together with brighter samples such as the Palomar-Green survey (Schmidt Green, 1983) to determine the luminosity function in four redshift slices from $z=0.3$ to $z=2.2$. |
BSP fit a varicty of models to this function ancl concluded that the rest-Lit model was pure luminosity evolution (PLE) in which the shape of the luminosity. function. was. paramoeterised bv (wo poweraws with a transition. between them. at a characteristic luminosity. | BSP fit a variety of models to this function and concluded that the best-fit model was pure luminosity evolution (PLE) in which the shape of the luminosity function was parameterised by two power-laws with a transition between them at a characteristic luminosity. |
The characteristic luminosity increased with redshift (we term this negative evolution. as the population appears to have become dimmoer with increasing cosmic time): in which à and 2 are the indices of the power-laws and AL*(2) describes the evolution: With the addition of two surveys extending to redshift 2«2.9 (Bovle. Jones Shanks 1991. Zitelli et al. | The characteristic luminosity increased with redshift (we term this negative evolution, as the population appears to have become dimmer with increasing cosmic time): in which $\alpha$ and $\beta$ are the indices of the power-laws and $M^{*}(z)$ describes the evolution; With the addition of two surveys extending to redshift $z < 2.9$ (Boyle, Jones Shanks 1991, Zitelli et al. |
1992) the above model has been slightly mocified such that there is à maximum redshift bevond which no evolution occurs. | 1992) the above model has been slightly modified such that there is a maximum redshift beyond which no evolution occurs. |
The most recent parameters of this model have been. presented by Bovle (1991) and are: à5 39.521ο. κ=45. Mic224 ena=19. | The most recent parameters of this model have been presented by Boyle (1991) and are; $\alpha=-3.9$, $\beta=-1.5$, $k=3.5$, $M^{*}_{0}=-22.4$, $z_{max}=1.9$. |
2913 ruled out any need for additional density evolution [or quasars with Ag23. | BSP ruled out any need for additional density evolution for quasars with $M_{B}
\le -23$. |
PLE can be interpreted as either representing theached evolution of individual objects in which a single population of quasars formed. at a single epoch and have been growing dimmoer ever since. or as thesfafishead evolution of the properties of successive populations. | PLE can be interpreted as either representing the evolution of individual objects in which a single population of quasars formed at a single epoch and have been growing dimmer ever since, or as the evolution of the properties of successive populations. |
BSP noted that the latter interpretation implies a conspiracy between birth and death rates. | BSP noted that the latter interpretation implies a conspiracy between birth and death rates. |
However | However |
null | firstpage–lastpage 2009 \begin{document} |
an orbit similar to that of the companion of 51 Pegasi (Mavor&Queloz 1905}. | an orbit similar to that of the companion of 51 Pegasi \cite{MayorQueloz95}) ). |
Thus. shortvy after the detection of the periodicity iu the radia-velocity signal. we began plotometric observations of IID1166155 with the Te τι automatic plooclectric telescope (APT) at Fairborn Observatory i1 Arizona to search for possible transits of the companioi across the disk of the star. | Thus, shortly after the detection of the periodicity in the radial-velocity signal, we began photometric observations of 166435 with the T8 m automatic photoelectric telescope (APT) at Fairborn Observatory in Arizona to search for possible transits of the companion across the disk of the star. |
We acquired 326 Stromuuere1 b and y observations with a two-channel precision photOuneter on the APT hetween 1998 June aud 2000 June. | We acquired 326 Strömmgren $b$ and $y$ observations with a two-channel precision photometer on the APT between 1998 June and 2000 June. |
Tιο observations were reduced differentially with respect o the comparison star 99 Ter 66775. 1165908. FTV). corrected. for atmospheric extinction with wiehtl extinction cocficicuts. and transformed to the Stronunercn system. | The observations were reduced differentially with respect to the comparison star 99 Her 6775, 165908, F7V), corrected for atmospheric extinction with nightly extinction coefficients, and transformed to the Strömmgren system. |
External precision of a siugle observation with the 0.80 m APT averages 0.0011 mae. | External precision of a single observation with the 0.80 m APT averages 0.0011 mag. |
Further doetiis of the automatic-telescope operations and cdata-reductio1 procedures can be found in Πο(1999). | Further details of the automatic-telescope operations and data-reduction procedures can be found in \cite{Henry99}. |
. The individual photometric observations are avalable ou the Tennessee State University Automated Astronomy Group weld | The individual photometric observations are available on the Tennessee State University Automated Astronomy Group web. |
TN Periodogreuii analysis of the cutire set of 326 observations aken together reveals a photometric period of 3.79950.0005 dd. Thus. the photometric and racial-velociv periods agree within their respective uncertainties. | Periodogram analysis of the entire set of 326 observations taken together reveals a photometric period of $3.7995 \pm 0.0005$ d. Thus, the photometric and radial-velocity periods agree within their respective uncertainties. |
For or photometric analysis we adopt. as reference. the radial-velocitv period of 3.7987-dayv aud Ly 150996.5. the radial-velocity imaxiumun of the hest- sne-ctre model. | For our photometric analysis we adopt, as reference, the radial-velocity period of 3.7987-day and $T_0=2450996.5$ , the radial-velocity maximum of the best-fit sine-curve model. |
The phoonietrie Observations are dividedd iutoi five SYOUpDs as SLOW rin Table 1. where we eive the results of least-squares. sine-curve fits on the raclal-velocity period to the five Strónumueren y data sets. | The photometric observations are divided into five groups as shown in Table 1, where we give the results of least-squares, sine-curve fits on the radial-velocity period to the five Strömmgren $y$ data sets. |
We also list the periods derives frou the five individual «ata sets | We also list the periods derived from the five individual data sets. |
T1ο photometric anmplituces vary from 1nunag to munae: the ical magnitudes of the five cata sets have a ranee of about )nuuae. | The photometric amplitudes vary from mag to mag; the mean magnitudes of the five data sets have a range of about mag. |
Although the five lioit curves ayproNimate simusoids. tle rus vaτος are. nevertheless. πεnuewhat higher than the mimae precision of tvpica] observations. | Although the five light curves approximate sinusoids, the rms values are, nevertheless, somewhat higher than the mag precision of typical observations. |
This is prinarily the effect of slight evee-to-cevele changes in the light. curves within cach data set: the ruis values increase for the later data sets since the later data sets are longer axd coutaiu more eveles. | This is primarily the effect of slight cycle-to-cycle changes in the light curves within each data set; the rms values increase for the later data sets since the later data sets are longer and contain more cycles. |
Data set l. which covers ouly 10 davs or 2.7 cveles. has the largest number of observaIOUS sj1ος a this time we were makine repeated Ilueasuremienuts eacli night to search for transits. | Data set 1, which covers only 10 days or 2.7 cycles, has the largest number of observations since at this time we were making repeated measurements each night to search for transits. |
These data are plotte in tI left aud rieht hird panels from the top o| 58. | These data are plotted in the left and right third panels from the top of 8. |
Basco on the photometry iieutioned above. we did not expect to fineL coherent light variability ou this 20110:l. | Based on the photometry mentioned above, we did not expect to find coherent light variability on this period. |
Tustead. we foud a smoothly varving. near yvosuusxvidal lieit curve with :| period matching the radia-velociv period aud a ΙΙ. ucar phase 0.25. where we had hoped to find μηs. | Instead, we found a smoothly varying, nearly sinusoidal light curve with a period matching the radial-velocity period and a minimum near phase 0.25, where we had hoped to find transits. |
frame) at 6.5+0.1 keV, rms width c = 0.95-Ε0.15 keV and EW = 0.25+0.11 keV. Next, we fitted the narrow absorption feature visible in figure 1 with a gaussian shaped absorption line, again with energy, width and equivalent width free. | frame) at $\pm$ 0.1 keV, rms width $\sigma$ = $\pm$ 0.15 keV and EW = $\pm$ 0.11 keV. Next, we fitted the narrow absorption feature visible in figure 1 with a gaussian shaped absorption line, again with energy, width and equivalent width free. |
The best-fitobserved line energy was 8.18+0.10 keV, with an rms width of <100 eV, and an EW of 170+60 eV. The addition of this gaussian absorption line improved the overall fit to x? /dof = 515/580, an improvement at 99.3 per cent confidence by the F-test (Bevington and Robinson 1992). | The best-fitobserved line energy was $\pm$ 0.10 keV, with an rms width of $\leq$ 100 eV, and an EW of $\pm$ 60 eV. The addition of this gaussian absorption line improved the overall fit to $\chi^{2}$ /dof = 515/580, an improvement at 99.3 per cent confidence by the F-test (Bevington and Robinson 1992). |
We then examined the EPIC data for other spectral features, excluding the region ~1.8-2.3 keV where calibration residuals associated with the Si K and Au M edges remain. | We then examined the EPIC data for other spectral features, excluding the region $\sim$ 1.8–2.3 keV where calibration residuals associated with the Si K and Au M edges remain. |
The strongest feature is a possible absorption line at ~3.0 keV; fitting this with a gaussian line, again with energy and equivalent width free, further improved the 2-11 keV fit, to y?/dof of 508/578 (98 per cent confidence). | The strongest feature is a possible absorption line at $\sim$ 3.0 keV; fitting this with a gaussian line, again with energy and equivalent width free, further improved the 2–11 keV fit, to $\chi^{2}$ /dof of 508/578 (98 per cent confidence). |
A note of caution is appropriate here, since the ~3 keV feature is only significant in the pn data. | A note of caution is appropriate here, since the $\sim$ 3 keV feature is only significant in the pn data. |
We reject a second feature of similar depth, apparent near 3.3 keV, where a gaussian fit (figure 3) shows this ‘absorption line’ to be marginally too narrow to match the EPIC resolution. | We reject a second feature of similar depth, apparent near 3.3 keV, where a gaussian fit (figure 3) shows this `absorption line' to be marginally too narrow to match the EPIC resolution. |
Finally, we retain only the most convincing absorption features, at ~8.18 keV and ~3.02 keV, in Table 1, which lists the observed and AGN rest frame energy of each line, together with their most likely identifications and corresponding outflow velocities. | Finally, we retain only the most convincing absorption features, at $\sim$ 8.18 keV and $\sim$ 3.02 keV, in Table 1, which lists the observed and AGN rest frame energy of each line, together with their most likely identifications and corresponding outflow velocities. |
Since the Fe K-shell dominates X-ray absorption above ~7 keV, with a series of resonance and weaker satellite lines (eg Palmeri 22002) leading up to the absorption edges of He-like FeXXV at 8.76 keV and H-like FeXXVI at 9.28 keV, the most likely interpretion of the ~8.18 keV feature, where the narrow profile indicates a line rather than an edge, is with the primary resonance absorption line in H- or He-like Fe. | Since the Fe K-shell dominates X-ray absorption above $\sim$ 7 keV, with a series of resonance and weaker satellite lines (eg Palmeri 2002) leading up to the absorption edges of He-like FeXXV at 8.76 keV and H-like FeXXVI at 9.28 keV, the most likely interpretion of the $\sim$ 8.18 keV feature, where the narrow profile indicates a line rather than an edge, is with the primary resonance absorption line in H- or He-like Fe. |
These alternative identifications indicate an outflow velocity of ~0.22c or ~0.26c, respectively. | These alternative identifications indicate an outflow velocity of $\sim$ 0.22c or $\sim$ 0.26c, respectively. |
The absorption feature at ~3.02 keV has a most probable association with the primary resonance absorption line of He- or H-like S (as S K-shell absorption is likely to to be dominant in a highly ionised absorber in this energy range). | The absorption feature at $\sim$ 3.02 keV has a most probable association with the primary resonance absorption line of He- or H-like S (as S K-shell absorption is likely to to be dominant in a highly ionised absorber in this energy range). |
We note the implied outflow velocities from these alternatives are consistent with the values deduced for the ionised Fe line, lending support to the overall interpretation. | We note the implied outflow velocities from these alternatives are consistent with the values deduced for the ionised Fe line, lending support to the overall interpretation. |
In summary, the detection of absorption lines in the EPIC spectra of pprovides intriguing evidence of an ionised outflow, with a velocity, depending on the line identifications, in the range ~0.20-0.26c. | In summary, the detection of absorption lines in the EPIC spectra of provides intriguing evidence of an ionised outflow, with a velocity, depending on the line identifications, in the range $\sim$ 0.20–0.26c. |
To attempt to reduce this uncertainty, and find supporting evidence for this conclusion, we extend our search in Section 5 to the simultaneous RGS observation ofPG0844+349. | To attempt to reduce this uncertainty, and find supporting evidence for this conclusion, we extend our search in Section 5 to the simultaneous RGS observation of. |
. Extending the 2-11 keV power law fits for both pn and MOS spectral data to 0.3 keV shows very clearly (figure 2) the strong soft excess indicated in earlier observations. | Extending the 2–11 keV power law fits for both pn and MOS spectral data to 0.3 keV shows very clearly (figure 2) the strong soft excess indicated in earlier observations. |
In the EPIC data this can be adequately modelled with the addition of 3 black body emission components, at kT=~70 eV, ~130 eV and ~330 eV, fitting the ‘gradual soft excess’ seen in aand which is apparently typical of higher luminosity AGN (Pounds and Reeves 2002). | In the EPIC data this can be adequately modelled with the addition of 3 black body emission components, at $\sim$ 70 eV, $\sim$ 130 eV and $\sim$ 330 eV, fitting the `gradual soft excess' seen in and which is apparently typical of higher luminosity AGN (Pounds and Reeves 2002). |
Based on this fit we obtain an average 0.3-10 keV flux for oof 1.1x107!! erg s! οπι3, corresponding to a luminosity of ~1044 erg s! (Ho=75 ! kmss!). | Based on this fit we obtain an average 0.3–10 keV flux for of $1.1\times10^{-11}$ erg $^{-1}$ $^{-2}$, corresponding to a luminosity of $\sim 10^{44}$ erg $^{-1}$ $ H_0 = 75 $ $^{-1}$ $^{-1}$ ). |
The 2-10 keV flux was 3.8x10? erg s! cm?, with a corresponding luminosity of 3x1015 erg s!. | The 2–10 keV flux was $3.8\times10^{-12}$ erg $^{-1}$ $^{-2}$, with a corresponding luminosity of $3\times 10^{43}$ erg $^{-1}$ . |
Compared with the oobservation of PG12114+143, the soft excess in iis found to be significantly ‘hotter’, as is evident from a comparison of figure 2 here with the similar plot for | Compared with the observation of PG1211+143, the soft excess in is found to be significantly `hotter', as is evident from a comparison of figure 2 here with the similar plot for |
For many years it was believed that globular clusters (GCs) follow with excellent precision the canonical zero-order approximation for the formation of simple stellar populations. where a chemically homogeneous gas cloud collapses under the action of its own potential well forming stars of different masses at the same time. | For many years it was believed that globular clusters (GCs) follow with excellent precision the canonical zero-order approximation for the formation of simple stellar populations, where a chemically homogeneous gas cloud collapses under the action of its own potential well forming stars of different masses at the same time. |
However. recent observations have shown that this hypothesis is quite far from reality. | However, recent observations have shown that this hypothesis is quite far from reality. |
One of the peculiarities observed is a spread in the abundance of light elements in stars of all GCs studied to date. | One of the peculiarities observed is a spread in the abundance of light elements in stars of all GCs studied to date. |
This phenomenon is very well represented by the O-Na anticorrelation stars). | This phenomenon is very well represented by the O-Na anticorrelation . |
. This anticorrelation differs in detail from one GC to the next. and its extension seems to depend on the present-day GC mass. but it does not seem to depend on the cluster's metallicity (?). | This anticorrelation differs in detail from one GC to the next, and its extension seems to depend on the present-day GC mass, but it does not seem to depend on the cluster's metallicity . |
. Due to the fact that massive GCs have deeper potential wells. it is commonly thought that the spread of Fe-peak elements would only be present in the most massive GCs. | Due to the fact that massive GCs have deeper potential wells, it is commonly thought that the spread of Fe-peak elements would only be present in the most massive GCs. |
However. spectroscopic evidence has revealed heavy element variations in GCs which are less massive than others which do not show a spread in these elements. | However, spectroscopic evidence has revealed heavy element variations in GCs which are less massive than others which do not show a spread in these elements. |
This ts the case of M22 (NGC 6656) and NGC 1851. with masses of ~5.4x10?M... and ~5.6x10M... respectively. which show heavy element variations. while other GCs (such as the extensive studied M3. with a mass ~8.4x 10M.) do not show this In fact. while M22 shows an iron spread between —].9 and -1.6. as well as spreads in other elements(?????).. NGC 1851 harbors two groups of stars with different [Ba/Fe] ratios. but. depending on the authors. with a uniform [Fe/H] or with a spread in metallicity of A[Fe/H]~0.2 dexGCs). | This is the case of M22 (NGC 6656) and NGC 1851, with masses of $\sim 5.4\times 10^5 M_\odot$, and $\sim 5.6\times 10^5 M_\odot$, respectively, which show heavy element variations, while other GCs (such as the extensive studied M3, with a mass $\sim 8.4\times 10^5 M_\odot$ ) do not show this In fact, while M22 shows an iron spread between ${\rm [Fe/H]} \approx -1.9$ and $-1.6$, as well as spreads in other elements, NGC 1851 harbors two groups of stars with different [Ba/Fe] ratios, but, depending on the authors, with a uniform [Fe/H] or with a spread in metallicity of $\Delta{\rm [Fe/H]}\sim 0.2$ dex. |
. From a deep study of their color-magnitude diagrams (CMDs). these GCs show that the simple stellar population hypothesis is ruled out. as revealed by an observed subgiant branch (SGB) split(??). | From a deep study of their color-magnitude diagrams (CMDs), these GCs show that the simple stellar population hypothesis is ruled out, as revealed by an observed subgiant branch (SGB) split. |
. Another GC presenting an SGB split i$ NGC 6388(?).. with a mass of L4»109A... | Another GC presenting an SGB split is NGC 6388, with a mass of $1.4\times 10^6 M_\odot$. |
have pointed out that such SGB splits could be related to an enhancement in CNO abundances. or an internal age difference of a few Gyr. | have pointed out that such SGB splits could be related to an enhancement in CNO abundances, or an internal age difference of a few Gyr. |
However. have shown that for NGC 1851 the C+N+O content would be constant between the two groups with different [Ba/Fe]. | However, have shown that for NGC 1851 the C+N+O content would be constant between the two groups with different [Ba/Fe]. |
Moreover. have pointed out that such SGB splits may also be caused by variations in the helium abundance. though only at moderately high metallicity. | Moreover, have pointed out that such SGB splits may also be caused by variations in the helium abundance, though only at moderately high metallicity. |
For the moment. NGC 2808 (1.4x10M.) seems to be a fairly special case. with at most a small spread mn metals ([Fe/H|]=-1.10+ 0.03) but a wide O-Na anticorrelation(?). | For the moment, NGC 2808 $1.4\times 10^6 M_\odot $ ) seems to be a fairly special case, with at most a small spread in metals ${\rm [Fe/H]}=-1.10 \pm 0.03$ ) but a wide O-Na anticorrelation. |
. Indeed. this GC has three populations observed down to the main sequence (MS). which suggests different helium abundances. ranging from Y=0.25 to Y=0.37(2???). | Indeed, this GC has three populations observed down to the main sequence (MS), which suggests different helium abundances, ranging from $Y=0.25$ to $Y=0.37$. |
. This last peculiarity was in fact predicted by from a study of the cluster’s horizontal branch (HB) morphology. and recently tested by using high-resolution far-UV and optical images. as well as using visual and near-IR spectroscopy. | This last peculiarity was in fact predicted by from a study of the cluster's horizontal branch (HB) morphology, and recently tested by using high-resolution far-UV and optical images, as well as using visual and near-IR spectroscopy. |
Another GC which may share properties similar to NGC 2808's ts NGC 2419. with a present-day mass of 1.4x10°Ms2:2s: see also? ?)). | Another GC which may share properties similar to NGC 2808's is NGC 2419, with a present-day mass of $1.4\times 10^6 M_\odot$; see also ). |
From photometry. the case of w Cen. the most massive Galactic GC (3x 10°M..). has been known for a long time(??).. although only recent studies have unveiled some of its most intriguing properties. | From photometry, the case of $\omega$ Cen, the most massive Galactic GC $3\times 10^6 M_\odot$ ), has been known for a long time, although only recent studies have unveiled some of its most intriguing properties. |
This GC has at least three separate. well-defined red giant branches (RGBs). an extended HB morphology. and a large number of subluminous extreme HB stars22222222222). | This GC has at least three separate, well-defined red giant branches (RGBs), an extended HB morphology, and a large number of subluminous extreme HB stars. |
Recent (HST) observations have also shown a triple MS(???).. in addition to at least four SGBs(?). | Recent (HST) observations have also shown a triple MS, in addition to at least four SGBs. |
. From spectroscopic studies. this GC also shows a complex behavior. | From spectroscopic studies, this GC also shows a complex behavior. |
Based on spectroscopically derived chemical compositions of MS stars. have suggested that the bluest MS ts highly enriched in helium (Y= 0.35). while metallicities of SGB and RGB stars show a wide distribution. from [Fe/H]~-2 to ~—0.4therein)... | Based on spectroscopically derived chemical compositions of MS stars, have suggested that the bluest MS is highly enriched in helium $Y=0.35$ ), while metallicities of SGB and RGB stars show a wide distribution, from ${\rm [Fe/H]} \sim -2$ to $\sim -0.4$. |
These different metallicity groups present some intriguing properties. including a smaller spread in the abundances of some elements (e.g.. C. N. Ca. Ti. and Ba: | These different metallicity groups present some intriguing properties, including a smaller spread in the abundances of some elements (e.g., C, N, Ca, Ti, and Ba; |
uncertainties by a constant correction factor. | uncertainties by a constant correction factor. |
Our result on the underestimation of the uncertainties in a linear-bisector calculation is supported in the previously mentioned paper bv Storm et al. (2005). | Our result on the underestimation of the uncertainties in a linear-bisector calculation is supported in the previously mentioned paper by Storm et al. . |
.. As the six Cepheids studied by them are in an LAIC cluster. we can be confident. that they are at the same distance. | As the six Cepheids studied by them are in an LMC cluster, we can be confident that they are at the same distance. |
The scatter in their distances is then an estimate of the true uncertainty in the linezr-bisector distance to the cluster. | The scatter in their distances is then an estimate of the true uncertainty in the linear-bisector distance to the cluster. |
This scatter was found by Storm et al. | This scatter was found by Storm et al. |
lo be twice the formal errors of the linear-bisector distances to the Cepheids. within the range of results determined here. | to be twice the formal errors of the linear-bisector distances to the Cepheids, within the range of results determined here. |
Other mathematical approaches have been used to solve the surface brightness equations for cistance aud radius. | Other mathematical approaches have been used to solve the surface brightness equations for distance and radius. |
Ordinary least-squares assumes no error o1 one variable and all errors on the other. | Ordinary least-squares assumes no error on one variable and all errors on the other. |
Iiverse fit leasi-squares assumes (he reverse. | Inverse fit least-squares assumes the reverse. |
The limitations of ordinary linear least-squares solutions (direct. and. inverse) include not only unclerestimation of the errors. but also possible svstematic bias in the resulting distances and radii as discussed by Laney Stobie and Gieren et al. (1997). | The limitations of ordinary linear least-squares solutions (direct and inverse) include not only underestimation of the errors, but also possible systematic bias in the resulting distances and radii as discussed by Laney Stobie and Gieren et al. . |
. The linear-bisector. least-squares calculation achieves a solution in-between (he two other least-squares calculations (direct and inverse). | The linear-bisector, least-squares calculation achieves a solution in-between the two other least-squares calculations (direct and inverse). |
Moreover. the linear-bisector error bar roughly corresponds (to the difference between (the (wo least-squares solutions. | Moreover, the linear-bisector error bar roughly corresponds to the difference between the two least-squares solutions. |
Maximum likelihood uses information on errors on one or both variables to choose a result between the {wo results of linear least-squares (direct ancl inverse fits), | Maximum likelihood uses information on errors on one or both variables to choose a result between the two results of linear least-squares (direct and inverse fits). |
As a result. maximum likelihood cannot differ by more than one linear-bisector sigma from ordinary least-squares and even less from a linear-bisector fit. | As a result, maximum likelihood cannot differ by more than one linear-bisector sigma from ordinary least-squares and even less from a linear-bisector fit. |
As we have shown that this linear-bisector sigma is aboul 1/3 of the Bavesian sigma. the masxinnun likelihood results should also be about 1/3 of the Bavesian values. | As we have shown that this linear-bisector sigma is about 1/3 of the Bayesian sigma, the maximum likelihood results should also be about 1/3 of the Bayesian values. |
Our resulis [or the linear-bisector solutions thus suggest that the maximum likelihood method would vield distances and radii in (he infrared surface brightness method that are unbiased if the uncertainties in the data are well understood. | Our results for the linear-bisector solutions thus suggest that the maximum likelihood method would yield distances and radii in the infrared surface brightness method that are unbiased if the uncertainties in the data are well understood. |
Unfortunately. these uncertainties are often not well understood. | Unfortunately, these uncertainties are often not well understood. |
First. Barnes et al. | First, Barnes et al. |
showed that «quoted uncertainties in Cepheicl photometry ancl radial velocities are usually underestimatec. | showed that quoted uncertainties in Cepheid photometry and radial velocities are usually underestimated. |
second, maximunm-likelihood caleulations usually adopt ai approximation for the unicertaintv in the displacements advocated by Balona [ον equally spaced velocity data. | Second, maximum-likelihood calculations usually adopt an approximation for the uncertainty in the displacements advocated by Balona for equally spaced velocity data. |
Our work demonstrates that. this approximation does not apply to typical unequallv-spaced. radial velocity. curves: i£ it did apply. the linear-bisector method would have vielded uncertainties in distance and radius close to those of the Bavesian MCMC ealeulation. | Our work demonstrates that this approximation does not apply to typical unequally-spaced radial velocity curves; if it did apply, the linear-bisector method would have yielded uncertainties in distance and radius close to those of the Bayesian MCMC calculation. |
Thus we expect the maximunm likelihood uncertaintiesto be underestimated as are the linear-bisector uncertainties and lor the same reasons. | Thus we expect the maximum likelihood to be underestimated as are the linear-bisector uncertainties and for the same reasons. |
TGD and WILJ gratefully acknowledge financial support for this work from McDonald | TGB and WHJ gratefully acknowledge financial support for this work from McDonald |
observed in a number of pulsars (seee.g.,Askegar&etal.2003;Contopoulos&Spitkovsky 2006). | observed in a number of pulsars \cite[see
e.g.,][]{Askegar01,DeshpandeRankin,Rankin03,ContopSpitkovsky06}. |
. The following discussion takes place in the corotating frame, where we define electromagnetic fields by equations (10)) and (11)). | The following discussion takes place in the corotating frame, where we define electromagnetic fields by equations\ref{eq:corotEfield}) ) and \ref{eq:corotBfield}) ). |
In this frame there is transverse particle drift with velocity of order vagÉLXB/B?. | In this frame there is transverse particle drift with velocity of order $v_{\rm drift}\sim \vec{E}_{\perp}\times
\vec{B}/B^2$. |
This drift velocity can be thought of as the minimal velocity of particles, ignoring motion along the magnetic field lines. | This drift velocity can be thought of as the minimal velocity of particles, ignoring motion along the magnetic field lines. |
The transverse electric field E, reverses sign through the axis of maximum potential drop, and so there is rotation of field lines about this axis. | The transverse electric field $\vec{E}_{\perp}$ reverses sign through the axis of maximum potential drop, and so there is rotation of field lines about this axis. |
In general, this axis is not in the direction of the magnetic moment, and in fact does not even need to be a straight line. | In general, this axis is not in the direction of the magnetic moment, and in fact does not even need to be a straight line. |
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