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Figure { shows how the naenitude uncertainties decrease as the brieltuess of the star diucreases (towards the right). | Figure 4 shows how the magnitude uncertainties decrease as the brightness of the star increases (towards the right). |
The ower. thicker line in the figure was derived from the original. uuquautized nuages and shows tha the statisical unecrtainty ou the magnitudes decreases from o=0.23 for the faintest detectable sars (width m = 20). to σ=0.015 for stars tha are 2 maeuitudes brighter. | The lower, thicker line in the figure was derived from the original, unquantized images and shows that the statistical uncertainty on the magnitudes decreases from $\sigma = 0.23$ for the faintest detectable stars (with m = 20), to $\sigma = 0.018$ for stars that are 3 magnitudes brighter. |
The line has a slope close to 1.0 (iu log - log coordinates) as expected in the limiting case where the noise is dominate by the sky. aud hence the sienal-o-nolse ratio of the measurement increases in direct proportion to the signal. | The line has a slope close to 1.0 (in log - log coordinates) as expected in the limiting case where the noise is dominated by the sky, and hence the signal-to-noise ratio of the measurement increases in direct proportion to the signal. |
The other 3 lines in Figure. Lowere derived from the same nuages after they were firs quautize and compressed with successively coarser q values of L.0. 0.5. and 0.25. when also applviung the subtractive ditherus option. | The other 3 lines in Figure 4 were derived from the same images after they were first quantized and compressed with successively coarser q values of 1.0, 0.5, and 0.25, when also applying the subtractive dithering option. |
The verical displacement of these lines shows that the measurement errors are systematically larecr in the quantized images as conpared to he iieasureimeuts m the origina nuage. | The vertical displacement of these lines shows that the measurement errors are systematically larger in the quantized images as compared to the measurements in the original image. |
This increase is relatively simall for qo l. but increases rapidly for coarser quantization values. | This increase is relatively small for q $\geq$ 1, but increases rapidly for coarser quantization values. |
Figure 5 shows a similar plot of he Standard Deviation of the stellar ceutroid measurements. iu units of pixels. as a function of the magnitude of the star aud the q quantization factor. | Figure 5 shows a similar plot of the Standard Deviation of the stellar centroid measurements, in units of pixels, as a function of the magnitude of the star and the q quantization factor. |
In the original unuquautized images. the positional uncertainty decreases from σ=0.21 pixels for jo n = 20 stars. to σ=0035 pixels for 1ο stars that are 3H magnitudes brighter. | In the original unquantized images, the positional uncertainty decreases from $\sigma = 0.24$ pixels for the m = 20 stars, to $\sigma = 0.035$ pixels for the stars that are 3 magnitudes brighter. |
The )ositional nieasureukut uncertainties are larger iu je quantized mage by about the same factor as 10 increase in the magnituο uncertainties shown oeu Figure 1. | The positional measurement uncertainties are larger in the quantized image by about the same factor as the increase in the magnitude uncertainties shown in Figure 4. |
Tn order to better quantity the effects shown iu jese 2 figures. Table 1 sunnuarizes how the noise aud the measurement uncertaimlos Inerease as the inages are quantized more coarsely. | In order to better quantify the effects shown in these 2 figures, Table 1 summarizes how the noise and the measurement uncertainties increase as the images are quantized more coarsely. |
The measured )ereentage merease in the VAD noise level in the quantized mages is eiven in column 2 aud agrees exactly with the prediced value. from equation 10.. | The measured percentage increase in the MAD noise level in the quantized images is given in column 2 and agrees exactly with the predicted value from equation \ref{eq:fractionalnoise}. |
More strikingly. the maenitude and position uncertainties for the fainest stars also increase by about the same factor as the toisc, as shown in columns 3 and. | More strikingly, the magnitude and position uncertainties for the faintest stars also increase by about the same factor as the noise, as shown in columns 3 and 4. |
This ueans that equation 10 also provides a good esnuate of how much the neasurement unucert:ünties of the faintest objects iu an image. which are lianited by the backerouud roise, will increase when the image is quautized. | This means that equation \ref{eq:fractionalnoise} also provides a good estimate of how much the measurement uncertainties of the faintest objects in an image, which are limited by the background noise, will increase when the image is quantized. |
Finally. σοι 5 aud 6 in Table 1 give the corresponding increase i measurenmoeu uncertainties Or stars that are 5 magnitudes. or a factor of LOO. brighter than the image cetection threshold. | Finally, columns 5 and 6 in Table 1 give the corresponding increase in measurement uncertainties for stars that are 5 magnitudes, or a factor of 100, brighter than the image detection threshold. |
As expected. these bvieghter objects are relatively ess affected by quantization because he iuhereut DPoissonuiui noise iu the brighter pixels is larger hau the spacing hetween the quantized levels. | As expected, these brighter objects are relatively less affected by quantization because the inherent Poissonian noise in the brighter pixels is larger than the spacing between the quantized levels. |
Also. the small formal statistical errors on the naguitudes and posiious of the brighter stars (which are less than (LOOL of a naguitude or pixel. respectively. iu this Case] are often iusiguificant compared to the svstematic errors in the absolute calibration of the measurements. | Also, the small formal statistical errors on the magnitudes and positions of the brighter stars (which are less than 0.001 of a magnitude or pixel, respectively, in this case) are often insignificant compared to the systematic errors in the absolute calibration of the measurements. |
For comparison. Figures 6 aud 7 show that if the quantized pixels are not direred then he measurement orYOYs are larecr lan in the dithered case shown previously in Figures { and 5. | For comparison, Figures 6 and 7 show that if the quantized pixels are not dithered then the measurement errors are larger than in the dithered case shown previously in Figures 4 and 5. |
It is strikine tha while dithering provides a uodest increase πι he accuracy of the ceutroid neasurenieuts. it ereatly iniproves the magnitude neasurements. | It is striking that while dithering provides a modest increase in the accuracy of the centroid measurements, it greatly improves the magnitude measurements. |
Futwer juvestigation has shown hat the larecr nuvgnitude errors are almost entirely caused by a craluatic increase iu the errors in the SExtractor ]vackeround estimate around each star if the quaiitized mage is not dithered. | Further investigation has shown that the larger magnitude errors are almost entirely caused by a dramatic increase in the errors in the SExtractor background estimate around each star if the quantized image is not dithered. |
SExtractor asstucs hat the best estimate of the true backeround level is given by the peak in the histogram of the pixcJ values (1.60. the mode) aud | SExtractor assumes that the best estimate of the true background level is given by the peak in the histogram of the pixel values (i.e., the mode) and |
colponcuts. and = 70.100 IK aud (Πο) 2 for the wing material near IRAS LA. Low amass star forming site suggested to be a binary svsteni separated by ~ 0.37 (Colicuetal.LOSL).. | components, and = 70–100 K and ) = for the wing material near IRAS 4A. Low mass star forming site suggested to be a binary system separated by $\sim$ $\arcsec$ \citep{Cohen84}. |
Single Gaussian cussion profiles were detected im both transitions with a central velocity typical of the region at ~ σιsi. Mori | Single Gaussian emission profiles were detected in both transitions with a central velocity typical of the region at $\sim$ 6.4. |
arty-Schievenetal.(1995) used aud2oy.. 218 GIIZ) aud CS observations to derive a kinetic teniperature of ~ LO Iv and a spatial deusity ofcnr. | \citet{MS95} used and, 218 GHz) and CS observations to derive a kinetic temperature of $\sim$ 40 K and a spatial density of. |
.. Robertset.al.(2002) and Roberts&Millar(2007) examined transition ratios of several species. ποιοπιο the (150 Giz) and (72 GIIz) trausitious ofToCO.. aud quote a bolometric teniperature of 97 I& for the region. | \citet{Rob02} and \citet{RM07} examined transition ratios of several species, including the (150 GHz) and (72 GHz) transitions of, and quote a bolometric temperature of 97 K for the region. |
Because the transitions emploved by Moriarty-Schieveuetal(1995) are biased toward temperatures < 50 Iv and the LVC prediction of our observed transition ratio supports a higher kinetic temperature. we have assumed 100 I for our analysis while also testing temperatures of LO and 150 Is. Additional measurements lave been conducted for L1551 for the transition bv Aravaetal.(2006)... | Because the transitions employed by \citet{MS95} are biased toward temperatures $\leq$ 50 K and the LVG prediction of our observed transition ratio supports a higher kinetic temperature, we have assumed 100 K for our analysis while also testing temperatures of 40 and 150 K. Additional measurements have been conducted for L1551 for the transition by \citet{Ara06}. . |
The spatial deusitv of IBS 5 las Όσοι estimated: usine several methods. | The spatial density of IRS 5 has been estimated using several methods. |
Butueretal.(1991) fouud an average volue deusitv of within au aneular radius of 1378 by modeling Eu-ufrared eiission. while Fulleretal.(1995). estimate usus OO emission. | \citet{But91} found an average volume density of within an angular radius of $\farcs$ 8 by modeling far-infrared emission, while \citet{Ful95} estimate using O emission. |
M«larty-Schievenetal.(1995). derived the significantly higher deusity of using transition ratios of the CS moleciule. | \citet{MS95} derived the significantly higher density of using transition ratios of the CS molecule. |
The best-studied region of massive star formation to date (see review by Cenzel&Stutzikd 1989)) aud extremely bright iun ai plethora of chemical species (brigbtuess temperatures from Orion-KL are 2 5« that of anv other source in our studv). | The best-studied region of massive star formation to date (see review by \citealt{GS89}) ) and extremely bright in a plethora of chemical species (brightness temperatures from Orion-KL are $>$ $\times$ that of any other source in our study). |
This is the ouly source in our sanuple for which previous observations of the transition of have been made (Wilsonetal.1980:Myers.&BustonBastienetal. 1985). | This is the only source in our sample for which previous observations of the transition of have been made \citep{Wil80,MB80,Bas85}. |
. Outflows. shocks. aud. turbulence arising from) newly formed stars in the region lave led to a complex velocity structure comprised of several distinct conponents (Blakeetal.1987).. at least three of which are captured iu our observations: the hot core ~ 6]anο, Ar> 10st. ~ 300 KR). compact ridge o8si. Apo Lhe. ~ 135 K) and extended ridge ~ 9103. Aw ~ Elaus. ~ 135 KR). cach with spatial density 2((I15)) > (Mauguuetal.1993:Mig&Wootten1993). | Outflows, shocks, and turbulence arising from newly formed stars in the region have led to a complex velocity structure comprised of several distinct components \citep{Blake87}, at least three of which are captured in our observations: the hot core $\sim$ 6, $\Delta\nu >$ 10, $\sim$ 300 K), compact ridge $\sim$ 8, $\Delta\nu$ $\sim$ 4, $\sim$ 135 K), and extended ridge $\sim$ 9–10, $\Delta\nu$ $\sim$ 4, $\sim$ 135 K), each with spatial density ) $\gtrsim$ \citep{Man93,MW93}. |
. Ow J= lLspectrum exhibits an anomalously intense velocity component at ~ 10.31... which also appears as a much smaller coutribution iu the J=3 trausition. | Our $J=4$ spectrum exhibits an anomalously intense velocity component at $\sim$ 10.3, which also appears as a much smaller contribution in the $J=3$ transition. |
A feature at this velocity exists LO” north of our observed position (αποσπάetal.19903... leading us fo suspect a pointing error. | A feature at this velocity exists $\arcsec$ north of our observed position \citep{Man90}, leading us to suspect a pointing error. |
Poiuting was checked prior to observing. aud the Oriou-KL sceau was followed bv observations of OMC-2 IRS Land NGC 2021. | Pointing was checked prior to observing, and the Orion-KL scan was followed by observations of OMC-2 IRS 4 and NGC 2024. |
If a pointing error is to be blamed. it should manifest itself in the subsequent observations of OMC-2 IRS 1 and NGC 2021. but we could find no evidence for this. | If a pointing error is to be blamed, it should manifest itself in the subsequent observations of OMC-2 IRS 4 and NGC 2024, but we could find no evidence for this. |
The possibilities of an unidentified line or rest frequeucy error were also ruled out. | The possibilities of an unidentified line or rest frequency error were also ruled out. |
This spectrum is the average of ouly two Guutually consistent) scans conducted cing a single run. | This spectrum is the average of only two (mutually consistent) scans conducted during a single run. |
We suggest that this cussion arises from the extended ridge. which is typically observed around ~ 9st. but occasionally as high as 10st. | We suggest that this emission arises from the extended ridge, which is typically observed around $\sim$ 9, but occasionally as high as 10. |
. This interpretation.- and the physical parameters derived frou it. should be applied cautiously for the reasons explained above. | This interpretation, and the physical parameters derived from it, should be applied cautiously for the reasons explained above. |
Because of the disparity between the relative iuteusitv of the 10.3 feature in cach trausition. oulv the transition ratio for the hot core component could be fit by the LVG model. aud the extraordinarily biel iutensities precluded the determination of useful density limits for the other components through the procedure described ii refLTE.. | Because of the disparity between the relative intensity of the 10.3 feature in each transition, only the transition ratio for the hot core component could be fit by the LVG model, and the extraordinarily high intensities precluded the determination of useful density limits for the other components through the procedure described in \\ref{LTE}. . |
Broad SiO enmissiou coupled with relatively weak and CIT;OOITemission iudicates the presence ofau cherectic outflow in its earliest phase. sugeestiug that | Broad SiO emission coupled with relatively weak and OHemission indicates the presence ofan energetic outflow in its earliest phase, suggesting that |
A natural first step in explaining the observed emission line properties is to investigate whether the line ratios can be explained by photoionization from a single dominant star, an approach adopted qualitatively by Peimbertetal.(1975). | A natural first step in explaining the observed emission line properties is to investigate whether the line ratios can be explained by photoionization from a single dominant star, an approach adopted qualitatively by \citet{pei75}. |
. We will follow their discussion except that we shall use the spectral types summarized in Goudis(1982) and use the Q(H) values from the recent study by Heapetal.(2006). | We will follow their discussion except that we shall use the spectral types summarized in \citet{gou} and use the Q(H) values from the recent study by \citet{srh06}. |
. There are many candidate ionizing stars within Barnard's Loop and its extension into theBubble,, the brightest six in the LyC are the brightest Trapezium O6 V star with an expected LyC luminosity of 6 x 1045 photons1, the cooler (spectral type O9.5 II) 6 OOri with an expected LyC luminosity of 5.6x1045 photon ως OOri (O9.5 Ib) at 5.6x1045 photon ,, and z Ori (spectral type O9 IIT) at 6.6x1055 photons~!, plus aand σ Ori (both 09.5 V and 1.5x1045 photon s~')). | There are many candidate ionizing stars within Barnard's Loop and its extension into the, the brightest six in the LyC are the brightest Trapezium O6 V star with an expected LyC luminosity of 6 x $^{48}$ photons, the cooler (spectral type O9.5 II) $\delta$ Ori with an expected LyC luminosity of $5.6\times 10^{48}$ photon, $\zeta$ Ori (O9.5 Ib) at $5.6\times 10^{48}$ photon , and $\iota$ Ori (spectral type O9 III) at $6.6\times 10^{48}$ photon, plus and $\sigma$ Ori (both 09.5 V and $1.5\times 10^{48}$ photon ). |
It is unclear if e OOri belongs in this list. | It is unclear if $\epsilon$ Ori belongs in this list. |
With a spectral type of BO Ia it is cooler than the stars studied by Heapetal.(2006). | With a spectral type of B0 Ia it is cooler than the stars studied by \citet{srh06}. |
. Vaccaetal.(1996) did include one star (HD 37128) of this spectral type in their study and found it to be be 6000 K cooler than an O9.5 Ia spectral type star (HD 30614) and would therefore have a much lower LyC luminosity. | \citet{vacca} did include one star (HD 37128) of this spectral type in their study and found it to be be 6000 K cooler than an O9.5 Ia spectral type star (HD 30614) and would therefore have a much lower LyC luminosity. |
Since ¢ OOri has a spectral type of O9.5 Ib, we conclude that e OOri's LyC luminosity is much lower than 5.6x1035 photon aand we will not include it in our tally. | Since $\zeta$ Ori has a spectral type of O9.5 Ib, we conclude that $\epsilon$ Ori's LyC luminosity is much lower than $5.6\times 10^{48}$ photon and we will not include it in our tally. |
The location of these stars are shown in Figure 1. | The location of these stars are shown in Figure 1. |
Although there are many additional stars of later spectral type and less luminous stars are distributed throughout the inner Orion constellation, their contribution to photoionization ofBarnard's Loop must be minimal. | Although there are many additional stars of later spectral type and less luminous stars are distributed throughout the inner Orion constellation, their contribution to photoionization ofBarnard's Loop must be minimal. |
For a summaryof the properties of these stars see Table 1.1.IV of Goudis (1982). | For a summaryof the properties of these stars see Table 1.1.IV of \citet{gou}. . |
can be calculated. | can be calculated. |
Although the RV semi-amplitude K and the period P are determined by the RV curve. Mig. Mi and sin?{ remain free parameters. | Although the RV semi-amplitude $K$ and the period $P$ are determined by the RV curve, $M_{\rm sdB}$, $M_{\rm comp}$ and $\sin^3i$ remain free parameters. |
Nevertheless. the masses can be constrained by assuming tidal synchronisation (see also Napiwotzki et al. 200100. | Nevertheless, the masses can be constrained by assuming tidal synchronisation (see also Napiwotzki et al. \cite{napiwotzki5}) ). |
Combining the orbital parameters with an estimate of the sdB mass and with the determination of its vjsin/ and surface gravity. allows the mass of the invisible companion to be constrained. | Combining the orbital parameters with an estimate of the sdB mass and with the determination of its $v_{\rm rot}\sin{i}$ and surface gravity, allows the mass of the invisible companion to be constrained. |
The mass of the sdB primary is taken from the population synthesis models (Han et al. 2002.. 2003)) | The mass of the sdB primary is taken from the population synthesis models (Han et al. \cite{han1}, , \cite{han2}) ) |
that predict a mass range of My 00.37 — MM... for sdBs in binaries. which experienced a common envelope ejection, | that predict a mass range of $M_{\rm sdB}$ 0.37 – $_{\rm \odot}$ for sdBs in binaries, which experienced a common envelope ejection. |
The mass distribution shows a sharp peak at a mass of about 0.47M. (see Fig. | The mass distribution shows a sharp peak at a mass of about $0.47\,{\rm M_{\odot}}$ (see Fig. |
22 of Han et al. 2003)) | 22 of Han et al. \cite{han2}) ) |
ranging from 0.43 to MM... | ranging from 0.43 to $_{\rm \odot}$. |
This theoretical mass distribution is consistent with analyses of close binary systems (e.g. Geter et al. 2007)) | This theoretical mass distribution is consistent with analyses of close binary systems (e.g. Geier et al. \cite{geier1}) ) |
as well as asteroseismic analyses of pulsating sdBs (see Charpinet et al. | as well as asteroseismic analyses of pulsating sdBs (see Charpinet et al. |
2008 and references therein). | \cite{charpinet} and references therein). |
If the rotational period of the sdB primary is synchronised the rotational velocity v=2gRap/P can be calculated. | If the rotational period of the sdB primary is synchronised the rotational velocity $v_{\rm rot}= 2 \pi R_{\rm sdB}/P$ can be calculated. |
The radius of the primary is given by the mass radius relation R=JMagG/g. | The radius of the primary is given by the mass radius relation $R = \sqrt{M_{\rm sdB}G/g}$. |
The measurement of the projected rotational velocity vj,sin/ therefore allows us to constrain the inclination angle i. | The measurement of the projected rotational velocity $v_{\rm rot}\,\sin\,i$ therefore allows us to constrain the inclination angle $i$. |
For the most likely sdB mass μμ=0.47M.« the mass function can be solved. and both the inclination angle and the companion mass can be derived. | For the most likely sdB mass $M_{\rm sdB}=0.47\,{\rm M_{\odot}}$ the mass function can be solved, and both the inclination angle and the companion mass can be derived. |
The errors are caleulated by chosing the most extreme values for the input parameters within their respective error limits. | The errors are calculated by chosing the most extreme values for the input parameters within their respective error limits. |
In order to account for the theoretical uncertainity in sdB mass. we adopted the predicted mass range for the sdB (0.43—0.47 M.) and calculated the lower limit for the companion mass under the assumption that μμ=0.43M... | In order to account for the theoretical uncertainity in sdB mass, we adopted the predicted mass range for the sdB $0.43-0.47\,{\rm M_{\odot}}$ ) and calculated the lower limit for the companion mass under the assumption that $M_{\rm sdB}=0.43\,{\rm M_{\odot}}$. |
The error budget is dominated by the uncertainties in the vj;sin; and logg measurements. | The error budget is dominated by the uncertainties in the $v_{\rm rot}\sin{i}$ and $\log{g}$ measurements. |
The mass function provides a lower limit to the mass of the invisible companion of 0.35M... | The mass function provides a lower limit to the mass of the invisible companion of $0.35\,{\rm M_{\odot}}$. |
In the case of a white dwarf primary it 1s impossible to hide the contribution of a main sequence star even of the lowest mass in optical/NIR spectra since these are mtrinsically faint. | In the case of a white dwarf primary it is impossible to hide the contribution of a main sequence star even of the lowest mass in optical/NIR spectra since these are intrinsically faint. |
This is not the case for sdB stars. | This is not the case for sdB stars. |
A main sequence companion with a mass lower than 0.45M.« can not be excluded because its luminosity would be too low to be detectable in the spectra (Lisker et al. 2005)). | A main sequence companion with a mass lower than $0.45\,{\rm M_{\odot}}$ can not be excluded because its luminosity would be too low to be detectable in the spectra (Lisker et al. \cite{lisker}) ). |
This is the reason why the companions’ nature still remains unknown for most of the 280 sdB systems in the catalogue of Ritter Kolb (2009)). | This is the reason why the companions' nature still remains unknown for most of the $\approx$ 80 sdB systems in the catalogue of Ritter Kolb \cite{ritter}) ). |
Additional information is needed. | Additional information is needed. |
No spectral features of a cool main sequence star are present in the optical spectra of 6687. | No spectral features of a cool main sequence star are present in the optical spectra of 687. |
Furthermore. Farthi et al. (2005)) | Furthermore, Farihi et al. \cite{farihi}) ) |
included 6687 in a near-infrared imaging survey to search for low-luminosity companions to white dwarfs and found no evidence for an infrared excess which could be caused by à cool main sequence companion. | included 687 in a near-infrared imaging survey to search for low-luminosity companions to white dwarfs and found no evidence for an infrared excess which could be caused by a cool main sequence companion. |
As the lower limit for the mass of 6687's companion derived from the mass function. is lower than 0.45M... additional information is needed to clarify its nature. | As the lower limit for the mass of 687's companion derived from the mass function is lower than $0.45\,{\rm M_{\odot}}$, additional information is needed to clarify its nature. |
We made use of the gravity and projected rotational velocity to constrain the mass to 0.71703:M... | We made use of the gravity and projected rotational velocity to constrain the mass to $0.71_{-0.21}^{+0.22}\,{\rm M_{\odot}}$. |
The companion therefore can not be ἃ main sequence star but has to be a white dwarf. | The companion therefore can not be a main sequence star but has to be a white dwarf. |
Taking into account the possible mass range it is very likely to be of C/O composition, | Taking into account the possible mass range it is very likely to be of C/O composition. |
Its mass exceeds that of an average white dwarf. | Its mass exceeds that of an average white dwarf. |
6687 isonly the fourth sdB star. for which the white dwarf nature of the companion could be shown unamibiguously. | 687 isonly the fourth sdB star, for which the white dwarf nature of the companion could be shown unamibiguously. |
The others have been discovered by analysing ellipsoidal light variations 11930+2752. 00101+039) or eclipses 00422-5421]. Orosz Wade 1999)) 1 | The others have been discovered by analysing ellipsoidal light variations $+$ 2752, $+$ 039) or eclipses $+$ 5421, Orosz Wade \cite{orosz}) ). |
1930-2752 and PG00101-039 have been confirmed with the method used here (Geier et al. 2007. 2008). | $+$ 2752 and $+$ 039 have been confirmed with the method used here (Geier et al. \cite{geier1,geier2}) ). |
The derived companion mass was caleulated. under the assumption of orbital synchronisation. | The derived companion mass was calculated under the assumption of orbital synchronisation. |
Since. theoretical synchronisation timescales. for hot stars with radiative envelopes are not consistent (Zahn 1977:; Tassoul Tassoul 1992)). empirical evidence for orbital sychronisation in sdB binaries is needed. | Since theoretical synchronisation timescales for hot stars with radiative envelopes are not consistent (Zahn \cite{zahn}; Tassoul Tassoul \cite{tassoul}) ), empirical evidence for orbital sychronisation in sdB binaries is needed. |
Geier et al. (2008)) | Geier et al. \cite{geier2}) ) |
found such evidence by detecting a variation in the lighteurve of the sdB+WD binary 00101+039. which could be identified as ellipsoidal deformation of the sdB. Since the orbital period of 001014039 ts 0.57 d. sdB binaries with shorter periods like 6687 are very likely synchronised as well. | found such evidence by detecting a variation in the lightcurve of the sdB+WD binary $+$ 039, which could be identified as ellipsoidal deformation of the sdB. Since the orbital period of $+$ 039 is $0.57\,{\rm d}$ , sdB binaries with shorter periods like 687 are very likely synchronised as well. |
Recently van Grootel et al. (2008)) | Recently van Grootel et al. \cite{vangrootel}) ) |
performed an asteroseismic analysis of the pulsating sdB binary Feige48 and for the first time proved orbital sychronisation in this way. | performed an asteroseismic analysis of the pulsating sdB binary Feige48 and for the first time proved orbital sychronisation in this way. |
The orbital period of Feige 48 (0.36 d) is very similar tothe one of 6687. | The orbital period of Feige 48 $0.36\,{\rm d}$ ) is very similar tothe one of 687. |
Furthermore. the atmospheric parameters of 6687 (Tay=24300 K. logg= 5.32) indicate that it has already evolved away | Furthermore, the atmospheric parameters of 687 $T_{\rm eff}=24\,300\,{\rm K}$ , $\log{g}=5.32$ ) indicate that it has already evolved away |
is the number of photometric objects in sample i. | where $N^{\rm phot}_i$ is the number of photometric objects in sample $i$. |
This wherecircumventsNphet the issue of which photometric objects to cross-correlate against a set of spectroscopic objects in a chosen bin of redshift. | This circumvents the issue of which photometric objects to cross-correlate against a set of spectroscopic objects in a chosen bin of redshift. |
Clearly photometric samples which peak at very different redshifts from the spectroscopic sample are significantly down-weighted in the sum. | Clearly photometric samples which peak at very different redshifts from the spectroscopic sample are significantly down-weighted in the sum. |
Note that our method also down-weights both objects with unusual colours that might have multi-peaked PDFs and objects with poorly constrained photometry, such as near survey limits, where the PDF might be very broad. | Note that our method also down-weights both objects with unusual colours that might have multi-peaked PDFs and objects with poorly constrained photometry, such as near survey limits, where the PDF might be very broad. |
Since the binning is so far arbitrary we can consider the limit where each slice in Eq. (6)) | Since the binning is so far arbitrary we can consider the limit where each slice in Eq. \ref{eqn:wpvarweight}) ) |
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