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In our implementation, each grid point is moved slightly away from its nearest neighbor for a preset number of iterations. | In our implementation, each grid point is moved slightly away from its nearest neighbor for a preset number of iterations. |
The effect is illustrated in Fig. | The effect is illustrated in Fig. |
3 for a random 2D sampling of a Gaussian density profile. | \ref{smooth} for a random 2D sampling of a Gaussian density profile. |
The top panels shows the initial unsmoothed distribution while the lower panels showed the triangulated point distribution after the smoothing algorithm has been applied. | The top panels shows the initial unsmoothed distribution while the lower panels showed the triangulated point distribution after the smoothing algorithm has been applied. |
The plots in the right column show the neighbor distances as a function of radius. | The plots in the right column show the neighbor distances as a function of radius. |
In the smoothed grid, the distances are much less scattered and follows the Gaussian profile (shown as the light colored full curve) much more accurately than in the top panel. | In the smoothed grid, the distances are much less scattered and follows the Gaussian profile (shown as the light colored full curve) much more accurately than in the top panel. |
The smoothing strategy should not be exaggerated, i.e., moving the points too slowly and iterating trough too many stpdf, because the algorithm will then act as an annealing process and it will result in a perfectly regular grid where all variations in the point distribution due to the underlying density field is smeared out. | The smoothing strategy should not be exaggerated, i.e., moving the points too slowly and iterating trough too many stpdf, because the algorithm will then act as an annealing process and it will result in a perfectly regular grid where all variations in the point distribution due to the underlying density field is smeared out. |
By doing it right, however, a smooth grid can be obtained while the underlying density structure is still preserved in the grid. | By doing it right, however, a smooth grid can be obtained while the underlying density structure is still preserved in the grid. |
We have found empirically that by using 25 iterations and moving the closest neighbors about of the distance away from each other results in a sufficiently smooth grid that preserves the underlying physical structure well. | We have found empirically that by using 25 iterations and moving the closest neighbors about of the distance away from each other results in a sufficiently smooth grid that preserves the underlying physical structure well. |
During gridding of our source model, we also distribute a number of points randomly on the surface of a sphere surrounding our model. | During gridding of our source model, we also distribute a number of points randomly on the surface of a sphere surrounding our model. |
These points are also Delaunay triangulated and connected to the model grid points, but they do not represent anything except the surface of our computational domain. | These points are also Delaunay triangulated and connected to the model grid points, but they do not represent anything except the surface of our computational domain. |
Whenever a photon reaches one of these sink points, it is considered to have escaped the model. | Whenever a photon reaches one of these sink points, it is considered to have escaped the model. |
The photon transport itself goes along Delaunay lines only, from one point to another, which makes integration of Eq. | The photon transport itself goes along Delaunay lines only, from one point to another, which makes integration of Eq. |
1 particularly simple and very fast. | \ref{radtran} particularly simple and very fast. |
In the three-dimensional Delaunay triangulation, the expectation value for the number of lines attached to a grid point is approximately 16 (?) and the spatial sampling of J, is thus limited to this number of directions. | In the three-dimensional Delaunay triangulation, the expectation value for the number of lines attached to a grid point is approximately 16 \citep{ritzerveld2006} and the spatial sampling of $J_\nu$ is thus limited to this number of directions. |
However, we still need to trace a number of photons along each Delaunay line, not only in order to sample the frequency band properly, but also because we cannot conserve momentum stringently with a single photon on this grid. | However, we still need to trace a number of photons along each Delaunay line, not only in order to sample the frequency band properly, but also because we cannot conserve momentum stringently with a single photon on this grid. |
In principle, a photon passing a grid point froma certain direction should continue to travel in the exact same direction. | In principle, a photon passing a grid point from a certain direction should continue to travel in the exact same direction. |
This is in general not possible due to the random orientation of the Delaunay lines, so instead we choose one of the two outgoing Delaunay lines (£1 and £2) that make the smallest angle with the original direction of the photon. | This is in general not possible due to the random orientation of the Delaunay lines, so instead we choose one of the two outgoing Delaunay lines $\ell1$ and $\ell2$ ) that make the smallest angle with the original direction of the photon. |
The outgoing line is picked at random, but weighted by the ratio of the two angles, where Z;«4Z». | The outgoing line is picked at random, but weighted by the ratio of the two angles, where $\angle_1 < \angle_2$. |
The same procedure is used at all subsequent grid points (using the original momentum vector to determine the outgoing direction) until the photons escape the model. | The same procedure is used at all subsequent grid points (using the original momentum vector to determine the outgoing direction) until the photons escape the model. |
By sending a number of photons along each initial Delaunay line, we thus probe, not a single line of sight, but rather a cone, while still conserving momentum on average. | By sending a number of photons along each initial Delaunay line, we thus probe, not a single line of sight, but rather a cone, while still conserving momentum on average. |
An example of the photon propagation is shown in Fig. | An example of the photon propagation is shown in Fig. |
4 for a single point and a single direction. | \ref{photprop} for a single point and a single direction. |
Because of the relative low number of photons needed to probe the spatial directions, we can allow ourself to increase the number of photons used to sample different frequencies, while we still maintain a low (initial) number of photons per grid point. | Because of the relative low number of photons needed to probe the spatial directions, we can allow ourself to increase the number of photons used to sample different frequencies, while we still maintain a low (initial) number of photons per grid point. |
The inset in Fig. | The inset in Fig. |
4 shows the distribution of the location where the photons reach the surface of the grid. | \ref{photprop} shows the distribution of the location where the photons reach the surface of the grid. |
This distribution is reasonably well described by a Gaussian distribution around the intersection of the original momentum vector and the surface. | This distribution is reasonably well described by a Gaussian distribution around the intersection of the original momentum vector and the surface. |
The number of initial photons is a user-defined setting, but as a default value, we use five times the number of neighbor points, so that each neighbor is initially probed by five photons. | The number of initial photons is a user-defined setting, but as a default value, we use five times the number of neighbor points, so that each neighbor is initially probed by five photons. |
These photons are distributed evenly across a frequency range of «2σ with respect to the line center so that the median photon coincides with the local rest frequency. | These photons are distributed evenly across a frequency range of $\pm 3 \sigma$ with respect to the line center so that the median photon coincides with the local rest frequency. |
o is determined by the local turbulent velocity dispersion through the user-defined Doppler b-parameter. | $\sigma$ is determined by the local turbulent velocity dispersion through the user-defined Doppler b-parameter. |
Any given grid point will see more Delaunay connections coming from high density regions than from low density regions, simply because the grid point density is higher in high density regions. | Any given grid point will see more Delaunay connections coming from high density regions than from low density regions, simply because the grid point density is higher in high density regions. |
Because of this inhomogeneity in the angular distribution of Delaunay connection, care must be taken when averaging the radiation field using Eq. 6.. | Because of this inhomogeneity in the angular distribution of Delaunay connection, care must be taken when averaging the radiation field using Eq. \ref{jbar}. |
In our implementation, this equation reduces to a discrete sum where N is the number ofDelaunay neighbors. | In our implementation, this equation reduces to a discrete sum where $N$ is the number ofDelaunay neighbors. |
ω; is a weight that is proportional to the solid angle represented by the i'th Delaunay line. | $\omega_i$ is a weight that is proportional to the solid angle represented by the $i$ 'th Delaunay line. |
This angle corresponds strictly to a surface area on a unit sphere, but we use the area of the Voronoi facet that corresponds to the Delaunay line as a good approximation (within 10%)). | This angle corresponds strictly to a surface area on a unit sphere, but we use the area of the Voronoi facet that corresponds to the Delaunay line as a good approximation (within ). |
Figure 5 shows a comparison of J, between andRATRAN. | Figure \ref{flatdisk} shows a comparison of $J_\nu$ between and. |
. The input model is a thin flat disk with a density profile oc 77!. | The input model is a thin flat disk with a density profile $\propto r^{-1}$ . |
The radius is 500 AU and the height is 50 AU. | The radius is 500 AU and the height is 50 AU. |
noise in the observed. data. and necessitates the use of smoothing to the observed. data. | noise in the observed data, and necessitates the use of smoothing to the observed data. |
Mazeh Goldberg (1992) introduce an iterative algoritlim whose solution depends on the initial guess. | Mazeh Goldberg (1992) introduce an iterative algorithm whose solution depends on the initial guess. |
In the present work we [followed Tokovinin (1991. 1992) and constructed an algorithm = MANinnun LlIkelihood MáÁss. to derive the mass distribution of the extrasolar planets with a maximum likelihood approach. | In the present work we followed Tokovinin (1991, 1992) and constructed an algorithm — MAXimum LIkelihood MAss, to derive the mass distribution of the extrasolar planets with a maximum likelihood approach. |
aassunies that the planes of motion of the planets are randomly oriented in space and derives (he mass distribution directly by solving a set of numerically stable linear equations. | assumes that the planes of motion of the planets are randomly oriented in space and derives the mass distribution directly by solving a set of numerically stable linear equations. |
It does nol require anv smoothing of the data nor any iterative algorithm. | It does not require any smoothing of the data nor any iterative algorithm. |
aalso offers a natural wav to correct for the undetectecl planets. | also offers a natural way to correct for the undetected planets. |
The randonmness of the orbital planes of the discovered planets were questionecl recently bv Ilan et al. ( | The randomness of the orbital planes of the discovered planets were questioned recently by Han et al. ( |
2001). based on the analvsis of Hipparcos data. | 2001), based on the analysis of Hipparcos data. |
However a lew very recent studies (Pourbaix 2001: Pourbaix Arenou 2001: Zucker Mazeh. 2001a.b) showed that the Llipparocos data do not prove the nonrandomness of the orbital planes. allowing us to apply ito the sample of known minimun masses of (he planet candidates. | However a few very recent studies (Pourbaix 2001; Pourbaix Arenou 2001; Zucker Mazeh 2001a,b) showed that the Hipparocos data do not prove the nonrandomness of the orbital planes, allowing us to apply to the sample of known minimum masses of the planet candidates. |
In the course of preparing this paper lor publication we have learned about a similar paper by Jorissen. Mavor Ucdry (2001) that was posted on the Astroplivsies e-Print Archive (astro-ph). | In the course of preparing this paper for publication we have learned about a similar paper by Jorissen, Mayor Udry (2001) that was posted on the Astrophysics e-Print Archive (astro-ph). |
Like Heacox (1995). Jorrisen et al. | Like Heacox (1995), Jorrisen et al. |
derive first the distribution of the minimum masses and (hen apply (wo alternative algorithms to invert it (o the distribution of planet masses. | derive first the distribution of the minimum masses and then apply two alternative algorithms to invert it to the distribution of planet masses. |
One algorithum is a formal solution of an Abel integral equation and the other is the Richarcdson-Luev algorithm (e.g.. Heacox 1995). | One algorithm is a formal solution of an Abel integral equation and the other is the Richardson-Lucy algorithm (e.g., Heacox 1995). |
The first algorithm necessitates some degree of data smoothing and (he second one requires a series of iterations. | The first algorithm necessitates some degree of data smoothing and the second one requires a series of iterations. |
The results of the first algorithm depend on the degree of smoothing applied. and (hose of the second one on the nunmber of iterations performed. | The results of the first algorithm depend on the degree of smoothing applied, and those of the second one on the number of iterations performed. |
hhas no built-in [ree parameter. except the widths of the histogram bins. | has no built-in free parameter, except the widths of the histogram bins. |
In addition. Jorissen et ddicl not apply any correction to the selection effect we consider here. and displaved their results on a linear mass scale. | In addition, Jorissen et did not apply any correction to the selection effect we consider here, and displayed their results on a linear mass scale. |
We feel (hat a logarithmic scale ean illuminate some other aspects of the distribution. | We feel that a logarithmic scale can illuminate some other aspects of the distribution. |
Despite all the differences. our results ave completely consistent with those of Jorrisen et al.. | Despite all the differences, our results are completely consistent with those of Jorrisen et al., |
the sharp cutoff in the planet mass distribution at about 10AL. ancl (the small hieh-mass tail that extends up to about 20 iii particular. | the sharp cutoff in the planet mass distribution at about 10, and the small high-mass tail that extends up to about 20 in particular. |
seclion 2 presentsMAXLIMA.. while section 3 presents our results. | Section 2 presents, while section 3 presents our results. |
Section 4 discusses briefly our findings. | Section 4 discusses briefly our findings. |
In Fig. | In Fig. |
2 we show the distribution of the candidates from Table 1 (only stars no. | \ref{xyz} we show the distribution of the candidates from Table \ref{table:1} (only stars no. |
1 to 27) on top of the background of the 724 members from Paper I. We use the galactic rectangular coordinate system X,Y,Z with origin in the Sun, and axes pointing to the Galactic Centre (X), to the direction of galactic rotation (Y), and to the North Galactic Pole (Z). | 1 to 27) on top of the background of the 724 members from Paper I. We use the galactic rectangular coordinate system $X, Y, Z$ with origin in the Sun, and axes pointing to the Galactic Centre $X$ ), to the direction of galactic rotation $Y$ ), and to the North Galactic Pole $Z$ ). |
All classical white dwarfs except one are located within the tidal radius of the cluster. | All classical white dwarfs except one are located within the tidal radius of the cluster. |
All newly found candidates lie outside the tidal radius, hence they are no longer gravitationally bound to the cluster, but share the fate of hundreds of former main-sequence members that left the bound region. | All newly found candidates lie outside the tidal radius, hence they are no longer gravitationally bound to the cluster, but share the fate of hundreds of former main-sequence members that left the bound region. |
The five probable field white dwarfs (Nos. | The five probable field white dwarfs (Nos. |
23 to 27) are all at z » 10 pc. | 23 to 27) are all at z $>$ 10 pc. |
Of particular interest here is the distribution in the X, Y-plane. | Of particular interest here is the distribution in the $X,Y$ -plane. |
We note that all white dwarfs (except no. | We note that all white dwarfs (except no. |
16) follow the tilted distribution of the main-sequence stars. | 16) follow the tilted distribution of the main-sequence stars. |
By tidal interaction with the gravitational field of the Galaxy, stars can leave the cluster on both sides via the Lagrangian points L; and L2 of the Galaxy-cluster-star system, where L, is in the direction to the Galactic centre, i.e. towards larger (less negative) X, the Sun-facing side of the cluster, while Lz lies on the opposite side of the cluster centre. | By tidal interaction with the gravitational field of the Galaxy, stars can leave the cluster on both sides via the Lagrangian points $_1$ and $_2$ of the Galaxy-cluster-star system, where $_1$ is in the direction to the Galactic centre, i.e. towards larger (less negative) $X$, the Sun-facing side of the cluster, while $_2$ lies on the opposite side of the cluster centre. |
All white dwarfs outside the tidal radius (except nos. | All white dwarfs outside the tidal radius (except nos. |
16 and 22) populate the Sun-facing part of the cluster, and may have left it through L;. | 16 and 22) populate the Sun-facing part of the cluster, and may have left it through $_1$. |
The deficit of newly found candidates at longer distances from the Sun needs explanation. | The deficit of newly found candidates at longer distances from the Sun needs explanation. |
To investigate this we plot in Fig. | To investigate this we plot in Fig. |
3 the r', J and K, magnitudes as a function of the distance D from the Sun. | \ref{photlim} the $r'$, $J$ and $ K_s$ magnitudes as a function of the distance D from the Sun. |
The background points in Fig. | The background points in Fig. |
3 represent again the sample of 724 from Paper I. In the NIR distributions we note that the magnitude limit of the sample of 724 in the J and K, bands is at much brighter magnitudes than the 2MASS completeness limit of J = 15.8 and K, = 14.3 (see http://www.ipac.caltech.edu/2mass/releases/allsky/doc). | \ref{photlim} represent again the sample of 724 from Paper I. In the NIR distributions we note that the magnitude limit of the sample of 724 in the $J$ and $ K_s$ bands is at much brighter magnitudes than the 2MASS completeness limit of $J$ = 15.8 and $ K_s$ = 14.3 (see http://www.ipac.caltech.edu/2mass/releases/allsky/doc). |
For fainter red dwarfs or even brown dwarfs there was no optical counterpart in CMC1]4, i.e. in the CU subset to PPMXL. | For fainter red dwarfs or even brown dwarfs there was no optical counterpart in CMC14, i.e. in the CU subset to PPMXL. |
This is different with the white dwarfs. | This is different with the white dwarfs. |
We see from Fig. | We see from Fig. |
3 that the fainter, hitherto unknown Hyades white dwarfs all are well beyond the 2MASS completeness limit in J and K;. | \ref{photlim} that the fainter, hitherto unknown Hyades white dwarfs all are well beyond the 2MASS completeness limit in $J$ and $ K_s$. |
The photometric accuracy in the K, band at 16.0 typically is 0.25 mag, fainter ones have no accuracy estimate at all. | The photometric accuracy in the $ K_s$ band at 16.0 typically is 0.25 mag, fainter ones have no accuracy estimate at all. |
The situation is somewhat better in the J band. | The situation is somewhat better in the $J$ band. |
In the r' band the red and white dwarfs are comparable. | In the $r'$ band the red and white dwarfs are comparable. |
The faintest white dwarfs are near the completeness limit of CMC14 at r’ = 16.8. | The faintest white dwarfs are near the completeness limit of CMC14 at $r'$ = 16.8. |
With these remarks it becomes clear that we can reveal new Hyades white dwarfs beyond a distance of about 50 pc from the Sun in the CU subset of PPMXL only by chance. | With these remarks it becomes clear that we can reveal new Hyades white dwarfs beyond a distance of about 50 pc from the Sun in the CU subset of PPMXL only by chance. |
The PPMXL goes about 3 magnitudes deeper than its CU subset, so possibly white dwarfs with Hyades motion could be found therein. | The PPMXL goes about 3 magnitudes deeper than its CU subset, so possibly white dwarfs with Hyades motion could be found therein. |
However, PPMXL photometry in optical bands is from USNO-B1.0, and therefore inappropriate for this kind of work. | However, PPMXL photometry in optical bands is from USNO-B1.0, and therefore inappropriate for this kind of work. |
These distant white dwarfs could only be found by cross-matching PPMXL kinematic candidates with the catalogue from McCook&Sion(1999) if the latter would be complete down to the limiting magnitude of PPMXL. | These distant white dwarfs could only be found by cross-matching PPMXL kinematic candidates with the catalogue from \citet{1999ApJS..121....1M}
if the latter would be complete down to the limiting magnitude of PPMXL. |
where b and { ave (he Galactic latitude and longitude of the cluster. 0=R,4/D ancl By is the Suns Galactocentrie distance. | where $b$ and $\ell$ are the Galactic latitude and longitude of the cluster, $\delta = R_0/D$ and $R_0$ is the Sun's Galactocentric distance. |
ILowever. the cluster term is usually dominant. | However, the cluster term is usually dominant. |
Rough mean flux density (5,) estimates were made by assuming that the oll-pulse RAIS noise level was described by the radiometer equation. where Ti, is the total svsten temperature. C 1s the telescope gain. np—2 is the number of summed polarizations. and Av is the bandwidth. | Rough mean flux density $S_{\nu}$ ) estimates were made by assuming that the off-pulse RMS noise level was described by the radiometer equation, where $T\rmsub{tot}$ is the total system temperature, $G$ is the telescope gain, $n\rmsub{pol} = 2$ is the number of summed polarizations, and $\Delta \nu$ is the bandwidth. |
For the GBT 2GlIIz receiver. G—L9KJv! and Tin©23JI4Tuas. where Ty. is the contribution from the Galactic svuchrotron emission. | For the GBT $2\;
\GHz$ receiver, $G = 1.9\; \K\: \Jy^{-1}$ and $T\rmsub{sys} \approx
23\; \K + T\rmsub{sky}$, where $T\rmsub{sky}$ is the contribution from the Galactic synchrotron emission. |
This was caleulated by scaling the values from ? wilh a spectral index of —2.6. | This was calculated by scaling the values from \citet{hss+82} with a spectral index of $-2.6$. |
The typical uncertainty in these estimates of ο, is20%.. | The typical uncertainty in these estimates of $S_{\nu}$ is. |
We were able to record full polarization data for one observation per cluster. | We were able to record full polarization data for one observation per cluster. |
These were calibrated using the 25ILz noise diode on the GBT. and we searched lor a significant rotation measure (RAI) bv looking for a peak in the polarized flux. | These were calibrated using the $25\; \Hz$ noise diode on the GBT, and we searched for a significant rotation measure (RM) by looking for a peak in the polarized flux. |
We searched [rom —1000 to 1000radm.7. but in most cases. (he wwas not sulficient to constrain RAL | We searched from $-1000$ to $1000\; \mathrm{rad}\: \pmsq$, but in most cases, the was not sufficient to constrain RM. |
Thenotable exceptions are the (vo pulsars in NGC 6544. [or which we measure an average RM of ~158radi7. | Thenotable exceptions are the two pulsars in NGC 6544, for which we measure an average RM of $\sim 158\;
\mathrm{rad}\: \pmsq$. |
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