Bright Star Photometry  --- 1-D Profile Fitting

Introduction

Stars brighter than Ks ~ 5 mag are saturated even in Read 1 frames of 2MASS.  Therefore, some non-standard algorithms must be developed in order to obtain photometry of these stars.  The 1-D profile fitting algorithm reported here has been developed for this purpose, and has been adopted by the Version 3 pipeline of 2MASS.
 

Sample

The algorithm was developed by analyzing the 1-D profiles of a sample of 254 stars, including both stars saturated in 2MASS (209) and unsaturated stars (45). The unsaturated stars (~6 mags) are taken from 2MASS catalog.  The saturated stars are selected from ones that are  in the  "Bright Star Catalog" and that were observed in the J and K bands by Johnson et al. (1966, Comm. Lunar and Planet. Lab., 4, 99) and by Lee (1970, ApJ, 1970, 217). The selection criteria are as follows:
   1) no variable flag
   2) V < 3  for B stars
   3) V < 4  for A stars
   4) V < 5  for F stars
   5) V < 6  for G stars
   6) no V cut-off for stars later than G (e.g. K stars, M stars ...).

The photometric accuracy of the J and K band magnitudes of the saturated stars by Johnson et al. (1966) and by Lee (1970)  were claimed to be on the order of ~0.030 mag (1 sigma). Among the 209 saturated stars, the H band magnitudes were found for 110 from the CIO catalog. Given the heterogeneity of CIO, the 1 sigma error of the H band data is likely to be at least ~0.1 mag. Note that most of the variables are excluded by the selection criterion 1). For unsaturated stars, the J, H and Ks magnitudes are taken from 2MASS catalog. The accuracy of 2MASS magnitudes are better than 0.1 mag.
 
 

Model assumptions

The algorithm is based on the following assumptions:

       i) For a given camera (e.g. the J-band camera in northern observatory), counts
           (DN) by 'good pixels' are proportional to the brightness of the star in question.
      ii) The distribution of DN/pix is axial symmetric  (1-D model).
     iii) The 1-D PSF has the following function form (with 11 free parameters
            f0, f1, f2, f3, r0, r1, r2, r3, p1, p2, q):
             f=f0*exp(-(r/r0)^2) + f1*exp(-(r/r1)^p1) + f2*exp(-(r/r2)^p2) +  f3/(1+(r/r3)^2)^q .
     iv) The reference magnitudes (e.g. from Johnson et al. 1966) are accurate.
      v) Stars are non variables.
 
 

Data trimming

For each star in the sample, a corresponding Read 1 source was determined in each of the 3 bands (J, H, Ks). Read 1 data of pixels within a circle of of r = 60" (30pix * 2"/pix) centered on the peak of the source are collected from 2MASS database. The data are background subtracted and zero-point corrected. Then, 'bad pixels' are trimmed out. The `bad pixels' are defined as follows:

       i) Counts < 60 DN (the nominal 5 sigma rms).
      ii) Counts > 40000DN (the nominal saturation limit).
     iii) When more than 20 percent of pixels in an annulus (width=1pix) have
            counts < 60DN, all the pixels in that annulus are `bad'.
     iv) When more than 20 percent of pixels in an annulus have
            counts > 40000DN, all the pixels in that annulus are `bad'.
      v) Pixels in the adjacent annulus next to the one where more than 20
           percent pixels have counts > 40000DN, will also be 'bad'
           (due to asymmetry, these pixels are badly affected by the saturation).
     vi) For a given annulus, pixels with the logarithm of the counts higher
           than mean+2sigma or lower than mean-2sigma (i.e. '2sigma+'  pixels) are `bad'.

The remaining pixels are `good pixels'.
 

Fitting procedure

      i) In the first step, for each camera (total 6,  namely 3 bands times 2 hemispheres),
         `good pixels' of all stars are included to build a grand 1-D PSF. The counts
          of each star are normalized as following:    cnt_1=cnt*10^(-0.4*(m_ref-m_0)),
          where m_0 = 6, 5.5, 5 for J, H, Ks band, respectively.
     ii) An analytical 'best fit' function of the form given above (11 parameters) is
          manually determined (too many parameters for an automatic
          least square fit to converge). This is an iterative procedure, adjusting parameters
          by 'trial and fail' in order to obtain a good fit to the data. It should be noted that
          the final parameters of the analytical 1-D PSF are determined by
          the requirements given in v), although it is obvious that
          the parameters so determined should also give a good fit to the data
          of the 1-D PSF.
   iii) For each camera, the analytical grand 1-D PSF
          is then applied to the 'good pixels' (uniformly weighted)
          of each star, and an estimate of magnitude is obtained (m_fit)
          by the least square fit.
    iv) For each camera, the median and the standard dispersion of the
          magnitude deviation (m_fit - m_ref) are calculated for the sample,
          with the 3-sigma+ outliers excluded.
     v) Fine tune the parameters and repeat steps ii) --- iv) until
            (a) the median of the deviation is consistent with zero (i.e.  absolute value < 0.01);
            (b) the standard dispersion of the deviation is minimized;
            (c) the deviation v.s. magnitude plot is flat (i.e. no magnitude dependence);
            (d) no systematic trends (e.g. always going up or down, or bumps/dips at
                   the same r) in the plots of residuals (DN - model) v.s. radial distance r
                   for individual stars.
Note that the requirements (b), (c) and (d) are highly correlated.
 

Seeing Correction

Instead of constructing PSFs for different seeing shapes, we choose to do photometry for all stars observed by the same camera using a single PSF, and then correct the effect of different seeing conditions by providing a `seeing correction' (in magnitude). The correction is a continuous function of both the seeing shape and the brightness (magnitude):
                                 corr_see   =  a(m) + b(m)*ssh
where m is the magnitude, ssh the seeing shape. The function form of a(m) is:
                                 a = (a0 + a1*m + a2*m^3 + a3*m^5) * exp(-(m/m0)^n)
The function form of b(m) is the same:
                                 b = (b0 + b1*m + b2*m^3 + b3*m^5) * exp(-(m/m0)^n)
In order to avoid excessive values, a cutoff (abs(corr_see) < cut0) is applied.
 

Results


The final analytic 1-D PSFs of six cameras are specified by the parameters  given in the Table 1.
 
Table 1: Parameters of grand-1D PSFs 
Specification 
Parameters and plot 
Band
hemisphere
Stars 
included
outliers 
(excluded)
f0 
f1 
f2 
f3
r0
r1
r2
r3
p1
p2 
q
plot 
J 
s 
138
1
exp(10.05)
exp(8.02)
exp(5.85)
1.2
0.62
0.55
0.4
14
0.97
0.57 
0.9
figure 1
J 
n 
113
2
exp(9.97)
exp(8.26)
exp(6.27)
1.0
0.63
0.55
0.3
10
1.00
0.55
0.9
figure 2
H 
s 
81
2
exp(10.10)
exp(7.70)
exp(6.80)
1.6
0.61
0.50
0.4
8
0.95
0.65 
1.0
figure 3
H
n 
72
0
exp(10.23)
exp(8.30)
exp(6.60)
2.2
0.63
0.50
0.4
8
0.90
0.65 
1.0
figure 4
K 
s 
136
3
exp(10.02)
exp(8.10)
exp(6.40)
17
0.62
0.50
0.4
2
0.99
0.65
1.0
 figure 5
K 
n 
112
2
exp(9.97)
exp(8.72)
exp(6.05)
20
0.65
0.50
0.4
2
1.00
0.65
1.0
 figure 6

In Figure 7 we show the best-fit PSFs in a single plot (the H band and Ks band PSFs are shifted up by a factor of 10 and 100, respectively).
 
Figure 7

Standard dispersions are calculated for the deviation DMAG=(m_fit - m_ref), where for saturated stars in the J and K bands the m_ref is the magnitude taken from Johnson et al (1966)  or Lee (1970), and in the H band the CIO magnitude. For unsaturated stars m_ref is the magnitude taken from 2MASS database. Outliers of 3 sigma+ are excluded. Results are presented in Table 2. The `sigma_total' is the one sigma deviation of both saturated and unsaturated altogether. The `sigma_sat' is the one sigma deviation of saturated stars.  The `sigma_seecor' is the one sigma deviation of saturated stars after the seeing correction. Values of sigma_seecor can be regarded as our best estimates for the one sigma errors of the magnitudes derived using the algorithm presented here.
 
 
Table 2: Standard dispersions of magnitude deviation 
band 
hemisphere 
sigma_total 
number
sigma_sat 
sigma_seecor 
number
J 
S 
 0.08
138
0.09
 0.07
122 
J 
N 
 0.14
115
0.17
 0.13
87 
H 
S 
0.09
81
0.09
 0.09
65 
H 
N 
 0.14
72 
0.17
 0.13
44 
K
S 
 0.09
136
0.10
 0.08
120 
K 
N 
 0.12
112
0.15
 0.13
87

 
 

Parameters of seeing corrections, determined empirically, are listed in Table 3.
 
 
Table 3: Parameters of seeing corrections  
Band
hemisphere
a0 
a1 
a2 
a3
b0
b1
b2
b3
m0
n
J 
s 
0.0012
0.91
-0.16
0.017
-0.0013
-0.94
0.16
-0.018
4 
8
J 
n 
-0.056
0.93
-0.0054
-0.00023
0.055
-0.97
0.012
7.7E-5
5.5
8
H 
s 
0.98
0.82
-0.039
0.00041
-1.02
-0.90
0.043
-0.00044
7 
8
H
n 
1.42
0.74
-0.025
0.00029
-1.42
-0.74
0.025
-0.00029
7 
8
K 
s 
2.18
0.64
-0.14
0.0089
-2.24
-0.64
0.14
-0.0089
4
8
K 
n 
-0.0022
0.90
-0.11
0.0090
0.0020
-0.88
0.10
-0.0084
4
8

Plots of magnitude deviations (DMAG), before and after seeing corrections, versus see_shape are presented in the following figures. The corresponding seeing corrections are given. Stars of different magntudes and seeing correction for different magnitudes are color coded.
 
 
J band south 
BEFORE SEEING CORRECTION AFTER SEEING CORRECTION
Figure 8 Figure 9
J band north 
BEFORE SEEING CORRECTION AFTER SEEING CORRECTION
Figure 10 Figure 11
H band south 
BEFORE SEEING CORRECTION AFTER SEEING CORRECTION
Figure 12 Figure 13
H band north 
BEFORE SEEING CORRECTION AFTER SEEING CORRECTION
Figure 14 Figure 15
Ks band south 
BEFORE SEEING CORRECTION AFTER SEEING CORRECTION
Figure 16 Figure 17
Ks band north 
BEFORE SEEING CORRECTION AFTER SEEING CORRECTION
Figure 18 Figure 19

 

Table 4  lists results for all individual stars investigated here. Values like 99.99 and 99.999 mean `no data'. Columns are:

           (1) scan number as in 2MASS data base.
           (2) R.A. (2000) in degrees.           (3) Dec. (2000) in degrees.
           (4) - (6) J_ref, H_ref and K_ref are reference magnitudes taken either
                           from the literature (for saturated stars) or from 2MASS database
                           (for unsaturated stars).
           (7) - (9) J_fit, H_fit and K_fit are results from the fitting algorithm
                            presented here.
           (10) - (12) d_J, d_H, d_K are deviations:
                                 d_J = (J_fit - cor_J) - J_ref
                                  where cor_J is the seeing correction (column (19)). The fitting plot
                                  is attached to the deviation value of each band.
           (13) - (15) Xi_J, Xi_H, Xi_K are indicators of `goodness of fit'. The definition of,
                                  for example, Xi_J is the square root of the xi-square/N, where N is the
                                  number of pixels involved in the fit, and xi-square is the quadratic
                                  sum of deviations (in magnitudes) of individual pixels.  Hence, Xi_J
                                  indicates the average deviation (in mag) of individual pixels from the
                                  theoretical fit. A residual plot is attached to the Xi of each band.
            (16) - (18) Seeing shapes in J, H, K, respectively.
            (19) - (21) Seeing corrections in mag. For a given star the final J magnitude,
                                  for example,  from the fitting algorithm presented here should be
                                   J = J_fit - cor_J.
 

[Last Updated: 2001 Mar 21 by C. Xu.]

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