This is an update of the progress in the task of 1-d fittings of the
bright stars. The results (including this
report) can be found in skywave: /data3/2mass/ray2/results
(1) Profiles of five more stars have been provided by Raymond. All
have CIO K mags
(from K=-1.2 to K=1.5),
and 4 have CIO J mags (from J=-1.1 to J=1.7).
(2) By fitting the profiles of these very bright stars, I found that
the function I used in the
begining (two exponentials) is
not a good shot.
(3) After talking to Gene Kopan and Bill Wheaton, I decided to use
the so called
'Voigt function' (in "Astrophysical
Formulae", K.R. Lang, p220). Since this is an integral
function and not very easy to use, I
found, with some experiments, the following approximation
of the function which seems good enough
for our purpose:
H(a,b)
= exp(-bv^2) + 3.1*av/(sqrt(3.1416)*bv^2+3.1416)
where:
av = sqrt(log(2))*dv_l/(2*dv_d)
and
bv = 2*sqrt(log(2))*x/dv_d
(dv_d and dv_l are the FWHMs of the
central Gaussian and of the Lorentz wing, respectively).
(4) The fitting function is then the following 3 parameter (a(0), a(1)
and a(2)) function:
s = exp(a(0)-bv^2) + exp(a(0))*3.1*av/(sqrt(3.1416)*bv^2+3.1416)
where:
bv=2.*sqrt(alog(2.))*x/a(1)
and
av=sqrt(log(2))*a(2)/(2*a(1)) .
(5) After some experimenting, I found that I can further reduce the
number of parameters from
3 to 2, becuase there is a strong
correlation between a(2) and a(1). This indicates that,
for our data, the amplitude of
the central Gaussian and that of the Lorentz wing is highly
correlated. In fact, the parameter
av is found to follow the relation: av=exp(-a(0)*0.5+1.5).
This relation eliminates the need for
the third parameter a(2).
(6) The results are given in the table, and in the gif plots
found in this directory.
(7) Notes:
1) The amplitude is parameterized as
exp(a(0)). This is to ensure that it is always > 0.
2) There seems to be a good correlation
between CIO mags and mags from these fits,
though
the relation is not really linear.
3) The fit can still be improved. But
it takes a lot of time.
4) This algorithm needs to be tested
with more stars.
Cong
Table: Results
star# J-CIO
J_fit a_j(0)
a_j(1) K_CIO
K_fit a_k(0)
a_k(1) H_CIO
1
-1.1 -0.73
18.48 2.07
-1.2 -1.65
15.50 2.15
-1.3
2
0.6 0.49
15.68 1.84
-0.1 0.52
13.35 2.27
-0.1
3
1.31 14.81
1.94
0.1 1.50
12.30
2.41
4
1.7 2.52
13.70 1.92
0.5 1.74
12.18 2.28
5
1.5 2.29
14.20 1.67
1.5 2.55
11.67 1.99
1.6
Figures:
fit_star1_j.gif
fit_star1_k.gif
fit_star2_j.gif
fit_star2_k.gif
fit_star3_j.gif
fit_star3_k.gif
fit_star4_j.gif
fit_star4_k.gif
fit_star5_j.gif
fit_star5_k.gif
