This manual provides instructions for using CurveFit, an integrated Windows environment designed for analyzing the light curves of RS Canum Venaticorum binary stars and doing theoretical fits of circular dark spots to the distortion-wave curves of these stars. The use of this program, however, is not restricted to this class of stars. It can be used to fit any eclipsing, non-contact binary stellar systems and to match starspots to any photometric distortion wave.
Note that this program replaces the earlier suite of Dos light-curve fitting programs. Those programs are still available for those who would rather use Dos
The program runs on an IBM compatible microcomputer. The minimum suggested hardware configuration is at least a 486 system with 8 MB of RAM and a hard disk—a Pentium with 16 MB of RAM is recommended.
The program requires at least Windows 95--it will not run on earlier versions of Windows, but it will run on Windows 98, 2000, NT, and XP. When the program is run, the window shown in Figure 1 opens with drop-down menus for the program utilities and procedures.
Figure 1. CurveFit
Main Window
The
menu items are:
Tools – Contains the following options:
Add Parameters – This option has two sub-options
Curve
Fitting – Allows you to open
a data file and add the required parameters for the operation of the binary
light curve fitting procedure.
Spot
Fitting – Allows you to open
a data file and add the required parameters for the operation of the spot
fitting procedure.
Bin
Data – Used to bin the
data in a data file containing more than 200 data points to reduce the number
of data points to 200.
Fold Data – Used to combine data points of two or more files into a
single file.
Enter
Phase/Magnitude – Allows the
manual entry of phase and magnitude data.
The phase can be entered in phase units, degrees, Universal Time, or
Heliocentric Julian date.
Shift Data – Contains the following sub-options:
Max
Delta Mag = 0 – Shifts the values
of the delta magnitude of a data file so that the maximum delta magnitude is
equal to zero, the form expected by the binary curve fitting procedure. Also, if the eclipse is more positive in
magnitude, it inverts the values so the eclipse is more negative in magnitude.
Max
Delta Mag at Phase 0 – Shifts the
phase so that the maximum delta magnitude corresponds to phase zero.
Max
Delta Mag = 1.0 – Shifts the
values of the delta magnitude of a data file so that the maximum delta
magnitude is approximately equal to 1.0, the form needed by the spot fitting
procedure.
Graph – Contains two options:
Light
Curve – Plots delta
magnitude versus phase. Two data files
can be plotted to compare for example the modal curve with the actual
data. Both light curve and spot curve
data can be plotted. The graph can be
saved as a Windows bitmap file for further processing or pasting into other
applications
Spot – Plots the positions and sizes of star spots on a Mercator
projection of the star. The graph can
be saved as a Windows bitmap file for further processing or pasting into other
applications
Fitting – Contains two options:
Light
Curve - Fits input data of phase and delta
magnitude for an eclipsing binary star to a modal curve.
Spot – Fits input data of phase and delta magnitude to a modal
star spot curve.
Help – At present
contains only one option:
Help File – Refers you to this document, which is available as an html file and an Adobe pdf file.
About – Gives
version information for the program.
Preparation of Data Files for the
Light Curve Fitting Utility
It is important to use high-accuracy
photometric data (1% or better) if possible.
Use the following procedure:
1.
You need complete coverage over a short period of time—closely-spaced points over a few orbital
periods are best. The more spread out in
time the data is, the higher the likelihood that the starspots will move and/or
change in size. Approximately 100 data
points are needed to give a good fit without using excessive computer time. If more than 200 points are available, you
should
use the data binning utility to calculate the normal points. If normal points are available in the
original sources, then use them, provided they are not spread out to far in
time.
2.
Check to see whether the observations are in the instrumental system or
standard UBV system. For old photometry
done without filters, check to see if an effective wavelength is given. Watch out for "visual" photometry!
3.
Write the basic information for the source of the material in a data
log.
4.
If you need to manually enter the phase and delta magnitude data, you
can use the phase magnitude data entry utility. This will prompt you for the proper header information for the
data file and will then allow you to enter the phase and delta magnitudes of
the binary system you want to analyze.
You should use the following naming convention for the file produced:
starname#.org
where
# is a number used to differentiate
between different data files of the same star system, and org stands for the original data file. For example, if the system in question is XY Ursae Majoris, and
this is the first file that has been created for this system, then the file
name would be: xyuma01.org. Remember that unlike earlier versions, with
Windows 95 and above, the filenames are not limited to eight characters plus
extension. The format for the data file
is shown in Table 1. At the beginning
and end of the file are eight lines of information about the data. At the beginning of the file there is a
blank line after the eight information lines.
Note also that after the last phase and delta magnitude, a phase of
-99.0 and a delta magnitude of 0.0 is required to indicate the end of the data.
5.
Use the light curve graphing utility to plot out the points of the data
set and examine them to see if the shoulders outside of the eclipse have a
delta magnitude of 0.0. If not, the
magnitudes will have to be shifted so this is the case. Use the data shifting utility to do
this. You should give the output file
from the shifting utility the extension .sft
to distinguish it from the original data file.
Be sure to write down in the log book the offset (in magnitude)
calculated by the shifting utility.
This utility will also convert the data into the format of eclipses
having more negative delta magnitude if the original data has shows eclipses in
more positive delta magnitude. After
using the shifting utility, make another plot of the data to make sure the offsets
were done properly so the shoulders of the light curve have a delta magnitude
equal to 0.0.
6. If the primary minimum is far away from phase 0.0, use phase shifting utility to offset the phases, so that primary minimum corresponds closely to phase 0.0.
Preparation of the Input file for
the Light Curve Fitting Utility
To prepare the data file for input to the curve fitting utility, use the parameter adding utility. This utility will read in the designated file and then prompt you for the various parameters necessary for the curve fitting procedure. It will insert these parameters with heading, phase, and delta magnitude information into a new file using the name of the data file you entered, but will give it the extension .dat.
Table 2 shows the 16 parameters the curve fitting procedure
uses for optimizing the fit of a light curve of an eclipsing binary
system. Typically, only a few of these
parameters will be varied for any given run.
Astronomical data and physics are needed to estimate the values of some
of the input parameters.
Table 3 shows a sample listing of an
input file for the curve fitting procedure as produced by the parameter adding
utility. The first line of seven
numbers after the heading information are control parameters and their function
is as follows:
First - Tells the curve fitting procedure whether or not to print out the light curve which was read in. (0 = no print out, 1 = print out.)
Second - Tells the curve fitting procedure the total number of parameters (16).
Third - Tells the curve fitting procedure whether an eccentric orbit fitting will be used (0 = circular orbit). If so, fitting parameters 9 and 10 (eccentricity and mean anomaly at orbital phase zero) will be used; otherwise they will be ignored.
Forth - Tells the curve fitting procedure the number of iterations to use.
Fifth - Tells the curve fitting procedure whether to calculate surface fluxes using blackbody approximations (= 1) or your own values (= 0). Usually we use the blackbody approximation. In that case it is necessary to provide the effective temperatures of the two stars.
Sixth - Tells the curve fitting procedure whether or not a correction file will be used. This is a file generated by the star spot fitting utility that accounts (theoretically) for the distortion wave. For the first pass of the curve fitting procedure a correction file is not used.
Seventh - This tells the curve fitting procedure whether or not to print out a final light curve (0 = No, 1 = Yes).
Next comes a listing of the fitting
parameters and the initial steps in their values for the fitting search. Step sizes are normally
"guesstimates" of the expected order of accuracy of each parameter
value. The parameter adding utility
prompts you for the initial value of each of these and the step size. Note that the utility provides some default
values. U, the unit of light, should be nominally 1.0 for the combined
light of the two stars. L1 is the fractional
luminosity of the primary (hotter) star.
K = r2/r1
is the ratio of the radii of the secondary and primary stars. (Here primary refers to that star eclipsed
at the primary minimum.)
Next are the limb darkening
coefficients, u1 and u2, for the primary and
secondary star. You can infer these
from the measured (or assumed!) spectral types and they will typically be in
the range from 0.6 to 0.8. (See
Al-Naimiy, 1978, for a table of values, which is given in Table 4 at the end of
this manual.) The phase correction, Djo, is any offset to the primary minimum at phase 0.0; it can arise from period changes (or a poor
ephemeris!).
The parameter r1 (= rh)
is the radius of the primary star in units of the semi-major axis of the
orbital separation. The inclination, i, will normally be close to 90o
for an eclipsing system. The
eccentricity of the orbit, e, is
zero for circular orbits as is the longitude of periastron, v. (Actually v is not used but the mean anomaly at phase zero, Mo. These two quantities are easily related for
a given eccentricity--see Budding, 1974, Astro.
Sp. Sci., 26, 371.) Longitude of periastron, v, is what is customarily quoted for the appropriate element,
but Mo is a more
convenient parameter for this fitting algorithm.
The fractional luminosity of the
secondary, L2 (= Lc) is normally tied to L1 so that L1 + L2 = U 1.0, but this condition can be
relaxed. The mass ratio, q, is m2/m1.
The values of the next four
parameters depend on whether you are using the blackbody approximation or your
own values for the calculation of surface fluxes. If using your own values, you must give T1, the coefficient of gravity darkening for the
primary; T2, the
coefficient of gravity darkening for the secondary; E1, the luminous efficiency of the primary; and E2, the luminous efficiency
of the secondary. Typical values are
near unity. The formulae used to
calculate the gravity and reradiation flux parameters (T1, T2, E1, E2) are
given in Budding and Najim (1980).
If you are using the blackbody
approximation, then parameter 13 is T1,
the effective temperature of the of the primary (hotter!) star, and parameter
14 is T2, the effective
temperature of the secondary. You can
infer their values from spectral types (see Table 5), using the tables in Allen
(1973), Lang (1980), or Hayes (1978).
In many cases, we do not know the spectral type of the secondary, so it
is necessary to make a reasonable guess for its effective temperature. Parameter 15 is the effective wavelength of
the observations [note that the units are in Ångstroms (centimeters within the
procedure)]. Finally, parameter 16
plays the role of an "empirical albedo", which multiplies the
reflection factors E1 and
E2.
Normally, this should be kept at unity.
The next line of sixteen numbers
shows the selection of parameters to be optimized. Each digit corresponds, in order, to each of the 16 optimization
parameters. A "0" indicates
that the parameter will not be optimized (i.e. it will remain fixed). A "1" indicates that the
corresponding parameter will be optimized by
the Fitter routine. At most, you will usually optimize seven
parameters: U (#1), L1 (#2), K (#3), Djo (#6), r1 (#7), i (#8), and L2 (#11). Note
that L1 and L2 are coupled, so that for
#11, you will normally insert a "2", which tells
the Fitter routine that the
parameters #2 and #11 are coupled.
These sixteen numbers can thus have the following values:
0 - not altered
1 - optimize in a direct
way
2 - constrained to
optimized parameter by some internally set constraint.
The strategy to follow involves
fitting the "easiest" parameters first and then optimizing the
others. For the first run, it is
usually best to vary only U, the
reference luminosity, and Djo, the phase
correction, in order to fix these values first. U and Djo are sometimes referred to as the "fiducial"
parameters for the axes of the conventional Cartesian system, i.e.:
U ® y reference level
(1)
Djo ® x reference level
(0)
The next line gives another set of
six control parameters. The first
number (D) is the nominal error in
the observations; it should be checked
for each data set and hopefully is not larger than 0.01 magnitude. If in doubt, use 0.01 mag, which can be
easily rescaled. The second number is
the reduction of steps in progressive iterations when "homing in" on
an optimum. The third number is the
increase (augmentation) of step-size.
The fourth number is the difference in chi-square between iterations at
which the program will stop. The fifth
and sixth numbers (A2 and
A3) tell the optimizer
which routines to use. They are
involved in internal switches in the optimizing strategy: a search for a linear trend (to within A2) then vector search. If the chi-squared fails to improve (to
within A3), then switch
back to normal (parabolic) mode.
Generally, only D needs to be
changed in this line, but it may be of interest to experiment with some of the
other quantities to see what effect they have on the results.
The final line of sixteen control
integers tells the curve fitting procedure what order to vary the parameters in
its fitting process. Normally you can
leave these in the natural arithmetic order: 1, 2, 3, etc., but you can change
to another order if you want. It may be
a good idea, sometimes, to check a solution by approaching it from a different
route: i.e. by having a different sequence
for parameter optimization.
For subsequent runs, you can change
these various parameters with the Windows Notepad editor using the final values
from the previous calculation. If you
use another editor, such as WordPad or Word, be sure to use the plain text mode
so you don't introduce any spurious control characters into the file!
Running the Light Curve Fitting
Utility
Now you can use the curve fitting
utility, which will prompt you for the name of the input data file (the one
prepared using the parameter adding utility).
It will produce the following output files:
filename.out - main output information
filename.mod - theoretical light curve
filename.obs - observational points
filename.dif - difference curve used as
input for SPOT utility.
where
"filename" is the filename
of the input data file.
Once you have made an initial run of
the curve fitting procedure, plot the theoretical light curve (the .mod file) overlaid on the observed
light curve (the .obs file) to see
how well they match. Then edit the
fitter input file "filename.dat"
and replace the initial values of U
and Djo with those found in the first run. Hold Djo fixed for the next run, and optimize for U (#1), L1 (#2), K
(#3), r1 (#7), i (#8), and L2 (#11, you should put a 2 here to indicate that L1
and L2 are coupled). Note:
the main fitting is an even function about the zero phase point, so Djo is usually fairly independent of all other parameters for a
uniformly spaced data set corresponding to a genuine likeness to the underlying
model (i.e. intrinsically symmetric). U is, however, not so independent in
principle; it could correlate strongly
with E2 for example. Getting values for these six parameters are
about all you should reasonably hope for from a photometric solution. The masses and temperatures of the stars
should come from other lines of evidence (i.e. spectroscopy).
If the next run finds some
reasonable value for i, you can fix
it for subsequent runs. Use the output
from the previous run as input for the next.
It should not take more than three or so runs to get a reasonable set of
fits. If it does not, that should warn
you that something is amiss. Check your
initial run for just U and Djo by doing a plot with
filename.mod
and filename.obs super-imposed. Your initial model curve should not look too
bad in basic outline, though there may be significant misfits around the
minima. Figure 2 shows a plot made
using the CurveFit graphing utility.
Figure 2. Plot of Theoretical
Light Curve with Observed Light Curve Superimposed
When you are satisfied with the
results, make a copy of the final
filename.dat
file called "filenameC.dat". The "C" stands for
"clean", and it will be used for the CLEAN run.
Generally speaking, the spot fitting
utility operates similarly to the curve fitting utility--the main difference
lies in the different fitting functions.
Hence, the main features of the input file layout for the spot fitter
routine are quite similar to those of the main fitter routine.
Star Spot Fitting Utility Input
File Format
The curve fitting utility output file "filename.dif"
contains the difference between the theoretically derived light curve and the
actual one--that is, the so-called distortion wave (if there is one!).
Graph this file to see if it contains any systematic trends or looks like
a scatter diagram. If it seems that the points fall randomly about zero, then
either no spots exist or they are uniformly distributed in longitude.
If any trends appear, you can estimate by eye the longitude of a minimum
and whether one or two minima (one or two spot groups) exist.
Operation of the Spot Fitting
Utility
Do a first run of the spot fitting
utility at a fixed latitude of 45 degrees
( = 0.785 radians) with a guess for the spot's radius and
longitude. The radius size should be
approximately equal to the square root of the wave amplitude (0.10
radians). If two minima are not clearly
visible, start with only one spot group.
The fitting parameters are given in
Table 6; there are 11 possible. They are:
(1) longitude of
spot 1, a1, in radians.
(2) latitude of spot
1, b1, in radians.
(3) inclination of
the system, i, in radians (use
output from
the Fitter routine).
(4) radius of spot
1, r1, in radians.
(5) unit of light, U (normally = 1.0, use output from
the Fitter routine).
(6) the intensity of
the spot, Kl ( = 0.0 for a "black" spot). Klis defined as
the flux in the spot divided by the flux in the photosphere at the effective
wavelength, l.
(7) the limb darkening coefficient, u, of the spotted star (typically 0.70).
(8) the fraction
luminosity of the hotter star, L1
(use output from the curve fitting procedure).
(9) the longitude of
spot 2, a2, in radians.
(10) the latitude of spot 2, b2, in radians (usually set to b1).
(11) the radius of spot 2, r2, in radians.
Once you have found some reasonable values with a first run, the strategy is to vary a1, b1, r1, a2, b2, and r2 for the best fit. Note that two minima in the distortion wave, or one minimum with an asymmetrical shape indicate two spots. Once these parameters are optimized, you must carefully examine the error matrix to see if a "good" solution is achieved (in a chi-square sense). Beware of indeterminate values of the latitude and spot size (which are interrelated)! Once a good fit has been achieved, plot out the observational data points and theoretical curve (Figure 3).
Figure
3. Plot of Theoretical Spot curve
with Observed Spot Curve Superimposed
File Structure for
Spot
Fitting Utility
The spot fitting utility generates
the following output files:
filenameS.out - Main output file
filenameS.mod - Theoretical fit
filenameS.obs - Observational points
filenameS.cor - Correction curve for curve fitting procedure
Table 7 gives a sample input file (filenameS.dat) for the spot fitting utility. The first line of five numbers contains the control parameters. The first parameter tells SPOT whether or not to print out the input light curve (1 = yes, 0 = no). The second parameter is the number of unknowns (= 11). The third parameter gives the system eccentricity (0 = circular): in this case a dummy variable--but it should always be set to zero. The fourth parameter gives the number of iterations (normally = 10). The fifth parameter gives the number of spots (0, 1, or 2). Following these are the eleven fitting parameters (1-11, Table 6). The line after these indicates which parameters are to be optimized (0 = no, 1 = yes). The next line contains variables that will be fixed, except for the first, D, which is the nominal error in the data. The next line provides the order in which the parameters will be optimized (1 to 11). Then comes the input data in phase (degrees) and intensity.
Star Curve Fitting Utility with
Correction File
Using the best fitting parameters
from the earlier runs of the star curve fitting utility, copy them into a file
called "filenameC.dat"
("C" for
"clean"). Also rename the
filenameS.cor
to filenameC.cor. Change the
correction parameter from "0" to "1" so that the
correlation file output from the spot fitting utility will be read in. Then run the curve fitting utility for the
usual five parameters U, L1, r1, K, and i.
L1 + L2 = U should be set as a constraint, unless there are some grounds for
thinking there may be third light, i.e.
U - (L1 + L2)
= L3 > 0. Djo will be
reasonably determined independently. E2 might be a possible
seventh in case of an anomalous reflection effect--but note the case of BH
Vir!
This run may take a little longer as
the values adjust to that for a "clean" light curve. The results should have a smaller chi-square
than the uncleaned run. When plotted,
the regions affected by the distortion wave should have a noticeably better
fit. Plot out (Figure 4) the
theoretical curve (filenameC.mod)
and the corrected data (filenameC.obs). As a check, also plot out the new difference
curve (filenameC.dif). It should appear as a scatter diagram around
zero, with the magnitude of the scatter on the same order as the errors in the
observations. If there are still
systematic trends, a second run-through may be needed.
Figure
4. Plot of theoretical
light curve with corrected light curve superimposed
You can now use the spot graphing utility to plot the location of the spot(s) on a Mercator projection of the star's surface. The utility asks for the latitude, longitude, and radius of the spot. It allows you to plot more than one spot, so you can enter the results from several different data sets. In this way, you can graphically display the evolution of the spot groups on a given primary star.
After the last iteration, some
further numerical operations will appear in the main output file "filename.out" (both from the spot
fitting utility and the star curve fitting utility). These are connected with the setting up of the curvature Hessian
and the error matrix. Some detailed
notes about this are provided in Budding and Najim (1980). A fuller and readable background is provided
in Bevington's (1969) book Data Reduction
and Error Analysis for the Physical Sciences.
The main point here is to test for
determinacy of the sought parameters and also to provide more realistic formal
errors than the interim formal errors determined with each iteration. The final list of formal errors takes into
account the inter-correlations between parameters. The curvature Hessian must be positive definite for a valid
optimum. Ideally, it should be
dominated by its central diagonal (well-determined solution).
Guides to the character of this
Hessian, and therefore the nature of the solution, are provided by the
eigenvalue and eigenvector list; and also the Hessian's inverse--the error
matrix. The standard deviation error
assessments are determined in a direct way from the leading diagonal
solution. If any such element is
negative, it indicates a breakdown in determinacy. This is usually caused by "asking the light curve to tell
you more than it knows"--i.e. seeking to determine too many parameter
values. The presence of correlation
effects between different parameter values has a highly contributory effect
here.
The same effects are show in a
different way by the eigenvalue list.
The eigenvalues must all be positive for a valid optimal
"unique" solution. (Actually,
"unique" has a somewhat restricted meaning here). The eigenvalues represent the axes in a
principle axis transformation of the error ellipsoid. The eigenvectors indicate the orientation of these axes with
respect to the scaled parameter axes.
The main point here is to establish
where the essential determinacies of the problem really lie. Usually, for example, we find that the
largest eigenvalue is closely oriented towards the axis of the "unit of
light" parameter. This just
confirms "Murphy's Law"; the light curve tells you best the thing you
are least interested in--i.e. the mean out-of-eclipse light level.
References
Allen,
C. W.: 1973, Astrophysical Quantities (Third Edition). Athlone Press (London), p. 206.
Al-Naimiy,
H. M.: 1978, Astrophys. Sp. Sci., 53, 181.
Bevington, P. R.:
1969, Data Reduction and Error
Analysis for the Physical Sciences, McGraw Hill Book Co., NY.
Budding,
E.: 1974, Astrophys. Sp. Sci. 26,
371.
Budding,
E.: 1977, Astrophys. Sp. Sci. 48,
207.
Budding, E. and Najim, N. N.: 1980, Astrophys. Sp. Sci. 72, 369.
Budding, E.: 1993, An Introduction to Astronomical Photometry, Cambridge University Press (Cambridge).
Hayes, D. S.: 1978, The H-R Diagram, edited by A. G. D.
Philips and D. S. Hayes, Dordrecht, p.
70.
Lang,
K. R.: 1980, Astrophysical Formulae, Springer Verlag, (Berlin), p. 564.
Table 1. Example of Original File
RT Andromedae
Date: 11, 12 Nov, 12 Dec UT 1987
Observatory: Capilla Peak 61 cm CCD camera
Wavelength: V (using Mould "V" filter 537.5 nm)
Comparison star: BD 52 3384 (SAO 35208)
Error: 0.01 mag ?
Source: Capilla archives N=96
11 Nov, 12 Dec partially cloudy
0.2590 1.1860
0.2670 1.1840
0.2720 1.1890
0.2860 1.1900
0.3020 1.1850
0.3120 1.1870
0.3260 1.1830
0.3370 1.1700
0.3490 1.1740
0.3590 1.1640
0.3710 1.1590
0.3820 1.1600
0.3930 1.1550
0.4030 1.1410
0.4160 1.0900
0.4260 1.0900
0.4470 1.0310
0.4670 0.9480
0.4870 0.9160
0.8100 1.1800
0.8270 1.1540
0.8360 1.1490
0.8480 1.1410
0.8710 1.1370
0.8770 1.1330
0.8900 1.1300
0.8990 1.1160
0.9110 1.0380
0.9170 1.0020
0.9240 0.9570
0.9530 0.6480
0.9580 0.5750
0.0340 0.8350
0.0410 0.9010
0.0460 0.9550
0.0520 1.0010
0.0570 1.0290
0.0640 1.0900
0.0700 1.1010
0.0780 1.1230
0.0840 1.1370
0.0920 1.1380
0.1660 1.1910
0.1720 1.1760
Table 2. Standard
Eclipsing Binary Fitting Parameters
1. U - the reference luminosity
2. Lh
= L1 - the fractional luminosity of the hotter
(primary) star
3. K =
r2/r1 - the ratio of the radii
4. u1 - the coefficient of linear
limb-darkening for the primary star
5. u2 - the coefficient of linear
limb-darkening for the secondary star
6. Djo
- the phase correction
7. rh
= r1 - the radius of the primary star
(in units of the semi-major axis of the
orbital separation)
8. i - the orbital inclination
9. e - the eccentricity of the orbit
10.
Mo - the mean anomaly at phase zero
11.
Lc = L2 - the
fractional luminosity of the cooler (secondary) star
12.
q = m2/m1 -
the mass ratio
13.
T1 - the coefficient of the gravity-darkening for the
primary star (or the temperature
of the primary star)
14.
T2 - the coefficient of the gravity-darkening for the
secondary star (or the
temperature of the secondary star)
15.
E1 - the luminous efficiency for the primary star (or
effective wavelength of the
observations in Ångstroms)
16.
E2 - the luminous efficiency for the secondary star (or
the "empirical albedo",
normally kept at unity)
Notes: The final main
(publishable) output from this can be expected to be:
(1)* The
geometric parameters (independent of the wavelength) r1, r2
(= K x r1), and i.
(2)* The
fraction luminosities L1/U, L2/U.
(3)
The adopted parameters (temperatures and wavelength if blackbody approximation,
otherwise limb darkening coefficients, gravity darkening coefficients and
luminous efficiencies; and the mass ratio).
Usually eccentricity = 0 is adopted.
(4)* Any
possible correction to the zero point of the phase, if significant.
(5)* The
actual reference out-of eclipse apparent magnitude of the system (obtained from
the reference luminosity used as a correction to the initially adopted delta
magnitudes).
* (all with error
measures)
Table 3. Sample Listing of FITTER Input File
Star: RT And
Date: 11-12 Nov, 12 Dec 1987
Observatory: Capilla Peak (CCD camera)
Wavelength: R (Mould filter - 667.0 nm)
Comparison star: BD 52 3384 (SAO 35208)
Error: 0.01 mag
Source: data from Capilla
N = 99
0 16 1 10 1 0 0
0.9800000 0.0020000
0.8700000 0.0050000
0.7200000 0.0050000
0.4900000 0.0050000
0.7300000 0.0500000
0.0000000 0.1000000
0.3100000 0.0050000
1.5376345 0.0043633
0.0220000 0.1000000
2.9700000 0.1000000
0.1000000 0.0050000
0.6500000 0.0050000
6250.0000000 100.0000000
4900.0000000 100.0000000
0.0000667 0.0000010
1.0000000 0.0100000
1 1 1 0 0 1 1 0 0 0 2 0 0 0 0 0
0.0100 0.9000 1.1000 0.0200 0.1000 0.1000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.82200000 -0.08300000
0.82900000 -0.08000000
0.83900000 -0.09100000
0.84900000 -0.09300000
0.87200000 -0.10100000
0.87800000 -0.12600000
0.89100000 -0.13600000
0.90000000 -0.14000000
0.91200000 -0.18500000
0.91900000 -0.22000000
0.92600000 -0.26400000
0.95400000 -0.56600000
0.96000000 -0.62800000
0.03700000 -0.33500000
0.04200000 -0.29300000
0.04700000 -0.24800000
0.05300000 -0.20600000
0.05800000 -0.17000000
0.06500000 -0.13500000
0.07200000 -0.10100000
0.08000000 -0.09000000
0.08600000 -0.07200000
0.09400000 -0.07400000
0.16800000 -0.05000000
0.17300000 -0.03600000
0.18100000 -0.02700000
0.23400000 -0.03600000
0.24100000 -0.01100000