Part X
Data Reduction

Introduction
To create useful data out of the observed raw data, data reduction must be used. To reduce the data the system calibration, zero points and color transformation coefficients, must be known and figured in. If All-Sky photometry is done, the extinction for each session must be determined. Extinction is a function of the transparency of the atmosphere and the air mass. For Differential photometry a seasonal average can be used if the program and comparison stars are not too far apart.

Because reducing raw data counts can be very time consuming and tedious, a computer program that handles this is a must. To calibrate your equipment some of the data reduction must be done manual. Once the data reduction transformation coefficients have been determined a computer program can use those to crank out accurate magnitudes. The following is presented to show what goes into the reduction of the raw data to obtain the data reduction transformation coefficients and create useful magnitudes.

Air Mass
Air mass X is defined as the Secant or (1/Cos) of the angle Z between the star's position and the zenith. This represents the amass of air between the observer and the edge of the atmosphere in space. See Figure 39.

Figure 39
Air Mass

By definition, a star at the zenith (Z= 0 degrees) has an air mass of X= 1.0000.

Air Mass = X
At Z= 30 degrees from the zenith, X= 1.1547.
At Z= 45 degrees from the zenith, X= 1.4142.
At Z= 60 degrees from the zenith, X= 2.0000.
At Z= 75 degrees from the zenith, X= 3.8637.

X= Sec (Z) = 1/Cos (Z)

As can be seen, as the star approaches the horizon, the computed air mass approaches infinity. This is why stars close to the horizon are very difficult to work with. The real air mass does not approach infinity, however, due to the curvature of the Earth, and the above relationship becomes inaccurate.

Because this is not an exact solution (primarily due to the curvature of the atmosphere) a correction factor should be included The resulting improved equation is:

X= Sec(Z) * (1 - 0.0012 * Sec (Z) * Sec^2 ( Z - 1 ) )

Where:

Sec (Z)= 1 / (Sin (Lat) * Sin (D) + Cos (Lat) * Cos (D) * Cos (H))

and

Lat = Observatory's Latitude
D = Star's Declination
H = Star's Hour Angle ( the local sidereal time minus the starŐs RA)

This is easy to figure if you know the star's hour angle. The value of Z can be measured directly with the use of a protractor and a string with a weight. Simply measure the angle the telescope makes with the string (vertical or zenith) and subtract it from 90 degrees.

However, this is not always easy to do, not very accurate, and requires additional data logging during the observations. It's usually best to use a computer program to determine the air mass.

Extinction Coefficients
It is important to determine the extinction coefficients (i.e., all the k coefficients). If one is starting from scratch, both first- and second-order coefficients can be determined using the same data set. This is accomplished by obtaining several observations of a pair of blue and red stars. While for differential photometry a seasonal average is sufficient, for All-Sky photometry the Extinction must be determined.

Suggested red-blue pairs are listed in this program. The pair of stars does not need to have previously determined UBV magnitudes. Only a large difference in instrumental color is required. Choose a pair of stars that will transverse a large range of air mass during the observing time available (i.e., stars with declination close to the celestial equator).

Second Order Coefficients k"v and k"bv
The second-order extinction coefficients, k"v and k"bv, for each star of the red-blue pair are found by:

vored = vred - (k'v + k"v * (b - v)red) * Xred

voblue = vblue - (k'v + k"v * (b - v)blue) * Xblue

(b - v)ored = (b - v)red - (k'bv + k"bv * (b - v)red) * Xred

(b - v)oblue = (b - v)blue - (k'bv + k"bv * (b - v)blue) *Xblue

where

vored = Visual red star instrumental mag corrected for atmospheric effects
(b - v)ored = (b - v) red star instrumental mag corrected for atmospheric effects

Xred = Red Star Air mass at observation

vred = -2.5 * Log10 (averaged V filter Red star reading)
bred = -2.5 * Log10 (averaged B filter Red star reading)
ured = -2.5 * Log10 (averaged U filter Red star reading)

voblue = Visual blue star instrumental mag corrected for atmospheric effects
(b - v)oblue = (b - v) blue star instrumental mag corrected for atmospheric effects

Xblue = Blue Star Air mass at observation

vblue = -2.5 * Log10 (averaged V filter Blue star reading)
bblue = -2.5 * Log10 (averaged B filter Blue star reading)
ublue = -2.5 * Log10 (averaged U filter Blue star reading)

k'v = First order visual extinction coefficient
k'bv = First order (b-v) extinction coefficient

k"v = Second order visual extinction coefficient
k"bv = Second order (b-v) extinction coefficient

Then, by subtracting:

(vored - voblue) = (vred - vblue) - k'v *(Xred - Xv) - k"v * ((b - v)red * Xred - (b - v)blue) * Xblue)

((b - v)ored - (b-v)oblue) =
((b - v)red - (b - v)blue) - k'bv * (Xred - Xblue) - k"bv * ((b - v)red * Xred - (b - v)blue * Xblue )

These equations are greatly simplified by using D to denote the difference between similar quantities for the red and blue star

Dv = vred - vblue
Dvo = vored - voblue

D(bv) = (b - v)red - (b - v)blue
D(bv)o = (b - v)ored - (b - v)oblue

and recognizing that

Xred ~ Xblue (or simply X, the mean air mass of the red-blue observation).

Then, the above equations become:

Dv = k"v * D(bv) * X + Dvo
k"v = Dv/(D(bv) * X + Dvo)

D(bv) = k"bv * D(bv) * X + D(bv)o
k"bv = D(bv) / (D(bv) * X + D(bv)o)

For a more accurate determination, several red/blue pairs can be measured and a plot of Dv versus D(bv) * X made.
The slope of the line is k"v.

A similar plot of D(b-v) versus D(b-v) * X can be made. The slope of that line is k"bv.

See Figures 40 and 41.

Figure 40
Determination of Second Order Extinction Coefficient k"v

Figure 41
Determination of Second Order Extinction Coefficient k"bv

First Order Coefficients k'v and k'bv
Having determined the k" coefficients, we can solve the first-order coefficients. Because these are independent of the star's color, either the blue or red star can be used, but not both at the same time!

The following uses the blue star:

vblue - k"v *(b - v)blue * Xblue = k'v * Xblue + voblue

k'v = (vblue - k"v * (b - v)blue * Xblue - voblue) / Xblue

(b - v)blue - k"bv * (b - v)blue * Xblue = k'bv * Xblue + (b - v)oblue

Again, for a more accurate determination, several blue stars can be measured and a plot of (vblue - k"v * (b-v)blue* Xblue) versus Xblue can be made.

The slope of the line is k'v.

Similarly, a plot of the quantities ((b-v)blue - k"bv * (b-v)blue * Xblue) versus Xblue has a slope k'bv.

Figures 42 and 43 show examples of these plots.

Figure 42
Determination of First Order Extinction Coefficient k'v

Figure 43
Determination of First Order Extinction Coefficient k'bv

Although there are other techniques to calculate first-order coefficients, the method described here has the advantage that no other observations are needed except for those of the red-blue pair.

Returning to the reduction of U filter data, we can write the relationship:

(u - b)blue = k'ub Xblue + (u-b)oblue

Then a plot of (u - b)blue versus Xblue has the slope k'ub.

Again, a sample plot is shown below in Figure 44.

Figure 44
Determination of First Order Extinction Coefficient k'ub

If the second-order extinction coefficients have temporal stability, they need not be re-determined on every night. Instead, a mean value for each k" can be adopted while solving only for the first-order coefficients. This is best done by observing the rising (or setting) of a single star with an instrumental color, (b - v), close to zero and applying last three equations in exactly the form as previously described.

Transformation Coefficients

Introduction
Once the raw data have been logged they must be reduced to be useful. If an analog system was used, observations must be corrected for any different gain settings. If a photon counting system was used the counts should be in counts per second and Dead Time corrections must be applied.

Basic Procedure

1. Average the readings for a given star + background for a given filter and for a given series of observations.
2. Average the readings for the background (sky),(if more than one background reading was taken, for a given filter and series.
3. Subtract the averaged background from the averaged star + background reading. The result is the partially reduced data for that star. The resulting data will be the raw v, b and u magnitudes and must be converted to v, b, and u by multiplying the log10 of these numbers by -2.5.
4. Repeat steps 1 - 3 for each filter.
5. Repeat steps 1 - 4 for each star.
6. To determine the extraterrestrial instrumental magnitudes:

Note:
These are extinction-corrected transformed magnitudes.

Where:

Note:
The zero points drop out when doing differential photometry and the extinction coefficients also drop out if the stars are close together (less than a degree).

Refer to ASTRONOMICAL PHOTOMETRY by Henden and Kaitchuck and PHOTOELECTRIC PHOTOMETRY OF VARIABLE STARS by Hall and Genet for details on determining transformation coefficients, extinction coefficients, and zero points.

The above data reduction works for All-Sky and Differential photometry. Differential photoelectric photometry is the same except the sequence of the data taking is important. All references for the program or variable star are to a comparison star. The procedure for Differential photometry is as follows:

1. Calculate Mcomp, the average of the two comparison star magnitude readings (C1 and C2) that bracket (in time) the variable star's magnitude.

Mcomp= (C1 + C2) / 2

2. Determine the differential magnitude (Mdiff) by subtracting the averaged comparison star's magnitude (Mcomp) from the program star's magnitude (Mprg).

Mdiff = Mprg - Mcomp

3. Determine the magnitude (M) of program star by adding the differential magnitude Mdiff to the standard (published) value for the comparison star's magnitude (Mstd).

M= Mdiff + Mstd

An example:

C1= 5.37
C2= 5.34
Mprg= 4.75
Mstd= 5.45

Mcomp= (C1 + C2)/2 = (5.37 + 5.34) / 2 = 5.355
Mcomp= 5.355

Mdiff= Mprg - Mcomp = 4.75 - 5.355 = -0.605
Mdiff= -0.605

M= Mdiff + Mstd = -0.605 + 5.45
M= 4.845

Repeat steps 1 - 3 for each color (filter) and each variable star and check star data set.

Dates and Times

Introduction
It is necessary to accurately log the date and times of your observations. All times should be in Universal Times and dates should be in Julian Date format. If very precise timing is needed, the date should be in Heliocentric Julian Date format.

Universal Time (UT)
Since observers in different parts of the world may be observing the same star or object simultaneously, a means of using a standard time is necessary. It was decided that the time should be referenced at the zero degree longitude mark where it passes through Greenwich England. The time at that point is called Universal Time (UT). It formally was called Greenwich Mean Time (GMT) and by the military, Zulu Time (ZT). Since various places around the Earth have different time zones, a correction must be made to convert local time to Universal Time. All times are based on standard time and not daylight savings times. For Eastern Standard Time (not daylight savings) EST convert to UT by subtracting 5 hours. For PST subtract 8 hours. For Swedish Winter Time (SWT) add 1 hour, For Japan Standard Time (JST) add 9 hours and for New Zealand Standard Time (NZST) add 12 hours.

Local Sidereal Time.
The Local Sidereal Time, or LST, is defined as the instantaneous hour angle of the Vernal Equinox. The LST is local in the sense that it depends on the location of the observer, specifically, the terrestrial longitude. A more practical definition of LST is that it is identical to the right ascension of a star on the observer's celestial meridian. Therefore, the following is an approximation for LST at a specified UT on a given date:

LST = GMS0hUT - LONG + 1.00274 * UT

where GMST0hUT is the Greenwich Mean Sidereal Time at 0h UT for the date as listed in a reference like the Astronomical Almanac, the observer's longitude is a positive value if west of Greenwich and negative if east, and the factor 1.00274 represents the ratio of the sidereal and solar rates. Usually, it is necessary to convert longitude from arc units to time units to complete the calculation. If the result is outside the range 0-24 hours, then add or subtract 24 hours.

Example:
What is the LST for an observer using the 1.3m telescope on Kitt Peak
(longitude = 111deg 35'52" W, latitude = 31deg 57'34" N) at 4h25m16s UT on September 9, 1990?

Converting from arc to time units,

111deg 35'52" is equivalent to 7h26m23s

From the Astronomical Almanac,

GMST0hUT = 23h11m08s

Then,

LST= 23h11m08s - 7h26m23s + 1.00274 x 4h25m16s
LST = 23h11m08s - 7h26m23s + 4h26m00s
LST = 20h10m45s

If a table giving GMST0hUT is not available, the LST can still be found by adopting a specific date as a reference point and generalizing the above equation in the form:

LST = 6h41m33s - LONG + 24h x mod(1.0027379093 x (JD - 2447892.5), 1)

where 6h41m33s is the GMST for Julian Date 2447892.5 (= 0h UT on January 1, 1990) and the current UT has been converted to JD.

The two-argument function mod(a,b) is the modulus function that is used in computer programs to find the fractional part of the quotient a/b (i.e., the remainder).

For example:

mod(1.1, 1) = 0.1
mod(9.9, 1) = 0.9

A problem inherent to this lengthy calculation is the need to maintain many significant digits. To keep time with a precision of one second, the JD must be specified to 12 significant digits (i.e., the nearest 0.00001 day)!

Example:
For the time and location given in the previous example, find the LST using the above equation.

4h25m16s UT on September 9, 1990

is equivalent to

JD 2,448,143.68421.

Then

LST = 6h41m33s - 7h26m23s + 24h x mod(1.0027379093 x (JD 2,447,892.5 - JD 2,447,892.5), 1)
LST = 6h 41m 33s - 7h 26m 23s + 24h x mod(251.87182,1)
LST = 6h41m33s - 7h26m23s + 20h55m34s
LST = 20h10m44s

Double Dates
No, this is not about another couple going out to dinner with you. Because observing is normally done at night, it is possible to start observing one day and finish the next. To avoid confusion, a double date is used. The first date is the date the night started and the second date the next day.

For example, I start my observing on Friday night 27 February 2004. The double date entry would be 24/25 February 2004. There is then no ambiguity as it is obvious the night in question is the night between 24 and 25 February 2004.

Julian Date (JD)
Julian Date is used to easily calculate the time between events which are separated by long periods. The beginning of Julian time is noon UT 1 January 4713 B.C.

Julian Date is the number of days that have elapsed from the beginning of Julian time. Julian Date is given in decimal form and not in hours, minutes, or seconds. The next Julian Day begins at noon Greenwich time or 12 hours Universal Time (UT). These dates and associated times are considered geocentric as they are referenced to the center of the Earth.

The Origin of Julian Date
Where did Julian Date come from? Contrary to some beliefs, Julian Date or Day has no connection with the Julian Calendar and was not named after Julius Caesar. So where did it originate? In 1583, Joseph Justus Scaliger developed the Julian Period. Scaliger took three cycles, the 28-year solar cycle, the 19-year lunar cycle, and the 15-year cycle of the Roman Indiction (used in calculating Easter) and multiplied them together. The resulting period (28 X 19 X 15 = 7980) is 7980 years. He then set about to determine the last time all three of these cycles passed through zero. It turned out to be the year 4713 B.C.

This is a very convenient time because all of recorded history has happened since then. All recorded astronomical events of interest occurred after this date. Astronomers found this to be a very useful reference and by using it, times between events were independent of day-of-the-week, month, or year.

Heliocentric Julian Date (HJD)
Because the speed of light is finite and the earth is traveling around the Sun with an orbital diameter of nearly 200 million miles, it is convenient to state a zero reference point for events. By using the center of the Sun (heliocentric) as a reference point all time measurements can be referenced to the same point. If this wasn't done, a star on the far side of the Sun would require nearly 16 minutes more for its light to reach the Earth than when the Earth was on the same side of the Sun as that star, approximately six months later. For stellar observations where timing to the minute or second is important, it is essential to correct the time to Heliocentric time.

For astronomical photoelectric photometry the use of Heliocentric Julian Date (HJD) allows precise timing information to be noted. One number can be used to reflect the precise timing (to the minute, second or microsecond, if needed). Heliocentric Julian Date is the Julian Date referenced to the center of the Sun and can be found using the following equations:

HJD = JD (Geo) + Hel Corr

The Hel Corr (Heliocentric Correction) can be determined by:

Hel Corr = T * R * (Cos (L) * Cos (A) * Cos (D)) + T * R * (Sin (L) * (Sin (E) * Sin (D) + Cos (E) * Cos (D) * Sin (A)))

Where:

T = Light travel time for one astronomical unit (499 seconds or 0.0057755 days)
R = Earth -Sun distance in astronomical units
L = Longitude of the sun
A = Star's right Ascension (in decimal hours)
E = Obliquity of the ecliptic = 23.45 degrees
D = Star's declination (in degrees decimal)

R and L must be found from the AMERICAN EPHEMERIS AND NAUTICAL ALMANAC for each observing night or the data can be entered into a data reduction software program and the HJD calculated automatically.

Manual Data Reduction
It is possible to do the data reduction manually. However, even with an electronic calculator, it can be a daunting and time consuming job. Using a computer with a software program design to do the reduction is much more efficient.

Software
Once the system's calibration fdata has been determined, raw data obtained during observing sessins can be entered into a software program to reduce ithem to standard data. HPO SOFT has created a program that will work on Macintosh OS (8.6 - 9.2), OSX and Windows operating systems.

Originally the program was developed in the early 1980s on a Sinclair ZX-81 computer using BASIC. Later it was programmed still in BASIC on a Z-100 computer. The first versions of the software reduced one set of data at a time. The program has progressed to the point where it handles all the raw data for each observing session, archives it, calculates air mass, HJD, reduces all the raw data and archives the reduced data. Lists of results can be displayed and printed. In addition, observatory data for multiple observatories, multiple observers, multiple different calibration data and extensive help are included. While currently only UBV data is currently handled, an expanded version that will include R and I data will be available in the future.

See: http://www.hposoft.com/Astro

Figure 45 shows a screen shot of the HPO SOFT UBV PEP Data Reduction Program Main Menu.

Figures 45 A - 45 D show screen shots of the HPO SOFT UBV PEP Data Reduction Program Sub menus, data entry and reduced data displays.

Figure 45
Data Reduction Program Main Menu

Figure 45 A
UBV RAW Data Menu

Figure 45 B
UBV RAW Data Entry

Figure 45 C
UBV Data Reduction Menu

Figure 45 D
UBV Data Reduction Display

Present Page Version as of 29 March 2004

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