Useful Formulas for Amateur SETI
The following is derived from a number of sources including:
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Practical Astronomy with Your Calculator by Peter Duffett-Smith
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The ARRL UHF/Microwave Experimenters Manual
Julian Date
The julian date is the number of days since Greenwich mean noon on the
first of January, 4713 B.C.
To compute the Julian Date:
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Convert local time to Greenwich Mean Time
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Let Y equal the year, M equal the month, D equal the day in decimal form.
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If M equals 1 or 2 then subract 1 from Y. and add 12 to M.
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Compute A. A=INT(Y/100)
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Compute B. B=2-A+INT(A/4). However, if the date is earlier than October
15, 1582 then B=0.
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Calculate C. C=INT(365.25*Y). If Y is negative then C=INT((365.25*Y)-.75).
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Calculate E. E=INT(30.6001*(M+1))
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Calculate JD (Julian Date). JD=B+C+D+E+1720994.5
Greenwich Sidereal Time (GST)
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Calculate JD (Julian Date) corresponding to 0 hours GMT for this date.
(This value should end in .5)
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Calculate UT. This is the GMT in decimal hours.
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Calculate T. T=(JD-2451545.0)/36525.0
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Calculate T0. T0=6.697374558+ (2400.051336*T)+(0.000025862*T2)+(UT*1.0027379093)
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Reduce T0 to a value between 0 and 24 by adding or subtracting
multiples of 24. This is the GST in decimal hours.
Local Sidereal Time (LST)
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Convert the GST to decimal hours and the longitude(L) to decimal degrees.
If longitude is west then L is negative.
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Calculate LST. LST=GST+(L/15)
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Reduce LST to a value between 0 and 24 by adding or subtracting multiples
of 24. This is the LST in decimal hours.
Hour Angle(HA) and Declination(DE) given the Altitude(AL) and Azimuth(AZ)
of a star and the observers Latitude(LA) and Longitude(LO)
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Convert Azimuth(AZ) and Altitude(AL) to decimal degrees.
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Compute sin(DE)=(sin(AL)*sin(LA))+(cos(AL)*cos(LA)*cos(AZ)).
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Take the inverse sine of sin(DE) to get the declination.
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Compute cos(HA)=(sin(AL)-(sin(LA)*sin(DE)))/(cos(LA)*cos(DE)).
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Take the inverse cosine of cos(HA).
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Take the sine of AZ. If it is positive then HA=360-HA.
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Divide HA by 15. This is the Hour Angle in decimal Hours.
Hour Angle to Right Ascension
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Convert Local Sidereal Time and Hour Angle into decimal hours.
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Subract Hour Angle from Local Sidereal Time.
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If result is negative add 24.
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This is the Right Ascension in decimal hours.
Parallax(p) to Distance(d) Conversion
d=1/p
Notes:
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Parallax is in arcseconds.
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Distance is in parsecs.
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1 parsec equals 3.2616 light years.
Relationship between the focal point(f), diameter(D) and depth(d) of a
parabolic reflector
f=(D2)/(16*d)
Notes:
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f,D,and d are all in the same units.
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The focal point is measured from the bottom of the reflector.
Gain of a parabolic reflector given the diameter(D), wavelength(W) and
efficency factor(k)
G=10*log(k*(pi*D/W)2)
Notes:
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G is the gain over an isotropic radiator.
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k is usually about .55
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D and W are in the same units.
An approximation for Beam Width(BW) given diameter(D) and wavelength(W)
BW=W/D
Notes:
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BW is in radians (multiply by 57 to convert to degrees)
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D and W are in the same units.
Doppler shift due to the earth's rotation.
Fd=Fo*K*COS(LAT)*COS(DEC)*SIN(HA)
Notes:
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Fd is the doppler shift due to the earth's rotation
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Fo is the frequency of observation
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LAT is the latitude of the antenna
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DEC is the declination of observation
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HA is the hour angle of observation in degrees
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K=pi*d/(c*t)
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d is the diameter of the earth (12756336 meters)
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c is the speed of light (3 x 108 meters/seconds)
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t is the number of seconds in a sidereal day (86197 seconds)
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K is 1.546111 x 10-6
Length of time a star remains in the beam of an antenna
T=13751*W/(D*COS(DEC))
Notes:
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W is the wavelength
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D is the diameter of the dish
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DEC is the declination of the star
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W and D are in the same units
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T is in seconds
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This is an approximation which breaks down if the dish is pointed near
+/- 90o declination
Converting noise temperature to noise figure
F=10*Log((T+290)/290)
Notes:
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F is in decibels
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T is in kelvin
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Log is base 10
Range at which a signal can be detected
R=8x10-6*(Pe*A/T)1/2* (t/B)1/4
Notes:
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R is in light-years
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Pe is the effective radiated power of the transmitter in watts
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A is the effective area of the receiving antenna in square meters
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T is the excess receiver noise temperature in kelvin
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t is the averaging time of the receiver in seconds
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B is the bandwidth of the signal in Hertz
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8x10-6 is a constant and calculated using the formula:
1/(LY*(4*pi*K)1/2)
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LY is a light-year in meters (9.4608x1015)
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K is boltzman's constant (1.38x10-23)
The Drake Equation
N=R*fs*fp*ne*fl* fi*fc*L
Notes:
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R is the average rate of star formation in the galaxy
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fs is the fraction of stars that are suitable for planetary
systems
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fp is the number of suitable suns with planetary systems
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ne is the mean number of planets that are located within the
zone where water can exist as a liquid
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fl is the fraction of such planets on which life actually originates
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fi represents the fraction of such planets on which some form
of intelligence arises
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fc is the fraction of such intelligent species that develop
the ability and desire to communicate with other civilizations
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L is the mean lifetime (in years) of a communicative civilization