Useful Formulas for Amateur SETI
The following is derived from a number of sources including:
Practical Astronomy with Your Calculator by Peter Duffett-Smith
The ARRL UHF/Microwave Experimenters Manual
Conversations with Malcolm Mallette and Dr. Paul Shuch. To both people I would like to say "thanks".
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:
Convert local time to Greenwich Mean Time
Let Y equal the year, M equal the month, D equal the day in decimal form.
If M equals 1 or 2 then subract 1 from Y. and add 12 to M.
Compute A. A=INT(Y/100)
Compute B. B=2-A+INT(A/4). However, if the date is earlier than October 15, 1582 then B=0.
Calculate C. C=INT(365.25*Y). If Y is negative then C=INT((365.25*Y)-.75).
Calculate E. E=INT(30.6001*(M+1))
Calculate JD (Julian Date). JD=B+C+D+E+1720994.5
Greenwich Sidereal Time (GST)
Calculate JD (Julian Date) corresponding to 0 hours GMT for this date. (This value should end in .5)
Calculate UT. This is the GMT in decimal hours.
Calculate T. T=(JD-2451545.0)/36525.0
Calculate T0. T0=6.697374558+ (2400.051336*T)+(0.000025862*T2)+(UT*1.0027379093)
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)
Convert the GST to decimal hours and the longitude(L) to decimal degrees. If longitude is west then L is negative.
Calculate LST. LST=GST+(L/15)
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)
Convert Azimuth(AZ) and Altitude(AL) to decimal degrees.
Compute sin(DE)=(sin(AL)*sin(LA))+(cos(AL)*cos(LA)*cos(AZ)).
Take the inverse sine of sin(DE) to get the declination.
Compute cos(HA)=(sin(AL)-(sin(LA)*sin(DE)))/(cos(LA)*cos(DE)).
Take the inverse cosine of cos(HA).
Take the sine of AZ. If it is positive then HA=360-HA.
Divide HA by 15. This is the Hour Angle in decimal Hours.
Hour Angle to Right Ascension
Convert Local Sidereal Time and Hour Angle into decimal hours.
Subract Hour Angle from Local Sidereal Time.
If result is negative add 24.
This is the Right Ascension in decimal hours.
Parallax(p) to Distance(d) Conversion
d=1/p
Notes:
Parallax is in arcseconds.
Distance is in parsecs.
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:
f, D, and d are all in the same units.
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:
G is the gain over an isotropic radiator.
k is usually about .55
D and W are in the same units.
An approximation for Beam Width(BW) given diameter(D) and wavelength(W)
BW=W/D
Notes:
BW is in radians (multiply by 57 to convert to degrees)
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:
Fd is the doppler shift due to the earth's rotation
Fo is the frequency of observation
LAT is the latitude of the antenna
DEC is the declination of observation
HA is the hour angle of observation in degrees
K=pi*d/(c*t)
d is the diameter of the earth (12756336 meters)
c is the speed of light (3 x 108 meters/seconds)
t is the number of seconds in a sidereal day (86197 seconds)
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:
W is the wavelength
D is the diameter of the dish
DEC is the declination of the star
W and D are in the same units
T is in seconds
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:
F is in decibels
T is in kelvin
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:
R is in light-years
Pe is the effective radiated power of the transmitter in watts
A is the effective area of the receiving antenna in square meters
T is the excess receiver noise temperature in kelvin
t is the averaging time of the receiver in seconds
B is the bandwidth of the signal in Hertz
8x10-6 is a constant and calculated using the formula:
1/(LY*(4*pi*K)1/2)
LY is a light-year in meters (9.4608x1015)
K is boltzman's constant (1.38x10-23)
The Drake Equation
N=R*fs*fp*ne*fl* fi*fc*L
Notes:
R is the average rate of star formation in the galaxy
fs is the fraction of stars that are suitable for planetary systems
fp is the fraction of suitable suns with planetary systems
ne is the mean number of planets that are located within the zone where water can exist as a liquid
fl is the fraction of such planets on which life actually originates
fi represents the fraction of such planets on which some form of intelligence arises
fc is the fraction of such intelligent species that develop the ability and desire to communicate with other civilizations
L is the mean lifetime (in years) of a communicative civilization
Location of the Sun on the Ecliptic
Subtract 2447892.5 from the Julian Date to get D
Multiply D by 0.017202792 to get N
Subtract 0.058732406 from N to get M
Add or subtract 6.2831853 to M to get a value between 0 and 6.2831853
Solve E-0.016713*sin(E)=M for E
Solve tan(V/2)=1.016855*tan(E/2) for V
Multiply V by 57.295779 and add 282.768422 to get W
Add or subtract 360 to get a value between 0 and 360
This is the Ecliptic Longitude. The Ecliptic Latitude is 0.
Ecliptic Coordinates given Right Ascension(RA) and Declination(DE)
Convert Declination(DE) to degrees
Convert Right Ascension(RA) to Hours then multiply by 15 to convert to degrees.
Compute sin(LA)=0.917464*sin(DE)-0.397819*sin(RA)*cos(DE).
Take the inverse sine of sin(LA).
This is the Ecliptic Latitude.
Compute y=0.917464*sin(RA)+0.397819*tan(DE).
Compute x=cos(RA).
Take the inverse tan of (y/x) and call it LO.
Add or subtract multiples of multiples of 180 degrees to LO so that LO meets the following conditions:
If y is positive LO lies between 0 and 180 degrees.
If x is negative LO lies between 90 and 270 degrees.
LO is the Ecliptic Longitude.
Notes:
0.917464 is the cosine of 23.44189 degrees
0.397819 is the sine of 23.44189 degrees
23.44189 degrees is the tilt of the Earth on it's axis
Doppler Shift do to Earth's orbit around the Sun relative to a given Ecliptic Latitude(LAE) and Longitude(LOE).
Fd=K*Fo*cos(LAE)*sin(LOE-LOS)
Notes:
Fd is the doppler due to the earth's orbit of the sun.
Fo is the frequency of interest.
LOS is the Ecliptic Longitude of the Sun.
LOE is the given Ecliptic Longitude.
LAE is the given Ecliptic Latitude.
K=v/c
c is the speed of light (3 x 108 meters/second)
v is the speed the earth orbits the sun (29800 meters/second)
K is 9.9333 x 10-5
Doppler Equation
Fd=Fo*(1-v/c)
Notes:
Fd is the doppler shifted frequency.
Fo is the frequency transmitted by the source.
v is the relative velocity in meters per second.
v is positive moving away the source.
c is the speed of light (3 x 108 meters/second)
Relativistic Doppler Equation
Fd=Fo*(1-(v/c)2)1/2/ (1-((v/c)*cos(A)))
Fd is the doppler shifted frequency.
Fo is the frequency transmitted by the source.
v is the relative velocity in meters per second.
A is the angle the observer is moving at relative to the source.
A is 0 moving toward the source and 180 moving away.
c is the speed of light (3 x 108 meters/second)