NORTON
ASTRONOMICAL EPHEMERIS PROGRAM PACKAGE
WARREN A. LEACH
OMSI Community Research Center
Portland,Oregon
ABSTRACT
________
NORTON is a RSTS BASIC-Plus package which gives
planet and asteroid positions for any date from
JANUARY 1, 1900. It has run under RSTS Versions 4
thru 6 (8K Job Max).
I _N _T _R _O _D _U _C _T _I _O _N _
from the computer's clock. Two dates can
be typed with 'to' in
between to indicate
Computers and linear programming are
a time period:
ideally suited to many of the
calculations encountered in Astronomy.
E.G. 'WHERE IS VENUS FROM
JAN 1,1976
The author collected and developed many
TO JAN
31,1976?'
of the formulae involved in Solar System
motions while a computer intern from The
Requests are stored as
indicators in file
Evergreen State College working for the
'SAM.DAT' and the
necessary program is
Kendall Planetarium at the Oregon Museum
chained. Information
requests answered
of Science and Industry (OMSI).
without chaining are
SIDEREAL TIME and
JULIAN DATE since these
are figured
Earlier versions performed individual
internally.
steps in the procedure necessary to
numerically represent a dynamic model of
Specific possible
operations include:
the Solar System. One program calculated
the slight variations in planetary orbits
Julian Date
Constellation
due to perturbations, some others located
Sidereal Time
Elongation
a planet from orbital elements and time,
Geocentric Distance
many interpreted local circumstances, and
Ascension
a few were graphics programs.
Angle
Velocity
Orbital Longitude
Right
Solar Distance
Hour
Declination
Altitude and Azimuth
As a Spring, 1974 lab project, the author
Obliquity of Ecliptic
linked most of the functions of
ephemeride generation programs with a
contains the equations to
word processor and a common data base.
using the
An inquiry is made with a loosely
generated by MNEPCH.BAS
formatted sentence or question:
The Mean,
Orbital Elements
E _P _H _E _M _1 _
represent a solar system
orbital elements
and stored in SAM.DAT.
Eccentric and True
Anomalies are derived
using Epoch Longitude
references. This
WHERE WILL MARS BE TOMORROW?
coordinates (Solar
will result in polar
distance and orbital
angle) needed to
VELOCITY AND ORBITAL LONGITUDE OF MERCURY
in the orbital
uniquely define position
plane.
converted to
ORBITAL ELEMENTS OF JUPITER
(X,Y,Z) and then
ON THE LINEPRINTER.
inclination to
These are
rectangular coordinates
rotated through orbital
the same reference, the
Earth's orbital
plane or Ecliptic. This
is changed to a
S _Y _S _T _E _M _ O _U _T _L _I _N _E _
coordinate system centered on the Earth
and its Equator: Right
Ascension and
Three programs comprise the basic
Declination. Other
calculations are
package. Each performs a specific
Velocity as a function of
Solar distance,
function described below. All are
the angle from the Sun as
seen from Earth
accessed by typing 'RUN NORTON'.
(Elongation) and the
localized coordinate
systems, Hour Angle,
Altitude and
Azimuth.
N _O _R _T _O _N _ is the program to accept and read
the input sentence. It is scanned for
M _N _E _P _C _H _ has
the equations for the annual
three particulars: Planet name, desired
variations in the shape
of orbits. The
output and date. If no planet name is
program is chained only
if the data file
found, all planets are assumed. If no
'SAM.DAT' contains
orbital elements for a
date is input, the current date is read
different year than
requested.
The formulae for the inner planet
Because values of the
orbital elements
elements are from the work of Simon
vary over extended
periods of time, they
Newcomb, the productive author of
are kept updated to
within a year of the
numerous volumes of T _h _e _ A _s _t _r _o _n _o _m _i _c _a _l _
request date. A job request far into the
P _a _p _e _r _s _ P _r _e _p _a _r _e _d _ f _o _r _ t _h _e _
U _s _e _ o _f _ t _h _e _
past will therefore cause MNEPCH
to be
A _m _e _r _i _c _a _n _ E _p _h _e _m _e _r _i _s _ a _n _d _
N _a _u _t _i _c _a _l _ A _l _m _a _n _a _c _ ,
chained. This
process is indicated by
which contain standards and techniques
the message:
for the U.S.Naval Observatory. The outer
'Calculating New
Epoch'
planet elements are read as program data
C _A _L _C _U _L _A _T _I _O _N _ O _F _ P _O _S _I _T _I _O _N _
for a 1975 epoch. Besides planets,
orbital elements for four asteroids are
Position in orbit is
determined for a
included. Additional space for more
date from Orbital
Elements and Epoch
elliptical orbits is provided for users
references. This is done
as follows:
to insert their favorites.
The mean angle traveled
per day (Mean
U _S _E _R _ P _A _R _T _I _C _U _L _A _R _S _
Motion) is defined:
N=360/P degrees
In the course of operation it is
=2*PI/P radians
necessary to know the user's place in
P is the Period of
one revolution
space and time:
in Mean Solar
days
Viewer's planet (usually Earth)
the task, but
Local Longitude and Latitude
connects period and
Hours from GMT
RSTS account number
all planets)
These are easily inserted in the 'User
Option' area in MNEPCH.
This is sufficient for
Kepler's Third Law
the mean distance:
A^3/P^2=C (constant for
For our Solar System:
C=K^2*(M0+M)
K is Gauss
Gravitation Constant
O _R _B _I _T _A _L _ D _E _F _I _N _I _T _I _O _N _
(.017202 is close enough)
M0 is one Solar mass
An orbit in space can be specified using
in solar units
the following orbital elements. The
first two define shape, three define
orientation and one is an epoch
reference.
M is planetary mass
The complete form is:
A^3/P^2=K^2*(M0+M)
In units of Solar Mass,
M0+M is quite
A
The mean distance is that between
the center and the furthest edge on
nearly one, so:
A^(3/2)/P*2*PI=K*2*PI
Taking the square root
and
an ellipse (semi-major axis).
by 2*PI
E
The measure of the oblateness or
Motion per day as a
out-of-round of an ellipse is
Major Axis 'A',
defined with B, the semi-minor axis:
(1)
B=A*SQR(1-E^2)
T _O _ T _R _U _E _ A _N _O _M _A _L _Y _
E=SQR((B^2-A^2)/A^2)
multiplying both sides
This gives the Mean
function of the SemiN=K*2*PI/A^(3/2)
M _E _A _N _
An approximate or Mean
Anomaly (angle
I
The angle between the plane of orbit
date can be found
and the Ecliptic is Inclination.
Longitude (L) for any
from Perihelion) for a
by taking the Mean
epoch (T1) and adding
Mean Motion per day
U
The angle between zero degrees and
Epoch (J-T1):
the Ascending Node is the Longitude
(2)
of the Ascending Node. A planet is
which position is
on or crossing the Ecliptic when at
to be found.
the Ascending or Descending Node.
of L.
(N) times the days from
M=N*(J-T1)+L-U-W
J is the date for
T1 is the Epoch date
L-U-W converts
longitude to an
W
The Argument of Perihelion is the
anomaly
angle between the Ascending Node and
Perihelion. It is different from
defined in an
the Longitude of Perihelion, which
the solar focus
is measured from the First Point of
circle with a
An Eccentric Anomaly is
intermediate step with
(the eccentric) and a
Aries by the relation:
major axis of
LONG.PERIH.=W+U
to the Mean
radius equal to the semithe orbit.
Its relation
Anomaly is defined by
Kepler's Equation:
L
The Mean Longitude at Epoch is the
(3)
reality reference for calculation of
Anomaly.
position in orbit. It is used in
EPHEM1 as an annual location from
from the
which true position is calculated.
presents no problem.
M=E1-E*SIN(E1)
Where E1 is the Eccentric
Finding the Mean Anomaly
Eccentric Anomaly
In practice, the opposite is required and
of Solar Distance and the
True Anomaly
the formula is not directly solveable for
for julian date J
follows:
E1. A non-algebraic technique is
X6=ATN(SQR(8*D^3)/3/K/(J-T9))
therefore needed. A table can be built
E2=2*ATN((TAN(S/2))^(1/3))
of the values from which interpolation
F=2*ATN(2/TAN(E2))
!TRUE ANOMALY
can resolve the answer. Isaac Newton
R=2*D/(1+COS(F))
!POLAR EQUATION
designed an interesting circular 'slide
! OF
ORBIT YIELDS R
rule' to solve for E1. The most
convenient solution for computer use is
T _R _A _N _S _I _T _I _O _N _S _ T _O _ U _S _A _B _L _E _ F _O _R _M _
arrived at with methods derived by
numerical analysis.
Conversion to rectangular
coordinates in
units of A.U. with the
Sun at (0,0,0)
S _O _L _U _T _I _O _N _ T _O _ K _E _P _L _E _R _' _S _
E _Q _U _A _T _I _O _N _
and referenced to the Ecliptic is
done
with orientational
elements.
The reverse function can be located in
X3=R*(COS(U)*COS(W+F))
this method credited to E.W.Brown.
Y3=R*(SIN(U)*COS(W+F))
Z3=R*SIN(W+F)*SIN(I)
For small E:
DEF FNS(Q)=ATN(Q/SQR(1-Q^2)) ! ARCSINE
Rotating through the
Obliquity of the
X0=ATN(E*SIN(M)/(1-E*COS(M)))
Ecliptic (E9) gives X,Y,Z
in terms of the
X2=SQR(1-2*E*COS(M)+E^2)
Earth's Equatorial Plane:
X3=FNS(-SIN(X0)^3/6/X2)
X5=X3
E1=M+X0+X3 !ECCENTRIC ANOMALY
Y5=Y3*COS(E9)Z3*SIN(E9)
Z5=Y3*SIN(E9)+Z3*COS(E9)
If E>.1 or great accuracy is desired:
X4=-(E*SIN(M+X0+X3))^3/6/X2
planets thus reduced,
X5=(E*SIN(M+X9))^5/120/X2
on the viewer's
E1=M+X0+FNS(X4+X5) !ECCENTRIC ANOMALY
convenient is
The position of the
a coordinate system based
planet is needed.
Most
Right Ascension and
Declination, measured
A more concise technique from W.M.Smart
and degrees
used in NORTON:
referenced to the
in 'Hours' (15 degrees)
respectively, and
E1=M+(E-E^3/8)*SIN(M)+E^2*SIN(2*M)/2
First Point of
+3/8*E^3*SIN(3*M)
Right Ascension is
Celestial Equator and the
Aries.
To calculate,
the slope of the
Equatorial X and Y
The Eccentric Anomaly yields the True
measured from the
Anomaly F:
X9,Y9,Z9:
E0=SQR((1+E)/(1-E))
F=2*ATN(E0*TAN(E1/2))
altitude up the Z
Or, also from W.M Smart:
X9)^2/(Y5-Y9)^2))
F=E1+(E+E^3/4)*SIN(E1)+E^2/4*SIN(2*E1)
+E^3/12*SIN(3*E1)
from Earth is
coordinates of a body
viewer's planet at
R8=ATN((Y5-Y9)/(X5-X9))
Declination is the
axis:
D8=ATN((Z5-Z9)/SQR((X5The distance to a planet
found from the
rectangular coordinates:
Orbital Longitude adds the angle from
Y9)^2+(Z5-Z9)^2)
zero degrees to the true anomaly:
O=F+(U+W)
between the Sun
R3=SQR((X5-X9)^2+(Y5Elongation is measured
and a body from Earth
using the Law of
The Distance from the Sun as a function
of angle is found from the Polar Equation
Q^2)/Q) !ARCCOS
for an ellipse:
R1^2)/(2*R*R3)
R=A*(1-E^2)/(1+E*COS(F))
distance
Or, more directly, from the Eccentric
distance
Anomaly:
R=A*(1-E*COS(E1))
(4)
the Solar
The position in space of a body then
defined in terms of orbital coordinates
Gravitation Constant
R,F, the orbital elements and date.
Cosines:
DEF FNC(Q)=ATN(SQR(1E6=FNC(R^2+R3^2R is Sun-Earth
R1 is Body-Sun
Velocity is a function of
Distance:
V=SQR(K*(2/R-1/A))
K is Gauss
L _O _C _A _L _
C _O _O _R _D _I _N _A _T _E _ S _Y _S _T _E _M _S _
P _A _R _A _B _O _L _I _C _ O _R _B _I _T _S _
H _o _u _r _
A _n _g _l _e _ is the time before a body is
The orbits of many comets are defined
in Zenith (straight up).
The hour angle
with a slightly different set of Orbital
of the Noon Sun is zero.
It is figured
Elements:
Time S9 from the
D
Perihelion distance, from focus
by subtracting Sidereal
Right Ascension R9:
R9-S9=HOUR
ANGLE
to vertex
T9
Perihelion Date and Epoch
is the angle above the horizon
A _l _t _i _t _u _d _e _
of a body.
measured from
I,U,W Orientational Elements
compass directions
Azimuth is
North corresponding to
(90 degrees is East).
They are found from
The process of finding position in terms
with the Hour
Spherical Trigonometry
Angle:
and unified graphic
interpretation is to
K4=FNS(-COS(D9)*SIN(S9-R9)/COS(K3))
be included in further
extensions.
L1 is local longitude
D9 is Declination
A _C _K _N _O _W _L _E _D _G _E _M _E _N _T _S _
FNS is Arcsine function
Guidance and advice
were received from
G _R _A _P _H _I _C _
I _N _T _E _R _P _R _E _T _A _T _I _O _N _
Peter
Langston, Evergreen
Programmer;
Garry Stasiuk, Director
of the Kendall
Plots of the curves defined by the
Planetarium;
Rusty
Whitney,
project
functions in this paper can be useful for
sponsor at OMSI; and the
Evergreen State
graphic output. A display of Right
College Library in
Olympia, Washington.
Ascension and Declination shows the
Earth's sky and with a background of
REFERENCES
stars results in a true planisphere. The
C _E _L _E _S _T _I _A _L _ M _E _C _H _A _N _I _C _S _, W.M.Smart,
equations of orbit can be used to create
Cambridge Press
a scaled Solar System model or orrery,
S _P _H _E _R _I _C _A _L _ A _S _T _R _O _N _O _M _Y _, W.M.Smart,
and plots using Altitude and Azimuth can
Cambridge Press (1971)
be useful to show a local view in
S _U _P _P _L _E _M _E _N _T _ T _O _ T _H _E _
A _S _T _R _O _N _O _M _I _C _A _L _
E _P _H _E _M _E _R _I _S _ A _N _D _ A _E _N _A _, H.M.Nautical
relation to the horizon.
Almanac
Office, London (1961)
M _E _A _S _U _R _E _M _E _N _T _ U _N _I _T _S _
G _L _O _S _S _A _R _Y _ _O _F _ _T _E _R _M _S _
Care must be taked to keep all unit types
Perihelion to a
consistent throughout all calculations.
Eccentric
For these programs, the Astronomical Unit
and second
is used for length and radians is the
True anomaly.
angular measure. Conversion factors are
used to alter to desirable output type:
the angle
ANOMALY is the angle from
planet.
The Mean and
anomalies are first
approximations to the
ARGUMENT OF PERIHELION is
12/PI HOURS=180/PI DEGREES=RADIANS
Node and
1 A.U.=92,957,000 Mi=149,600,000 KM
Similarly, all dates used must be in
terms of the same Epoch. The particular
angle from zero
choice of dates is a secondary
celestial sphere (also
consequence, but all must be consistent
Point of Aries) to
to the others for meaningful
Ecliptic measured
calculations. The Epoch base of NORTON
is January 0,1900.
between the Ascending
Perihelion.
ASCENDING NODE is the
degrees on the
called the First
the plane of the
along the orbit.
ASTRONOMICAL UNIT or A.U.
is the unit of
S _U _M _M _A _R _Y _
measurement convenient for Solar
System measurements.
It is the
Computation of astronomical ephemerides
Earth to the Sun.
and their interpretation of results is
proving feasible with the precision
projection on
available on most timeshared computers.
of the Earth's
Yet much work remains to design more
forms of output and better computation
techniques.
measure in degrees of
distance from the
CELESTIAL EQUATOR is the
the celestial sphere
Equator.
DECLINATION is the
a position above the
Celestial
Work completed at OMSI includes an
from Earth
animation of Comet Kohoutek and planets
coordinate).
with the graphics terminal coupled to a
16mm time lapse movie camera, predicting
of the orbital
local events for the Portland area, as
circular orbit has
well as providing convenient access to
zero. The
celestial circumstances over an extended
that figured
Equator, as viewed
(Geocentric
ECCENTRICITY is a measure
out-of-round.
A
an eccentricity of
Eccentric Anomaly is
from a focus of the
ellipse.
time period. Another possibility still
to be considered is computer guidance for
the Earth's
planet, comet or satellite projectors in
the Planetarium.
ECLIPTIC is the plane of
Orbit
EPHEMERIS a table of data
(usually
It has been the intent of the author to
the position of
put accurate astronomical data in easily
accessible form. With the addition of
other programs to generate other types of
specifications defining
output, the system will be made more
of an orbit in
useful. Past work with parabolic orbits
is being used for comet ephemerides
similar to current planetary and asteroid
distance between
stuff. Current work with local eclipse
object in Earth's
circumstances, rising and setting times,
numeric) concerning
celestial bodies.
ELEMENTS are
shape and orientation
space.
ELONGATION the angular
the Sun and another
Sky.
EPOCH An Epoch is a reference point in
time and space needed to allow a
open ended path
geometric model to represent a
comet.
phenomena in Nature. In NORTON, it
is comprised of a Mean Longitude and
orbital point to
a date.
PARABOLIC ORBIT is an
often defined by a
PERIHELION The closest
the Sun.
INCLINATION is the angle from the
angle to the
Ecliptic to the planet's orbit.
from zero degrees.
ORBITAL LONGITUDE is the
JULIAN DATE is a convenient means of
geocentric
handling dates over a long period.
measured East along
Dates are counted from an agreed upon
divided into 24
starting date. Astronomical Julian
Date (capital 'J') is referred to a
particular date, at noon, January
Right Ascension of
1,4173 BC.
(straight up).
RIGHT ASCENSION is a
position in orbit
coordinate system
the Celestial Equator
Hours.
SIDEREAL TIME is the
the observer's Zenith