Object Coordinates and Phenomena Calculation

Star Data Matrix
This is a convenient place to enter frequently required star locations. Each row contains the following data going across: Star name, Year epoch of data, 3 columns of RA (hr, min, sec), 3 columns of DEC (deg, min, sec), Proper motion in RA in arc-sec/yr (not RA seconds), Proper motion in DEC arc-sec/yr. Rows can be inserted at the end as needed to accommodate new stars.

Star Data Extraction
Notes: The following functions are used to extract data from the STARMAT table.
Input for all functions: Star Number in table. Header row is zero so first star is one.
 
Retrieve Star Name
Output: Star name from table.
 
Retrieve Epoch Year
Output: Epoch year from table may be integer or floating.
 
Retrieve RA of Star in Decimal Degrees
Output: RA of star in decimal degrees.
 
Retrieve DEC of Star in Decimal Degrees
Output: DEC of star in decimal degrees.
 
Retrieve Yearly Proper Motion in RA
Output: Proper motion in seconds of arc (same scale as DEC) per year.
 
Retrieve Yearly Proper Motion in DEC
Output: Proper motion in seconds of DEC per year.
 
 
Example: Retrieve star data.
Row of star to find data for:
Name:
Year:
RA:
DEC:
Proper Motion in RA in seconds of arc per year:
Proper Motion in DEC in seconds of arc per year:

Planet Data Matrix
The planet matrix is arranged by planet (missing Pluto) in order from the sun -- Earth data is the 3rd row after the headers. Data here assumes the standard epoch J2000. Each row contains: Mean heliocentric longitude at epoch, first order change in longitude per century and semimajor axis in AU. Dividing the values in the second column by the number of days in a Julian Century (36525) yields the mean daily motion of the planet in degrees at epoch J2000.
 
Pluto is absent because of its great inclination and eccentricity. Only enough data is included to perform low precision position calculations with the equations that follow the matrix. Minor planets can be added to the end of the table as long as their orbits have low eccentricity and inclination.

Resources: [AA: p. 141, pp. 200-2]
                 [ESAA: p. 704]

Planet Data Functions
Low Precision Helio and Geocentric Longitudes of the Planets
 
Notes: As Johannes Kepler could attest, calculating planetary positions would be a lot easier if the planets in our solar system traveled in purely circular orbits entirely in the plane of the ecliptic. With respect to the central sun (heliocentric) the daily motion would be a constant number of degrees per day. Calculating a planet's Earth centered (geocentric) location would take a little trigonometry once you knew the daily motion and distance from the sun for the Earth and planet in question. While not very exact, a ballpark estimate of planetary location can be made with the following equations if the above is assumed true. Accuracy falls off as the orbit deviates from a circle (eccentricity), increases in inclination to the ecliptic and is modified by perturbations. Therefore you should not use these equations to calculate comet orbits.
 
The first equation calculates heliocentric positions of a planet. This data is generated for the Earth and another planet for input into either of the three geocentric position equations that follow. The first geocentric equation handles the geometry for outer planets (Mars and outward) and the second for the inner (Mercury and Venus). The final geocentric equation (LONGe()) wraps the functionality of the prior two equations by using the semi-major axis parameter to determine if an inner or outer planet is in question.

Resources: [PAC: p. 72]

Low Precision Mean Heliocentric Longitude of Planet Equation
Input: Date of interest in Julian centuries from J2000, Planet number 1 = Mercury, ..., 8 = Neptune.
Output: Heliocentric longitude in degrees normalized to 0 - 360.

 

Low Precision Geocentric Longitude of Outer Planets
Input: Heliocentric longitudes of planet and Earth and semimajor axis of planet.
Output: Geocentric longitude of planet in degrees normalized to 0-360.

 

Low Precision Geocentric Longitude of Inner Planets
Input: Heliocentric longitudes of planet and Earth and semimajor axis of planet.
Output: Geocentric longitude of planet in degrees normalized to 0-360.

 

Low Precision Geocentric Longitude for Outer and Inner Planets Based on Semimajor Axis
Input: Heliocentric longitudes of planet and Earth and semimajor axis of planet.
Output: Geocentric longitude of planet in degrees normalized to 0-360.

 

Example: Low precision heliocentric longitude of the Earth when Spring begins.
 
Date of vernal equinox:
Low precision heliocentric longitude of Earth on above date:
 
By definition the geocentric longitude of the Sun at this instant is 0-degrees. The geocentric longitude of the sun always differs by 180-degrees from the heliocentric longitude of the Earth. So compare the result with the exact answer of 180-degrees.
 
Example: Low precision geocentric longitude at conjunction.
 
On Feb. 23, 1999 Venus and Jupiter were separated by a mere 1/3-degree -- able to fit together in a low-power telescopic field of view. It figures their geocentric longitudes must have been similar
 
Conjunction date:
Low precision geocentric longitude of Venus on above date:
Low precision geocentric longitude of Jupiter on above date:
 
The calculations place the planets in the area of the vernal equinox.

Rising and Setting
Sidereal Time of Rising and Setting
 
Notes: These calculations can determine the approximate sidereal time that an object is on the horizon. Sidereal time can be translated to UT to determine civil time of the event. The calculation here are an approximation based on a spherical Earth and do not take into account other factors like object motion, altitude and atmosphere refraction. These two factors alone can affect setting and rising time by several minutes.
 
From a particular location objects may be in the sky all the time or not at all. For such an object the rise and set functions will show an error. The WillRiseSet() function below can assist in determining if the error can be explained by an object that does not rise or set.
Resources: [PAC p. 36]; [TAA p. A12]

 

Core Function for Rise and Set
Used by the functions that follow.
Input: Latitude of observer and declination of object in degrees.
Output: Sidereal time in degrees.

 

Rise and Set Test Function
Determines if object will have a rising and setting.
Input: Latitude of observer and declination of object in degrees.
Output: String indicating if the phenomena can occur.

 

Sidereal Times of Rising and Setting
The first function determines rise time and the second set. The results are in degrees of sidereal time. Note that if these functions are used for objects with a large apparent motion against the background stars (moon, sun, comets, mercury, etc.) that the DEC and RA should be for the time close to the rising or setting. If you have a method of determining these coordinates for a set time you can determine rising and setting iteratively as follows:
  • Take the DEC and RA values for some point during the day in question and determine sidereal rise and set time using the functions below.
  • Use your method of determining the object DEC and RA for these times and substitute back into the rise and set equations.
  • Repeating this will yield better results until a point is reached where the values do not change.
 
Input: Latitude of observer, declination of object in degrees, right ascension of object in degrees.
Output: Sidereal time in degrees.

 

Example: Determine time of rising for Leonid meteor shower radiant.
 
Position of radiant in RA:
and DEC:
Position of observer in LAT:
and LONG:
Date of observation:
 
Will the radiant rise at all?:
Sidereal time of rise:
Universal Time of rise:  

 

Astro Utilities Electronic Book Copyright © 1999 Pietro Carboni. All rights reserved.