initial orbit determination

Double-R method for preliminary (initial) orbit determination using three sets of right-ascension and declination angles.

This Fortran program is designed to solve the problem of orbit determination from three optical sightings using the Double-R iteration method. The solution provides the state vectors (position and velocity) of the object at the second observation time (t2).

Input

• ra1, ra2, ra3: Right ascension at observation times t1, t2, and t3 (in radians).
• dec1, dec2, dec3: Declination at observation times t1, t2, and t3 (in radians).
• Mjd1, Mjd2, Mjd3: Modified Julian Date of observation times t1, t2, and t3.
• rSite1, rSite2, rSite3: Position vectors of observation sites in the Earth-Centered Inertial (ECI) frame (in kilometers).

Output

• r: Position vector at observation time t2 (in kilometers).
• v: Velocity vector at observation time t2 (in kilometers per second).

Execution

To run the program, execute the compiled binary in a Fortran environment. The program uses several helper functions from the doubler_functions module.

Helping Module "doubler_functions"

This Fortran module provides subroutines and functions for performing the Double-R iteration. The main subroutine doubler_itr executes the core Double-R iteration, while rv2ceo calculates classical orbital elements from position and velocity vectors. Additionally, there are utility functions such as site_vector, sidereal_time, and universal_time for handling observation site coordinates and time conversions.

• Subroutines:

  • doubler_itr: Executes the main Double-R iteration. It takes various inputs related to observations, station positions, and times and computes values like true anomalies, eccentricities, and differences between calculated and observed times.
  • degrad: Transforms angles between degrees and radians based on the specified type.

• Functions:

  • rv2ceo: Calculates classical orbital elements (semi-major axis, eccentricity, inclination, RA of Ascending Node, argument of perigee, true anomaly) from position and velocity vectors.
  • site_vector: Computes the observation site position vector in Earth-Centered Inertial (ECI) coordinates from geodetic coordinates.
  • sidereal_time: Calculates the local sidereal time from the Julian Date, UT, and East Longitude.
  • universal_time: Computes Universal Time (UT) from a given Julian Date.
  • CROSS: Calculates the cross product between two vectors.
  • angl2v: Calculates the angle between two 3D vectors.

• Additional Information: The module includes declarations for constants like Earth's radius, gravitational constant, and mathematical constants. Implicit none is used for variable declaration to ensure explicit variable typing. The module uses a selected real kind for numerical precision (ikind).

Iteration

The program uses the Double-r iteration method to refine the estimates of the state vectors. The iteration process is repeated until convergence or a maximum iteration count is reached.

Results

The final estimates of the state vectors (position and velocity) and classical orbital elements (semi-major axis, eccentricity, inclination, right ascension of ascending node, argument of perigee, and true anomaly) are provided.

Note

• The program is initialized with initial guesses for the position vector at t1 and t3.
• The iteration process refines these guesses to find the state vectors at t2.

Caution: This script is intended for educational purposes and may require adaptations for specific use cases or real-world applications.