flight_time_plan.txt

Algorithm for numberical approximation of flight time.

This document outlines a numerical approach for calculating the flight time of a constantly thrusting spacecraft on a hyperbolic orbit. The algorithm will be implemented in flight_time.py and used to calculate the time it takes for Einstein to reach the sun's focal point. This will be incorporated into the third scientific appendix.

There are two steps to the approach, each of which is outlined in greater detail. The continuous thrust of the engine is modeled as a chain of discrete, instantaneous increases in velocity interceded with coasting periods.

Step 1 - burn phase

The spacecraft's velocity changes instantaneously. The new orbit AND the state of the spacecraft along the orbit are calculated.

Step 2 - coast phase

The spacecraft coasts for a period of time. The spacecraft's orbit does not change but the state (position and velocity) change and are calculated.


Below is additional detail on the two steps.


Step 1 - burn phase

The spacecraft has a known orbit (a and e) position (r and theta) and velocity (v)

1. The spacecraft's velocity instantaneously increases a certain amount. The fuel on the
   craft is decremented. The velocity change and fuel change account for the reduction in mass.

2. The spacecraft's r and v are known. R has not changed during the instantaneous burn. 
   The semi-major axis is recalculated from the vis-viva equation

3. The eccentricity and true anomaly are simultaneously calculated by solving the radial form of the
   spacecraft's hyperbolic trajectory and the equation for flight path angle. The flight path
   angle is assumed to remain constant from the last time step.

4. The angle of flight is updated based on the new true anomaly and eccentrcity. The frame of reference
   is effectively rotated so that the 0 degree axis runs through the vertex of the hyperbola

The burn phase results in a new orbit (a and e) position (r and theta) and velocity (v and phi) 

Step 2 - coast phase

The spacecraft's orbit (a and e) remain the same and the position (r and theta) and velocity (v and phi)
are updated

1. Time is incremented a set amount and the spacecraft's step length is calculated based on the starting
   velocity

2. The angle of flight and the original r a new r is calculated by linear approximation - 
   treating the hyperbola as a curve of line segments 

3. The true anomaly theta is updated using the equation of the orbit

4. The velocity is updated based on the orbit and new r

5. The angle of flight (phi) is updated using the true anomaly and the eccentricity

The spacecraft's orbit (a and e) is unchanged while the position and velocity have been updated

The algorithm iterates between step 1 and step 2 until the burn is complete. After the burn is complete
the algorithm continues on step 2 for several more iterations with a progressively larger time step until
the flight angle falls close to zero. The algorithm then proceeds to step 3, which treats the spacecraft's 
orbit like a straight line with a center at the sun. In step 3 the time to reach a certain distance is 
solved analytically.