A.6.2.4 ΔV Analysis1

A.6.2.4 ΔV Analysis

The ΔV calculations are essential to the determination of the trajectory, indexing of flight path characteristics, and fuel mass determination. Due to the vital nature of these calculations, the equations for each applicable ΔV loss was determined very early in the design process and implemented within the trajectory codes. The equations used to determine each ΔV calculation are included below:1

The velocity required to maintain lower earth orbit is calculated using equation A.6.2.4.1 below.

/ (A.6.2.4.1)

μ is 3.986 x 1014 m3/s2 for earth and Rp is the radius of the earth plus the height of the periapsis (from earth’s surface).

The velocity required to overcome drag losses occurring within the earth’s atmosphere is calculated using equation A.6.2.4.2 below.

/ (A.6.2.4.2)

where D(t) is the drag experienced over every time step, m(t) is the mass of the rocket for every time step, and tb is the total burn time. It is noted that once the rocket leaves the atmosphere the D(t) term goes to zero and thus no more drag is accumulated.

The velocity required to overcome gravity losses experienced during flight is calculated using equation A.6.2.4.3 below.

/ (A.6.2.4.3)

where g(t) is the gravity force experienced by the rocket over the course of the flight for every time step, and γ(t) is the flight path angle of the rocket from horizontal over the course of the flight for every time step.

The ΔV earned from launching the spacecraft a certain location on the earth was calculated using equation A.6.2.4.4 below.

/ (A.6.2.4.4)

where Re is the radius of the earth [m] , φ is the latitude of the launch location and td is the total time in a day [sec]. For earth td is 86400 seconds and Reis 6376000 meters.

The trajectory team is also interested in determining whether the ΔV from the trajectory code is matching the ΔV split from the MAT codes. In order to determine this, a comparison is mode between the two codes using equation A.6.2.4.5 below.

/ (A.6.2.4.5)

where T(t) is the Thrust experienced by the rocket over the entire length of the flight for every time step. This equation allows the trajectory group to compare the ΔV split amongst the stages from the trajectory codes and the MAT codes.

The previous equation were combined from each stage and then the final ΔV determination is calculated using equation A.6.2.4.6 below

/ (A.6.2.4.6

This equation allows the team to make a determination about which trajectory is the most ideal and to give the propulsion group an estimation on the required ΔV consumption and thus the required fuel needed to reach the trajectory and lower earth orbit.

It is important to note that for ΔVthrust, ΔVdrag, and ΔVgrav these values were calculated using ode45 in Matlab. Since the ode codes were used separate from the launch vehicle determination, it was required that the ΔV requirements for each stage be separated until the entire code is run. Once this is complete the outputs from the separate stages are combined for a total ΔV determination for the entire flight path.

There is also consideration paid to the fact that while a trajectory may exist, the characteristics of that trajectory may not be ideal and may need to be corrected fro while in orbit. This consideration prompted a calculation of the ΔV required to circularize. This ΔV is calculated using equation A.6.2.4.7 below.

/ (A.6.2.4.7)

where Vcirc is the velocity required for a circular orbit based on the perioapsis of the obtained orbit, Vleo is the velocity of the obtained orbit, and γ is the flight path angle

The ΔV calculations are then used as a comparison tool for the optimization of the trajectory. It is clear to the trajectory group and the team in general that there is a correlation between required ΔV and cost of the rocket based on the fuel cost needed and the additional size required to carry excess amounts of fuel. In the initial version of the optimization code, ΔV was the determining factor in choosing the applicable launch parameters, steering coefficients and angles, and the initial launch angle based on this determination.

Later reviews of the trajectory revealed that there was significant change in the ΔV due to drag when the rocket was launched from an altitude instead of from the ground.

References:

1 Humble, Ronald W., Henry, Gary N., Larson, Wiley J., Space Propulsion Analysis and Design. The McGraw-Hill Companies Inc. St. Louis, Missouri, 1995.

Author: Scott Breitengross