4.1.2 Nominal Trajectorypg 1
In order for the 200g launch to be considered successful, the payload must reach an orbit of at least 200km according to the design parameters. Since the main consideration during this project is cost, it’s clear to us that a launch which achieves a nearly circular orbit is reuired. Upon reaching the required orbit, optimization occurs to refine the orbit to nearly circular conditions. We would able to meet these conditions successfully.
The following table describes the orbital parameters obtained for the 200g payload launch vehicle.
Table 4.1.2.1Orbit ParametersVariable / Value / Units
Periapsis / 677.39* / km
Apoapsis / 755.05* / km
Eccentricity / 0.0055 / --
Inclination / 28.5 / deg
Semi-Major Axis / 7092.22 / km
Period / 5944.0760 / sec
*Values are from the surface of the Earth.
The periapsis of the orbit is 677.39km, this exceeds the design requirements by 377.39km. This excess altitude will more than account for any errors associated with calculations and allows for a margin of error to compensate for unforeseen conditions during flight. The apoapsis for this launch is 755.05km, showing that we achieved a nearly circular orbit. An eccentricity of 0 indicates a perfectly circular orbit and we achieved an eccentricity of .0055. The inclination angle so we have an apoapsis difference of 36.43km. Project Bellerophone did not have a specified inclination to the orbit. Since we are launching from the Kennedy Space Center in Cape Canaveral, FL and launching directly east our orbit has an inclination of 28.5o. The semi-major axis is the distance from the center of the ellipse (center of the Earth) to the edge of the ellipse, and the semi-major axis for our orbit is 7167.36km. The period of an orbit is the time it takes for the satellite to make one complete revolution and the period for our orbit is 6038.7878 seconds which is about 1 hour and 40 minutes.
The velocity needed to reach our orbit is measured in the change of velocity also know as ΔV. Table 4.3.2.2 breaks down the ΔV budget for the 5kg case of Project Bellerophone.
Table 4.1.2.2ΔV BreakdownVariable / Value / Units / Percent
ΔVtotal / 9313 / m/s / --
ΔVdrag / 6 / m/s / 0.043
ΔVgravity / 1991 / m/s / 21.379
ΔVEarth assist / 411 / m/s / 4.413
ΔVleo / 7727 / m/s / 82.97
The ΔVtotal is a combination of all the other ΔVs. ΔVdrag refers to the velocity needed to overcome the drag which we will experience. This value seems incredibly low, and it should be that way because we are launching from a balloon at an altitude of 30km. Since we are launching from so high in the atmosphere the air density is relatively low and does not cause much resistance to the launch vehicle. ΔVgravity is the velocity needed to break gravity drag. ΔVEarth assist refers to the velocity the Earth’s spin is helping the launch vehicle reach orbit. This is the only ΔV which is helping us, the other ΔVs are velocities we need to overcome. ΔVleo is the velocity needed to achieve our orbit.
Figure 4.1.2.1 shows the entire orbit mentioned above.
Figure 4.1.2.1: Full orbit of 5kg payload.
(Scott Breitengross)
In order to obtain any orbit a steering law is needed to put the rocket on the correct path. We created and used a linear-tangent steering law for each stage. There are many other types of steering laws besides linear-tangent. Other types of steering laws include linear with any of the trigonometric functions along with polynomial with any of the trigonometric functions. Since we used a linear-tangent law we had to have two coefficients for each part of the law.
Table 4.1.2.3Coefficients for Steering LawVariable / Value / Units
a1 / -2.02785e-1 / --
b1 / 2.8636253e1 / --
a2 / -6.58045e-3 / --
b2 / 1.80044e0 / --
a3 / 3.22302e-19 / --
b3 / -4.66308e-1 / --
numbers refer to stage number
The linear-tangent steering law is calculated using Eq. (4.1.2.1) below.
Eq. 4.1.2.1
Where φ is the angle the launch vehicle is at, a is the constant mentioned above, t is time in seconds, and b is the other constant mentioned above.
These coefficients stay constant for each stage, but the angles in which they create change over time. In order to better understand the steering law and what the coefficients do table 3 describes the angles in which the vehicle is pointing at the end of each stage.
Table 4.1.2.4Angles from the Steering LawVariable / Value / Units
End of 1st stage / 42 / deg
End of 2nd stage / -25 / deg
End of 3rd stage / -25 / deg
Angles are the nose pointing based on the horizon
The angles in table 3 refer to the angle at which the nose of the rocket is pointing relative to the horizon. For example if the rocket were pointing directly east and parallel to the surface of the Earth then it would be 0o. Figure 4.3.2.2 shows how the angles for the steering law are defined.
Figure 4.3.2.2: Definition of steering law angles.
(Kyle Donahue)
Where br is pointing “up” or towards the sky and bθ is pointing east.
Figure 4.3.2.3 is of the trajectory part of the orbit mentioned.
Figure 4.1.2.3: Trajectory part of orbit for 5kg payload.
(Scott Breitengross)
The figure for the trajectory part of the orbit looks the way it does for several reasons. One thing to note is that the yellow dot is the launch site on the surface of the Earth, and the start of the red line should not correspond to that as we are launching from a balloon with an altitude of 30km. The shape of the trajectory is determined by the steering law which changes the angle. Another note on fig. 4.3.2.2 is that it is the nominal trajectory for the rocket but not necessarily the path the rocket will take.
The trajectory subgroup met the mission requirement of an orbit with a periapsis altitude of at least 300km. We accomplished our mission using a linear-tangent steering law and launching from a balloon.
Scott Breitengross (Kyle Donahue, Amanda Briden, Daniel Chua
Bradley Ferris, Allen Guzik, Elizabeth Harkness, Jun Kanehara)