4.3.2 Nominal Trajectory1
4.3.2 Nominal Trajectory
The nominal trajectory and orbit for the 5kg payload is directly related to lowering the cost of the launch vehicle. If the launch vehicle does not try to follow a nominal trajectory it will be over designed and over built. The mission requirements for project Bellerophone are to place a 5kg payload into an orbit with a minimum altitude of 300km. We meet the requirement of a minimum altitude and create an orbit which is near to circular. The final design uses a balloon as a launch platform which reduces the drag on the launch vehicle and this reduces the total velocity needed to get into Low Earth Orbit (LEO).
The following table describes the orbital parameters necessary to understand what orbit this vehicle is in.
Table 4.3.2.1Orbit ParametersVariable / Value / Units
Periapsis / 773.14* / km
Apoapsis / 809.57* / km
Eccentricity / 0.0025 / --
Inclination / 28.5 / deg
Semi-Major Axis / 7167.36 / km
Period / 6038.7878 / sec
*Values are from the surface of the Earth.
The periapsis of an orbit is closest approach to the surface of the Earth once it is in its orbit. For our vehicle the periapsis is 473.14km above the desired altitude. The apoapsis is the furthest distance from the surface of the Earth in which the vehicle will experience. For the mission a given apoapsis was not specified. The eccentricity is a measure on how circular the orbit is. For a circular orbit the eccentricity should be zero. This translates into a constraint on apoapsis, 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 KennedySpaceCenter 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 4.3.2.2ΔV BreakdownVariable / Value / Units / Percent
ΔVtotal / 9354 / m/s / --
ΔVdrag / 4 / m/s / 0.043
ΔVgravity / 2034 / m/s / 21.745
ΔVEarth assist / 411 / m/s / 4.394
ΔVleo / 7727 / m/s / 82.606
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 4.3.2.1 shows the entire orbit mentioned above.
Fig 4.3.2.1: Full orbit of 5kg payload.
(Kyle Donahue)
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 4.3.2.3Coefficients for Steering LawVariable / Value
a1 / -1.6117015e-1
b1 / 2.8636253e1
a2 / -5.0248315e-3
b2 / 1.3241455e0
a3 / 2.8623095e-19
b3 / -6.2486935e-1
numbers refer to stage number
The linear-tangent steering law is calculated using Eq. (4.3.2.1) below.
Eq. 4.3.2.1
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 4.3.2.4Angles from the Steering LawVariable / Value / Units
End of 1st stage / 24 / deg
End of 2nd stage / -32 / deg
End of 3rd stage / -32 / 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.
Fig 4.3.2.2: Definition of steering law angles.
(Amanda Briden)
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.
Fig 4.3.2.3: Trajectory part of orbit for 5kg payload.
(Kyle Donahue)
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.
Author: Kyle Donahue (Scott Breitengross, Amanda Briden, Daniel Chua
Bradley Ferris, Allen Guzik, Elizabeth Harkness, Jun Kanehara)