AAE 450 Project Bellerophon

A.6.2.5.2 Three-dimensional Plots1

A.6.2.5.2 Three-dimensional Plots

Plotting the three-dimensional plots that represent the Earth, the launch site and the trajectory of the rocket is the very final job of the trajectory code.In the developing phase of the project, we simply need the three-dimensional plots to judge if the test case gets into an acceptable orbit or not; however, the three-dimensional plots are the final “products” from the trajectory group, so they should be realistic and somewhat artistic as well at the end of the project since we have the final presentation in public.

Mapping toolbox of MATLAB makes it possible to produce a globe.In addition, there always exists the Sun, so we also implement the orientation of the Sun in the plots. In order to produce three-dimensional plots with those features, the following MATLAB commands are convenient: sphere, surf, plot3, rotate, view, colormap and light.

By usinglight command, we are able to specify the position of the light source. The distance between the Sun and the Earth is 1 [AU], which is approximately 150 million [km], and the axial tilt of the Earth is 23.44°, or 0.409 radians.

(A.6.2.5.2.1)

where is the vertical position from X-Y plane, is the distance between the Sun, and the Earth and is the axial tilt of the Earth.

In our Earth-centered inertial Cartesian coordinating system, the Sun is 61 million [km] above the plane of the equator of the Earth in summer of the northern hemisphere of the Earth, calculating by the equation (A.6.2.5.2.1), which approximates the vertical length by the arc length; we assume it is a good approximation for a technically non-serious calculation.

Once we are able to specify the position of the Sun by light command, it is not difficult to make it possible to specify the date and time, as the Earth rotates once every day and revolves around the Sun once every year; we simply use trigonometric functions.The only part that we need to pay attentionto is to make sure to specify the time in UTC, which corresponds to the longitude of 0°; that is why the time in Fig. A.6.2.5.2.1 is specified in UTC.

Fig. A.6.2.5.2.1: Orbit Trajectory at June Solstice, viewing from the latitude of 20° N.

(Junichi Kanehara)

Fig. A.6.2.5.2.2: Orbit Trajectory at December Solstice, viewing from the latitude of 20° S.

(Junichi Kanehara)

In the version 3.2 of AAE450_Trajectory_Plots, we are able to specify the time in our local time since it automatically computes UTC based on our local time and time zone. We need an attentive observation of the date and time for this additional feature; for example, 10p.m. on December 31st, or the day 365, in New York (Eastern Standard Time) is 3a.m. on January 1st, or the day 1, of UTC.

This way, we are able to produce realistic three-dimensional plots that feature the globes, the orientation of the Sun that is capable of representing the specified date and time, the location of the launch site, and the trajectory of the rocket as shown in Fig. A.6.2.5.2.1 and Fig. A.6.2.5.2.2.

Author: Junichi Kanehara