Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

A database was created in the final topic of Chapter 8 for the state of Orbiter at MECO, that is, its coordinates, altitude, velocity, bearings and mass.

The first topic of this chapter showed a chart of the altitude, for 8,000 seconds along the sub-orbit, that Orbiter would follow if no thrust were employed after MECO.

The apogee of that orbit reached about 182 km at about 2,370 seconds after MECO. It could be seen from the chart that the sub-orbit decayed after that time.

The first and second topics of this chapter developed the application of three computers to the design of the Course Commands that would guide Orbiter from MECO to Rendezvous. The first topic showed those commands in the context of the Input Table of our spreadsheet and are shown again next:

The table shows, in row 3, the 30 second delay after MECO in which Orbiter and the External Tank are separated.

Next there is a 70.32 second burn that is intended to take Orbiter into the altitude region of the ISS orbit. Such a burn could take place any time before the sub-orbit intersected the Earth. Why then?

Common sense suggests that waiting until Orbiter was loosing altitude would waste fuel and that the latest the first burn should happen is at apogee of the sub-orbit. See a comparison of a burn 30 seconds after MECO with one 2,370 seconds after MECO next:

Delaying the first burn delays reaching ISS altitude and requires a longer burn of the OMS engines, more fuel.

We had timed the launch from Kennedy to save fuel and arrive in orbit quite close to the ISS. On the way to our objective we had employed Adaptive Conditional Feedback to refine the Course Commands as they were executed so as to reach our objective quite closely. It is those refined commands that are used in our spreadsheet Input Table to derive various aspects of the mission.

The table shows the coasting interval of 2,451.68 seconds that takes Orbiter to near the apogee of the first burn. The final burn of 110.65 seconds takes Orbiter into the ISS Orbit quite close to the ISS.

**Launch may not occur Exactly When Scheduled**

Note that the total time taken to reach the ISS from MECO can be adjusted within fuel constraints by choices such as: choosing the time after MECO at which to initiate the first burn; using some radial burn time versus tangential burn time for the OMS engines in gaining altitude; choosing intermediate orbit altitudes a little higher or a little lower than that of the ISS; trading burn time with burn force when using the RCS.

We do not model the final navigation process that employs the manual controls of the RCS for docking.

See the, expanded, altitude and velocity results of the final, circularizing, burn next:

The controllables given to the solver in our 3D spreadsheet were the durations and azimuth angles of the two burns and the duration of the coasting period between the two burns. The objectives provided to the solver were apogee and perigee altitudes of the ISS and the inclination of the ISS orbit.

The Homann transfer orbit is an idealized orbit maneuver using two impulses of thrust that move a space vehicle in a vacuum from one circular orbit in a plane to a second circular orbit in the same plane. It consists one half of an elliptic orbit that connects the two circular orbits.

Our illustration of navigation from MECO to the ISS took Orbiter from an elliptic sub orbit to the elliptic ISS Orbit in an atmosphere using two non-impulsive burns.

In a practical situation needed corrections to the burns, their durations and their thrust directions would be made by our postulated Adaptive Conditional Feedback Control System.

Had Orbiter not docked with the ISS, it would circle the Earth in its own slowly decaying orbit. The path taken by an undocked Orbiter is shown on a projection of a selection of Earth's major cities next:

The path taken by the space station should differ slightly as it has a different mass, cross-section and drag coefficient.

The varying kinetic and potential energy, and their total, of Orbiter as it ascends from MECO into orbit is shown next:

It is seen in the left panel that potential energy rises and kinetic energy falls as altitude is gained.

In the right panel it is seen that the sum of potential and kinetic energy of Orbiter falls as it proceeds towards its orbit and the sum is seen to fall most rapidly during the burn intervals.

The total energy dropping during burn intervals might seem counter-intuitive but the reason for the energy drop is that the simulation calculates only the energy of the Orbiter. When Orbiter uses fuel to heighten its orbit, it ejects mass at high speed in the form of rocket exhaust.

Recall that potential energy is proportional to mass as well as height. The potential energy of Orbiter is reduced due to the use of fuel to gain height. The potential energy gained with height is more than offset by the loss in potential energy due to mass reduction.A residual loss in energy between orbital maneuvers is caused by air resistance.

Today's missions to the ISS may not proceed as well as did our mission. It may be necessary for Orbiter to go into a lower orbit so as to have greater velocity to catch up to the ISS. Alternatively a higher orbit might be needed to allow the ISS to catch up with Orbiter.

Suppose that at MECO it is determined that if Orbiter proceeded to ISS orbit it would likely arrive there about 10 seconds behind the ISS and an intermediate lower orbit is chosen.

For an ISS orbit at 302 km and corresponding velocity of about 7728.7 metres per second ISS would be about 77 km ahead of Orbiter.

In a 299 km orbit the velocity is about 7730.5 metres per second. In this orbit Orbiter could catch up and transfer to the ISS orbit in about 12 hours.

For interest, the altitude and velocity for over a 12-hour period in a near circular 299 km orbit is shown next:

Getting it right the first time and not requiring the use of an intermediate orbit would be a good thing for the astronauts.

**
Summing Up Hands-On Math **

Hands-On Math has been about exploring with mathematical models and through exploring gaining a deep understanding of the use of mathematics and of the space that we inhabit.

Physical highlights have been the modeling of the effects of gravity, buoyancy and resistance in our atmosphere, the art of ballooning, projectile behaviour and the behaviour of rockets in space.

Mathematical highlights include the techniques of a Solver to iteratively and closely find unknown values given known or objective values. With solver the absence of neat readily found mathematical solutions has not been an obstacle.

Procedural highlights include the use of Adaptive Conditional Feedback in a control application.

Without that much said about it, at the core of the mathematical procedures has been the use of numerical integration to solve awkward differential equations to get answers that make real world sense.

Great value and satisfaction is obtained when the individual constructs his own model. This is the reason that the spreadsheet was adopted for the modeling. It has been the objective of the writers to provide sufficient information to the reader for him to create the spreadsheet models.

For readers who choose not to construct some of the models, there is second hand experience to be gained by closely following the examples and, in some cases, using the two calculators that accompany Hands-On Math.

The spreadsheet version of one of the more complex calculators is provided as a download.

Calculators are provided with Hands-On Math so that the reader may carry out explorations of his own fancy with Gravity, Ballooning, Atmospheric Resistance and Projectiles without having constructed his own spreadsheets.

The complexity of the model of the three-dimensional rotating Earth and its atmosphere and its various forms ruled against providing user-calculators in that case. Considerable and, in the opinion of the authors, sufficient information about three-dimensional modeling has been provided in the text for a reader to construct his own three-dimensional models.

HOM provides a great many web references for further on-line exploration of the topics.

Last but not least, HOM has provided reference to many of the contributors and their contributions to Mathematics, Science and Engineering that has brought mankind to the verge of space.

The reader may have noted earlier in this topic that less fuel was required to reach ISS altitude from MECO in the case where the first burn occurred sooner rather than later.

Combine this observation with the current practice of having Orbiter and visiting Supply vehicles use their engines to boost the orbit of the ISS.

Would less fuel be required to be raised to orbit, with correspondingly lower mission costs, if Orbiter and the other vehicles just transferred boosting fuel to the ISS so that it could use its own engines for boosting itself on a much more frequent basis and thus experience much less loss of altitude between visits?

**
Finally**

The reader may wish to see the views of the authors regarding considerations that
should apply to the further exploration of space by man. These views can be
accessed via Postscript on the upper row of navigation tabs.

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