Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

**No Orbiter trajectory
timeline was found on any NASA website** visited by the writers. A website pseudo timeline
was found with an attribution to NASA. A screen image, shortened, of the
site is shown next:

** **

The site might create the impression that the data was real, but emails to William Harwood elicited the information that it was from a computer model of an Orbiter ascent produced at NASA and that the information was not available to the general public.

Given the absence of factual trajectory data, the writers decided to study this computer-model output in detail, and then from the study decided that it could be employed to obtain realistic conditions for Orbiter at the time of Main Engine Cutoff, MECO, a point in time just prior to the jettison of the Main Fuel Tank.

The computer simulation of Orbiter's trajectory concluded at MECO. Deriving the conditions at that point would the allow the authors to obtain both the trajectory of the fuel tank to its ocean splash-down and the trajectory of Orbiter from MECO to its rendezvous with the International Space Station.

The first step was to get the data into machine-readable form. A practical way of doing that is to use an OCR (Optical Character Reader) Scanner. OCR software is often included with popular scanners.

The image was saved in a file and standard image software was used to display and print a copy.

The printout required suitably large font so that the numerical figures were easily recognized by the OCR (Optical Character Reader) device. Once the image was scanned, its data was saved into a spreadsheet where the desired processing was done. See columns of that spreadsheet next:

Processing the data consisted of removing redundant row entries corresponding to SRB separation and MECO events. Next, the data was converted from standard units of measure (feet, etc) to metric units.

Examining the table of data closely reveals the time column as suspect since some duplicate values exist. We assume this is due to the fact that the simulation used a fractional time between calculations, which was rounded or truncated to the nearest integer. Doing a straight linear interpolation of the time from t = 0 to t = 510s, for each row, provides a new uniformly spaced time column, E.

For each row we calculate the radial distance as the radius of the Earth plus the altitude, Column N. The radial component of velocity, Vr, is obtained from the rate of change of radial distance with respect to time.

Since we are given the total velocity, Vi, we can get the tangent velocity, Vt, from (Vi2 – Vr2)0.5. This is in column P. Having Vt and Vr, we can get the elevation for the velocity vector as atan(Vr / Vt). This is Elev in column Q.

To obtain the azimuth of the velocity at MECO, we employ the following strategy.

Assume the STS has in fact reached an orbit where the inclination is close to that of ISS, 51.6 degrees.

We can obtain computer-modeled values attributed to NASA for the Latitude and Longitude of MECO from the web site where Latitude and Longitude are given within a few seconds of MECO.

Using the parameters that have been determined thus far, we can employ the expanded spreadsheet to get the actual azimuth of the velocity. Using the parameters along with an initial guess of the azimuth of 51.6 degrees and trying a few variations on this value will easily produce an azimuth such that the inclination is ~51.6. We expect an azimuth close to 51.6 since MECO is not far from the launch point.

The time taken from launch to MECO can be seen to be about 510.17 seconds. According to one NASA reference it will require about a further 43 hours to rendezvous with the ISS.

Approximately 30 seconds after MECO, the ET is freed from Orbiter and continues in a separate sub-orbital trajectory that leads to a splashdown in a remote region of either the Indian or Pacific Ocean.

We use our 3-dimensional modeling spreadsheet to calculate its trajectory.

For initial parameters we take values for the ET at MECO that are the same as for the Orbiter and assume they track with each other for a short time until separation. The primary difference between Orbiter and the ET simulations is due to different physical characteristics e.g. mass and shape. Also, the ET does not have any significant ability to propel itself with rocket motors. For the simulation we will assume that the parameters defining the physical aspects of the ET remain constant from MECO onward.

The ET mass is about 27,000 kg without propellant, (mass varies with ET model and there are several models). The mass of the propellant is about 730,000 kg. We will assume that the MECO event occurs as a controlled event before the ET has exhausted all propellant. This means a few percent of the propellant remains in the ET at MECO. For the simulation we will take the amount to be about 1.7% of the initial amount. This yields a total mass for the ET at MECO of about 40,000 kg.

The tank is shaped like a cylinder with a length of 46.88 m and diameter 8.40 m. Equating the ET cross sectional area with an equivalent sphere results in a sphere of radius 11.196 m. A detailed description of the tank reentry explains that the tank is deliberately tumbled for reentry to ensure its destruction into small, harmless pieces, as it passes through the lower atmosphere. To simplify the simulation we ignore destruction and account for the tumbling action by taking the Cd as 2.0, twice the value employed for Orbiter.

The following figure shows the trajectory for the chosen parameters. The simulation shows that the splashdown for the ET is in the Indian Ocean near the equator. For the set of parameters we have used, the splashdown occurs about 2400 seconds after MECO at Latitude 1.828 deg North and Longitude 64.934 East.

As noted earlier, the ET is destroyed before impacting Earth. We can convince ourselves that this is the case by looking at the energy involved in the simulation. At MECO, the ET total energy is about 390,767 kWh whereas at splashdown it is 1,366 kWh. The change in energy is due to air resistance losses that translate mostly into temperature increases to the air and to the ET material.

The ET consists of an enclosing shell as well as internal tanks, pumps, piping, electrical components and so forth. For this study we assume that the ET mass is entirely composed of its shell material. Although the shell is an aluminium-lithium alloy, from the previous link we note that lithium amounts to only a few percent of the alloy metals and therefore, for our study, we treat the ET as composed entirely of aluminium.

We assume, as a guess, that the ET absorbs half the energy loss, the other half going to the atmosphere. Since we assume that the ET is made totally of aluminium, we can estimate its temperature rise as a function of the energy loss. The formula for temperature change with energy is:

Change in energy = change in T * mass * specific heat of aluminium (= 0.9 kJ/kg)

Aluminium melts at 660 deg Celsius. At MECO its temperature has been reduced to a low value due to being in the upper atmosphere as well as carrying liquid hydrogen and oxygen for the rocket. We can assume its temperature is below 0 and the change in temperature is 700-Celsius degrees from that point to its melting point.

Change in energy = 700 * 27000 * 0.9 kJ = 1.7e10 J = 4,700 kWh

After absorbing 4,700 kWh of energy, the aluminium will start to melt with further energy input. With about half of 390,000 kWh available, this very approximate analysis shows there is ample energy to melt the ET even if only a fraction of the energy is transferred into the ET structure as heat.

The following charts show the altitude and energy profile for the ET trajectory, assuming the characteristics of the ET are constant (no disintegration). Note that from MECO to about 1,500 seconds the trajectory follows a sub-orbital path where the energy remains relatively constant. At that point however, the ET encounters the denser, lower atmosphere of Earth and energy is rapidly dissipated due to the resistance of the air. The ET speed slows considerably and near the end the trajectory tends to be in free fall towards the centre of the Earth at the terminal velocity of the ET. This is seen in the altitude graph as the linear portion after 2,000 s.

Gerald Bull used a super cannon to propel a dart shaped Martlet 2 to the standing record altitude of ~ 180 km. He later proposed to use a cannon to fire an object into orbit from Iraq. See The HARP Project and the Martlet.

If the proposed projectile were the record achieving Martlet 2, is there a mountain in Iraq that is sufficiently high that a super cannon placed on the mountaintop could propel it into an Earth orbit?

Newton's Thought Experiment, (ignoring atmospheric resistance), that he used to explain orbits, envisaged a cannon ball shot from a very high mountain that could attain an Earth orbit.

A simulation with the 3-D spreadsheet may shed some light on the practicality of both Gerald Bull's Dream and Newton's Thought Experiment when there is Earth's atmosphere to contend with.

Presume a mountain with Mount Everest's altitude, taken to be 8,848 metres, a much greater altitude than any Iraq mountain.

Choose as the object the Martlet 2 that still holds the 180 km altitude record. **
In the absence of atmosphere** it would require a muzzle velocity of ~7,904 m/s to
achieve a circular orbit at the altitude of 8,848 m. That velocity is much in excess
of the 2,100 m/s that we presumed as the value achieved in creating the 180 km altitude record.

It may be that in the future, considerably greater muzzle velocities will be achieved, perhaps employing an Electromagnetic Pulse to propel the object. See the Kirkcudbright gun that anticipates a muzzle velocity of ~4,500 m/s. This velocity would only succeed in driving a Martlet 2 about 2 degrees of a full 360 degree orbit before its orbit intersected Earth's surface.

Choose an equatorial orbit so that the Z coordinate of the orbit will be zero.

Stop the Earth and its atmosphere from rotating so that the rotation does not add to or subtract from muzzle velocity.

Choose the Martlet 2 parameters as 84 kg of mass, a radius of 0.065 m and a Cd of 0.0127, as were employed when modeling its altitude record achievement.

A tangential muzzle velocity of 12,000 metres/second was required to achieve very slightly over one orbit before impacting Earth's surface. See the X, Y and Phi, (angle around the equator), plots for this case next:

The Earth's surface is included in the X, Y plot for reference. Note that the orbit of the projectile is eccentric and that the time intervals of the Phi plot are an hour apart.

See the effect of increasing muzzle velocity to 12,950 m/s next:

Very few degrees of orbit have gained but the orbit eccentricity has increased greatly. Note that the time markers are now 10 hours apart.

Further experiments show that the effect of increased muzzle velocity is to ever increase the eccentricity with little increase in Phi.

As it is believed that nowhere in space does the density of the atmosphere become identically zero, even an extremely high velocity object will always travel in a near elliptical and slowly decaying orbit although the object may be attracted by or captured by the attraction of other space objects encountered during its travel.

This look at the effect of atmosphere does not detract from the value of Newton's airless Thought Experiment.

Nor does it detract from Bull's Dream. Although there is not much detail about the matter in the records of Bull's work, it is known that he envisaged that an object such as the Martlet 2 would release a smaller object at a given altitude that would behave as rocket powered and achieve a somewhat circular but decaying Earth orbit. We have not seen sufficient detail to model Bull's Dream.

NASA has been providing press kits for the shuttle missions such as, the aforementioned http://www.shuttlepresskit.com/STS-116/Presskit_STS116.pdf, from which the following pair of images has been excerpted.

The authors have located little detail on Orbiter's path from MECO to the ISS prior to the manual proximity operations phase. As a consequence, the authors made the decision to create a more detailed and, hopefully more intellectually satisfying, navigation scenario for that part of the mission.

Chapter 9 treats aspects of Orbiter's trip to the International Space Station.

The Chapter suggests a navigation system employing Adaptive Conditional Feedback Control that makes use of computers for modeling Orbiter's engines and the atmosphere, for the control of the engines and for providing the time varying coordinates of Orbiter and the ISS by means of a GPS-like system.

Some elementary orbital navigation techniques are presented.

Models are created both for a first-pass and for a delayed rendezvous of Orbiter and the ISS.

The Chapter concludes with a summary of the subject matter that Hands-On Math has addressed and raises a question for consideration.

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