A while ago I wrote a post entitled “How Apollo Got To The Moon”, which featured a few orbit graphics generated in Celestia, like the one above (which shows the orientation of Apollo 11’s departure orbit relative to the most intense region of the Van Allen Radiation Belt). I got a few enquiries about the data I’d used to plot the spacecraft orbits, and this is a long-delayed response to those enquiries.

As a preamble to this post, see my post “Keplerian Orbital Elements”, which describes in detail the various numbers used to define an orbit’s size, shape and orientation—the numbers I needed to plug into Celestia in order to draw the orbit in the illustration above.

This post is going to be about the NASA data sources I used, which don’t provide the orbital elements directly, but instead provide tables of state vectors, specifying the spacecraft’s position and velocity at a series of times. To plot the orbits the Apollo missions followed when they departed from, and returned to, the Earth, I need two specific state vectors. For departure, I want the state vector at Translunar Injection (TLI), which was defined to take place ten seconds after the Saturn S-IVB stage shut down its engines, having launched the Apollo stack on its way to the Moon. For the return orbit, I want the state vector for the Command Module at the atmospheric entry interface, defined as an altitude of 400,000 feet. In my next post on this topic, I’ll describe how to convert these state vectors into orbital elements.

A commonly used Apollo data source is Apollo By The Numbers (2000), a NASA publication compiled by Richard W. Orloff. Its data tables are widely available—they even have their own web pages on the NASA History website. The data I’m currently interested in appear under Translunar Injection and Entry, Splashdown, and Recovery. There’s also a useful page of Earth Orbit Data. But Apollo By The Numbers has the disadvantage of being a secondary source, containing a number of copying errors. So I went back to the primary sources—mission documents prepared immediately after each Apollo flight. These are available on-line as scans of the original typewritten pages, but the data are scattered across multiple sites, and links sometimes turn out to be broken. Even the NASA Technical Reports Server is missing some items, and is so full of oddly indexed material that it’s sometimes difficult to find even the material that is present. So I’ve spent some time recently compiling a collection of specifically Apollo-related documents at the Internet Archive, which I’ll link to as required below.

For the Apollo departure orbits, I took my TLI state vectors from the Postflight Trajectories prepared by Boeing. For atmospheric entry, my main source was the Mission Reports compiled at what’s now the Johnson Space Center.

The Postflight Trajectories include a pair of appendices charting reams of state vectors for the launch, Earth orbit and Translunar Injection phases on an almost second-by-second basis. The most useful tables for the TLI state vector are B-VII and C-VII (data in metric and imperial units, respectively). These have the look of being about as primary as you can get, given that they have the appearance of photo-reduced computer printouts, in contrast to the typewritten document to which they’ve been appended.

The relevant position data are the geocentric distance (GC DIST), longitude (LONG DEG E) and geocentric latitude (GC LAT DEG N)—a distance and two angles, which provide the spacecraft’s position relative to the centre of the rotating Earth. The relevant velocity data are the heading (HEAD DEG), flight-path angle (FLT-PATH DEG) and space-fixed velocity (SF-VEL)—again, two angles and a speed completely specifying the spacecraft’s velocity in three dimensions. The heading is the compass course, in degrees east of true north, along which the spacecraft is travelling; the flight-path angle is the angle above or below the local horizontal (parallel to the surface of the Earth directly below) in which it’s travelling; and the space-fixed velocity is its speed relative to a non-rotating Earth. This last value takes the spacecraft’s velocity relative to the surface of the Earth and augments it by the local rotation velocity of the Earth. That combined velocity determines how far the spacecraft will travel in its long elliptical orbit towards the moon—all the Apollo missions (and space missions in general) launched towards the east, to take advantage of the Earth’s rotation to give the spacecraft an extra boost.

These six numbers, together with the time and a knowledge of the Earth’s mass, are all that’s needed to derive an elliptical orbit that will be valid for the first few hours of Apollo’s departure from Earth.

But first we need to translate the time, given in seconds in the first column of the table above, into a format that’s meaningful in terms of orbital mechanics. The time given is what’s called the Range Time, or Ground Elapsed Time (GET)—the time since launch. Section 1 of the Postflight Trajectory report for Apollo 11 tells us that the mission launched on 16 July 1969, at 08:32:00 Eastern Standard Time. That corresponds to 13:32:00 Greenwich Mean Time. To make that date and time useful for plotting orbits, we need to convert it into a single number, the Julian Day (JD). My link gives you the necessary formulae to do that, but there are plenty of on-line calculators, too. There’s a suitable simple one here, which takes input in the form of the date and Greenwich Mean Time.* It only accepts a whole number of seconds, but all the Apollo missions launched on a whole number of seconds—most, like Apollo 11, on a whole number of minutes. I’ve nevertheless set the output to give the number of Julian Days to eight decimal places, to accommodate the TLI Range Time, which is quoted to a thousandth of a second in the table above.^†

The calculator tells us that launch (Range Zero) occurred on Julian Day 2440419.06388889. To that we need to add the Range Time of 10213.030 seconds (first dividing it by 86400, the number of seconds in a day). That gives us a total of 2440419.18209525 which, in the jargon of orbital mechanics, is the epoch of TLI.

The Postflight Trajectory documents also contain a summary table of the conditions at Translunar Injection, which include a smattering of Keplerian orbital elements—the inclination, descending node and eccentricity—which will provide good cross-checks on my own calculations^‡.

Unfortunately, there’s no similar data source for the state vectors at the time of atmospheric entry—the Postflight Trajectory tables end with the start of the Transposition, Docking and Extraction manoeuvre, on the way to the Moon. The only reports consistently providing relevant data are the Mission Reports^§, which span the time from launch to splashdown, and these seem to be the source for the tabulated data in Apollo By The Numbers. Here’s the table of entry conditions from the Apollo 11 Mission Report:

The “miles” in the table are in fact nautical miles, the equivalent of 400,000 feet. The table provides a time, longitude, velocity, flight-path angle and heading angle, but lacks the geocentric latitude and distance that I need. Instead it gives a geodetic latitude and an altitude.

The time is the Range Time, again, and I can convert it to Julian Days in the same way I did for TLI above. The mission elapsed time of 195h03m05.7s corresponds to 8.127149 days, giving me an entry epoch of 2440427.191038.

To calculate the missing geocentric latitude and distance requires some research into the finer points of Apollo coordinate systems, and a bit of geometry. To find the geocentric latitude of a point, we (figuratively) draw a line from the point to the centre of the Earth, and measure the angle between that line and the plane of the equator. To find the geodetic latitude, we drop a line at right angles to the local horizontal plane, and measure the angle that makes with the plane of the equator. On a perfectly spherical Earth, these two latitudes would be exactly the same, but because the Earth bulges at the equator, they’re slightly different. Here’s a diagram, with the flattening of the Earth greatly exaggerated:

The angle labelled ψ is the geocentric latitude, which is what we need for orbital mechanics; the angle labelled ϕ is the geodetic latitude, which is the latitude generally quoted in atlases and other geographical reference sources. The geocentric distance is the line r, connecting the centre of the Earth to the spacecraft, and the altitude is h, the distance between the surface of the Earth and the spacecraft, measured at right angles to the local horizontal plane.

Of course, the Earth’s surface isn’t a smoothly curving ellipsoid as in the diagram—but for the purpose of calculating geodetic latitude it’s treated as such. Various standard ellipsoids have been used by cartographers over the years, and by consulting the Project Apollo Coordinate System Standards, we can find out that the standard ellipsoid used throughout the Apollo missions was the Fischer “Mercury” Ellipsoid (1960). This ellipsoid was defined as having an equatorial radius (symbolized by a) of 6378166 metres. Its polar radius (b) was defined according to the ratio (a–b)/a, called the flattening (f), which Irene Fischer determined to be 1/298.3. And that’s all I need in order to calculate the geocentric latitude and distance, using the geodetic latitude and altitude.

First, I need to work out the eccentricity (e) of the Fischer ellipsoid^‖:

Then I need the length of R, which runs from the Earth’s axis to its surface in my diagram above, and is called the prime vertical radius.

The distance p, between the Earth’s axis and the spacecraft, measured parallel to the equator, is then:

And z, the distance between the equatorial plane and the spacecraft, measured parallel to the Earth’s axis, is:

Then the geocentric latitude is:

And the geocentric distance is:

Plugging in the Apollo 11 data from the table above, I get a geocentric latitude of -3.17 degrees, and a geocentric distance of 6500.02 kilometres. So now I have my state vector for atmospheric entry in the same format as for Translunar Injection, and I’m ready to calculate the orbital elements. Which I’ll do next time I return to this topic.

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* Actually Universal Time, the successor to Greenwich Mean Time for astronomical time-keeping; but GMT was still the standard during the Apollo missions.
^† This is ludicrously optimistic, of course, given the uncertainty of some other numbers that will feed into the eventual calculation—but I prefer to do my rounding at the end, rather than in the middle.
^‡ Similar summary data appear in the Mission Reports, as well as in the Saturn V Launch Vehicle Flight Evaluation reports from what’s now the Marshall Space Flight Center. The Mission Report data generally seem to be derived from the Postflight Trajectories, though there are some departures from that. The Flight Evaluation data are often very slightly different from the Postflight Trajectory data—I think because of differing emphases on radar tracking and telemetry in the two reports from different organizations with different responsibilities. But none of the other sources provide anything as immediately useful as the state vector tables in the appendices to the Postflight Trajectories. (In the main, Apollo By The Numbers seems to take its data for TLI and Earth Parking Orbit from that source, though not entirely consistently.)
^§ A number of earlier missions have Entry Postflight Analysis reports, which provide a more precise estimate of the entry state vectors than the Mission Reports, but this document was abandoned for Apollo 13 and doesn’t seem to have been reinstated.
^‖ This eccentricity is exactly the same property of the ellipsoid as the eccentricity that specifies the shape of an elliptical orbit, described in my post “Keplerian Orbital Elements”.