What precisely is downrange distance - how is it defined mathematically?

Downrange is the horizontal distance traveled by a spacecraft, or the spacecraft's horizontal distance from the launch site.

Spacecraft don't travel horizontally. I don't even know how the word "horizontal" and the word "spacecraft" can exist in the same sentence. Maybe the word "projected" would be helpful here?

This is simply the distance across the ground from the launch site.

This is the accepted and highly upvoted answer. If I had to write an equation from this explanation, or interpret what 500km downrange means, I'd be hard pressed. It's a great answer to get the general idea, but a bit wanting from a mathematical or orbital-mechanical perspective.

Is there an official or generally accepted, precise definition for how one would calculate downrange distance? I can imagine if I have an orbital plane, then downrange and altitude might be referenced to a sphere or an ellipsoid (Earth surface model) and thereby could be could have been used to define a position fairly precisely. Does such a definition exist?

To better illustrate why the question is not trivial, let's abstract the mathematics out of history temporarily. Imagine you would like to, or have been asked to calculate a down range distance. What is the first question you might ask yourself:

"If I have ECI coordinates of a launch site and a spacecraft, how would I calculate the downrange distance correctly? What model should I use for the Earth's surface? Or should I project along a sphere with the same altitude as the launch site?"

I'm looking for an authoritative answer, not what it probably means or could or might mean, how to "think about it as...", or "it doesn't matter" — it has certainly been relevant historically. Thanks!

EDIT: To elaborate on the previous sentence defining the scope of the question, a look at NASA's 254 page Apollo 11 Press Kit will show a dozen numerical downrange values with single digit nautical mile precision, (and one with decimal precision). To get a large downrange value to 1 nautical mile precision, one needs to choose a specific model for the shape of the Earth, at least sphere or ellipse.

• @RoryAlsop I'll encourage you to read my question again, and carefully. The answers to your questions are there, except "why do you want to" be cause that's irrelevant. It was done regularly, and done with precision. That's a fact. I am just asking for the math. Why not give it a few days and wait to see what answers may be posted by others.
– uhoh
Commented Apr 29, 2017 at 7:39
• @RoryAlsop I took some time and care to write the question, to make it clear I was after something specific. I did not ask the question "Is downrange distance really meaningful?" I asked "What precisely is downrange distance - how is it defined mathematically?" Those are both good questions, but I chose to ask the latter. I suppose that former is still available. If you look you'll see I ask a mix of both soft and hard questions. This one is tightly constrained on purpose.
– uhoh
Commented Apr 29, 2017 at 8:02
• @RoryAlsop Here's a good example of what seems like a common sense slam-dunk is actually worth asking and waiting for good answers: In “spacecraft talk” is nadir just a fancy word for “down”?.
– uhoh
Commented Apr 29, 2017 at 8:10
• well - I do see it as rude. I'm away. Commented Apr 29, 2017 at 8:19
• @uhoh: Compare with neighboring data and actual results. "Apollo 11 will enter the Earth's atmosphere (400,000 feet) at 195 hours and five minutes after launch at 36,194 feet per second. Command module touchdown will be 1285 nautical miles downrange from entry at 10.6 degrees north latitude by 172.4 west longitude at 195 hours, 19 minutes after Earth launch 12:46 p.m." - reentry speed down to 1fps, location down to 0.1 degree and time down to 1 minute, after over a week. Actual splashdown was 13°19′N 169°9′W. I wouldn't trust the other numbers more.
– SF.
Commented May 1, 2017 at 17:19

The definition would be straightforward: length of arc over Earth surface at sea level (altitude zero sphere, not actual ground), between point of launch, and point directly below ship's nadir (intersection of line connecting craft and Earth center, with altitude zero sphere.)

It doesn't need to be defined even that precisely (use average ground level instead of zero, or use direct straight line distance instead of arc) because it's not used in actual craft guidance - it's a piece of data helpful for ground observation crew, "range safety", recovery crew, reporters/photographers, air traffic control, and so on, and for these purposes accuracy of ~100 meters is perfectly sufficient, and at orbital altitudes it loses about all significance for space launches.

It's pretty important for rocket artillery and short range missile launches though. But that's not a space exploration topic.

If you need to write an equation, being given the coordinates, you normalize the coordinates to the surface (remove the altitude component) and then use whichever distance metric over the surface (as used in aviation, naval navigation, land navigation, artillery balistics) to determine the distance. Which exactly you use doesn't really matter, because, as I mentioned, there is no requirement for this to be precise.

• Yup - this was what Russell said too :-) Commented Apr 29, 2017 at 23:19
• Thanks, but I'm after what was done, not suggestions what I should do. Are you sure there was never any standard way? During the Apollo era just for example, each NASA engineer who happened across a situation where a number was needed or requested just made up their own personal method on the spot? No convention? I'm trying to establish if there was a standard way, if an accepted definition existed.
– uhoh
Commented Apr 30, 2017 at 1:16
• @uhoh: I'm pretty sure there were multiple standards, each per application. Downrange for flight control purposes (airspace clearance) derived directly from aviation standard definition of distance, downrange for short suborbital flights (sounding rockets) following artillery standards, downrange for sea recovery following nautical distance definition. I'm also pretty sure press releases are given what's available on hand from other applications without much thought which "downrange" is given.
– SF.
Commented May 1, 2017 at 16:59
• @SF. I am not sure what to do with one person's "pretty sure" without any kind of reference or citation or way to verify. At this point it's an opinion, and although a popular one, not a good stackexchange answer. From the beginning I've worked hard to make it clear I'm looking for more than an opinion. " I'm looking for an authoritative answer, not what it probably means or could or might mean, how to 'think about it as...', or 'it doesn't matter' ...". Can you find two references with two definitions to demonstrate there are multiple definitions?
– uhoh
Commented May 1, 2017 at 18:29
• @uhoh: Instead of "downrange distance", you should be looking up land distance definitions in general. These have good maths behind them, while downrange distance is just a trivial instance of these by fixing one end to the launchpad, and being so trivial, nobody ever bothers dwelling on it. It's like if you were asking "what precisely is the launch site air humidity?" - It's air humidity, as defined by general meteorology, and measured at the launch site. You won't find separate studies or definitions of that. There's a lot on air humidity in general, but this one is a trivial variant.
– SF.
Commented May 17, 2017 at 6:46

I may have overlooked this in the preceding replies but so far I think there is a conceptual oversight. There are actually two forms of "downrange distance" that are usually considered, whether for space launchers or ballistic missiles. Most of the respondents appear focused on the geodesic ("great circle") distance between the launch point and the current point in space. That is fine for some purposes, including the evening news. This distance may be thought of as purely geometric. However, for some purposes one must consider the ground track, the path coverage over Earth's surface as it rotates. The ground track is not a geodesic and can only be computed numerically since it involves the summation of the incremental distances covered at each instance of time. For short time spans, the ground track and geodesic distances are nearly indistuishable. For large spans of the true anomaly, the ground track will deviate significantly to one side or the other of the geodesic path. This is especially noticeable for, say, sounding rockets having long hang times above the surface. These can sometimes produce crazy looking curves, depending on launch point and apogee. So, I would first clear up what is desired. If a ground track is desired, then numerical integration (summation) is the only solution, which doesn't have to be complex to get a satisfactory practical result.

• Thanks for the insight! You can probably see there was some pushback and I still don't exactly understand why. The usage example that I later added (at the end of the question) is a press kit rather than an engineering document, and so it is hard to know exactly what was meant there. I'll see if I can find a more suitable example.
– uhoh
Commented Sep 27, 2018 at 21:02