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

Question

Wikipedia:

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?

Space Exploration Stackexchange:

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.

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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?"

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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!

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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.

Answer 1

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.

Answer 2

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.