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If one sees a meteor, is there any way to get even a rough approximation of its height, entry angle, size, or other characteristic without triangulation from another position?

If it appeared as a point source and got uniformly brighter, you'd know to take a step aside. And if it appeared on one horizon, traveled overhead, and disappeared over the other, you'd be able to say "Well, that was a shallow angle of attack." But short of those scenarios (which are, probably for the best, rare), is there anything?

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If you see it staying in one spot you can infer that it is moving directly along your line of sight, as you say. But in the more likely event that you see it move across the sky you can determine that it is moving in a given plane only. Unless you have some independent way of judging its size or distance you can't tell any more. This actually happens all the time when a bug flies in front of someone's camera and they think they've seen an interstellar visitor.

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If it stays in one spot, you will not even know it is a meteor moving "directly along your line of sight" unless it changes in brightess. Otherwise it could be some other celestial body. –  JKL Mar 26 '13 at 2:15
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Not an exact answer to your questions, but somewhat related ...

The passage of a meteor is quite quick so it would be difficult I'd think to determine many of those parameters with any accuracy. Unless of course it is observed from another location at the same time (triangulation as you say).

However, for slower moving objects (or those further away than a meteor), it can be done from a single observation point. The technique is called Diurnal parallax and can be used to calculate the distance to a nearby asteroid using simple trigonometry and a nice little piece of intuition.

We can find the distance to an object by measuring its parallax angle, i.e. the angle it makes compared with more distant background objects. This requires forming a baseline, (the base of our triangle).

Surprisingly we can do this with only one person standing in one position. We just have to assume that there is a virtual observer at the centre of the Earth watching the object at the same time. Between them and us is our baseline. This baseline changes as the Earth rotates, it becomes zero when the object is directly overhead (at transit), and is at its maximum 6 hours before and after (see image). The parallax angle therefore forms a sine wave over a 12 hour period.

By monitoring the target at its transit point and for a period of time before/after that point, we can determine its movement against background stars and therefore the parallax angle at each point in time, forming part of a sine wave depending on how long we observe it for (a few hours is usually enough). We do this a number of times over consecutive days, we can remove the targets own motion across the sky from this sine wave.

Together, with some mathematical magic, we can find the sine wave which best fits our observations and therefore determine the distance to the object reasonably accurately.

It gets a bit involved and requires some regression analysis but it can be achieved. The paper which describes the technique is written by Alvarez & Buccheim, 2012 and is available online.

How the rotation of the Earth creates our baseline to measure the parallax of an object.

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