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Let's say that Peter has a set of superb telescopes and he's watching the sky every night, for 3-4 months. He may not know it just yet, but there is an asteroid heading towards Earth, on a crash course. He is too lazy to plot an orbit, because he doesn't believe he could die like that.

Over the months the Earth travels quite a lot around the Sun, resulting in various amounts of parallax, depending on the distance. The location of the telescopes are also a factor, and so is the (rotation of the) axis of Earth.

To simplify this and to get rid of Earth radius parallax, Peter is a researcher on the South Pole with those mighty telescopes.
Over the course of months, what kind of a trajectory would a doomsday asteroid make? As it gets bigger/brighter (which is also relative, because the asteroid itself can rotate)/closer, how would its position change, compared to the stars in the background?
Would the change slow down or accelerate? Would it be a small piece of a spiral (I don't mean complete spirals, just the fact that it wouldn't be a parabola or a circle)? What do you think Peter would see if he animated its movement during the course of months on a virtual sky sphere?

The National Geographic article Asteroid Called ‘Spooky’ Will Buzz Earth on Halloween has an animation of the trajectory of asteroid 2015 TB145 (nicknamed Spooky because it looks like a skull).

16 minutes apart visual timelapse:
https://upload.wikimedia.org/wikipedia/commons/e/ec/2015_TB145_discovery.gif

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There is no simple answer to this question other than: you have to do the math.

Most often you would see the asteroid brightening more rapidly and moving across the sky faster as the days pass, but not necessarily. It could arrive directly along our orbital path and just brighten but show no motion in the sky. It could start out lit up fully (full phase) and then arrive with only partial lighting from the sun, so it does not necessarily brighten.

You generally would not know the phase, unless you calculate it, because you will not be able to resolve it until it gets really close.

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In sailing, you might see another boat on a heading that will cross yours (i.e., it is coming in from port or starboard). You obviously want to know if there's any risk of collision. The trick is that you take a bearing on the boat and note it. Then you wait a while (long enough for you to close the range but short enough that you're still quite far apart) and take its bearing again.

  • If it has decreased, the boat will pass safely in front of you
  • If it has increased, you will pass it front of it and it will be safely astern
  • If it is the same, you are on a collision course! So make an immediate turn...

This works for objects moving at a constant speed across a flat surface. Not sure if it also applies to accelerating objects on curved paths.

If it does turn out to be true, the object would get steadily brighter night by night, while holding the same position against the fixed stars.

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