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So the scenario is this, lets say an object such as a huge star or galaxy was to just 'appear' at a distance of 10 light years from Earth, and it is heading directly towards Earth at a constant speed of 1 light year every 10 years which works out to be a speed of 29,979.2458km/s (about 1/10 speed of light).

Also lets say that in this example the earth and the observer are stationary, so the object is simply moving directly towards the observer in empty space. Lets also say that we are able to physically observe the object using a telescope.

So the light from the object will take 10 years to reach us, at which point the object would have moved closer to us by a light year. So the moment the light reaches us, we'd see the object at a distance of 10ly but its true position would be 9ly away. Now the light from its current position will only take 9 years to reach us, so as we observe the object, we would perceive it to move from 10ly to 9ly in only 9 years.

This means that the object would appear to be moving faster than it actually is. If you continue to plot this on a graph it creates two lines that intercept at the point of collision. So the difference in distance between the true position of the object and the observers perceived position of the object decreases with time until the objects collide.

To simplify further, if the object travels at 29,979.2458km/s, it will take 100 years for it to travel 10ly and collide with earth. However as the observer misses the first 10 years, it will still appear at a distance of 10ly but it will only take 90 years to travel the 10ly distance until it collides with earth. This gives us an average speed of 33,310.273km/s.

My question then is would the object appear to be accelerating towards us even though its true speed is constant? And how can you tell if what we see when we observe objects in space is accurate given that the nature of light can cause illusions like this.

I had a look into the doppler shift equations for light, the math is a little beyond my current capability however i did notice that time dilation gets taken into account, is this essentially what that part of the equation is in reference to?