Unless you are using a perfectly aligned multi-thousand pound mount, or very short exposures you are often going to end up with some trailing of your stars. This can be from a very minor elliptical appearance of stars to long ugly trails. Whilst it isn’t possible to rid your image of severe trails the following is very effective on even quite badly affected stars.
The technique relies on creating a layer, either a duplicate background or a separate star layer, blending using darken mode and applying the offset filter.
Darken blend mode – in this mode over lying pixels are compared and the darker of the 2 pixels is the one displayed regardless of whether it is in the top or bottom layer.
Offset filter – this filter moves the layer horizontally and/or vertically by the specified amount.
As the stars in the layer above are offset in relation to the stars below dark pixels from the background become overlaid with the bright pixels below. The darken mode ensures that these bright pixels (which are star trails) are eliminated.
Create a duplicate layer and set the blend mode to darken. You will not see any change in the image since the pixels are all perfectly aligned.
Although most star adjustments are best carried out by creating a separate star layer this can cause problems using this technique since the offset results in mismatched backgrounds.
Choose the filter menu/other/offset. You will see that the offset has boxes for vertical and horizontal adjustment. It is easier to enter numbers into the box manually than it is to use the sliders. For the horizontal adjustment positive numbers shift the image to the right and negative to the left. In the vertical box positive move it down and negative up.
Move the image in the right direction – this sometimes takes a bit of trial and error. Don’t use an offset of more than 2 pixels since this will start to damage your image. If you need a larger adjustment accept the change then create a further duplicate layer and make further offsets.
Make sure the radio button for undefined areas is set to transparent (this determines what is done with waste pixels left over from the offset – best to just get rid of them).
You will probably find that as you come close to achieving a round star that a one pixel adjustment is too coarse. If this is the case go for an adjustment that slightly over does rather than one that underdoes. Close the offset window and go to edit on the menu bar. Select “fade offset”
The fade control is a very useful photoshop tool. It is available immediately after a change to the image and is used to reduce the change at a sub pixel accuracy. Leave the mode as normal and move the slider to the left away from 100%. This will provide fine adjustment to the offset to allow you to achieve an accurate result.
You should now have much rounder stars than you started with. Unfortunately the offset process can have some impact on the overall image, it will probably appear a little darker and maybe a little blurred. The stars will be dimmer and if you have made a large adjustment you may have lost some small stars. Don’t panic! Return to levels and make sure the black point is correctly adjusted. You will probably be able to brighten things up with a tweak to curves. The stars can certainly be livened up with techniques described in the next guide.
If you have the time and patience you can select the main target, invert the selection and create a new layer via copy. Duplicate the new layer and make offset adjustments without messing up your target. You can then manually select stars within the target, expand the selection by a few pixels and create a layer of your selection, duplicate it and apply another offset. How much effort you put in all depends on how keen you are to get the perfect image.

This is a single sub of a horsehead image taken over 5 minutes, unguided with a ZS66 on a windy night. It is an extreme example and would normally be rejected but I’ve used it so that you can see what can be achieved.
I shifted the duplicate layer 2 pixels to the left and up.
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This wasn't enough so I created a copy of the offset layer. Since I was getting close I zoomed in to get a more accurate idea of what was going on. I applied a further 2 pixels offset both up and leftwards. Shown below :-

This was about as close as I could manage but I applied a small fade to 94%

So here is the result, compare it with the initial image


Looking at the treated image you can see that although the stars are a better shape the image has lost some of it's vibrancy, not surprising given the liberties I have taken with it. This can be corrected with some further work in levels and curves/