Comet Drift Calculation

When we observe a comet with a telescope, we often see a fairly dim and fuzzy object. Especially, when the comet is close to its perihelion, we also observe that the comet moves rapidly through the field of view. With high magnifications, one can virtually see the comets movement in front of the stars in real time. This is quite impressive for the visual observer because it gives you a feeling of the real time events in our solar system. However, when it comes to photographical observations we are facing some difficulties. When the comets nucleus and coma is relatively dim, it can be difficult to track the comet directly and we have to track to the stars. But when we do this, the comet will constantly move through the field of view during integration. In this little essay I want to share my personal procedure on how to calculate the maximum integration time, so that the comet is still sharp while tracking to the stars.

As described above, the constant motion of the comet limits the exposure time. If we ignore the max. useful exposure time, we will most likely end up with an image like this:

The picture on the left side is a simulation created with Cartes du Ciel. It shows perfectly round stars, but the comet was drifting through the field of view during exposure.

This picture is useless, because the exposure time was too long.

Our aim should be that we end up with a single frame like the following one:

The picture on the left shows a sharp comet and the stars are sharp too. The exposure time was stopped exactly at the moment, when the comets drift would become visible for the CCD camera.

When we take multiple images like the one above and later stack those images, we will get a final image like this one:

The picture on the left side shows a perfectly sharp comet. It is a stack of multiple single images, centered to the comets nucleus. The stars are trails, but the comet is perfectly sharp.

We have gained the best imaging-capacity.

First of all, we need to know the current velocity of the comet relative to the stars. In order to get this information, we must take a look at the ephemerides. I personally choose Calsky.com for this task. Calsky provides the drift for Right Ascension (dRA) and Declination (dDEC) in ''/h (angular second per hour).

For example:

We want to image comet C/2013 R1 Lovejoy on Nov. 24, 2013. According to Calsky, the drift for Right Ascension dRA = 525.1 "/h and dDEC = 17.0 "/h. Based on this information, we have to calculate the total drift.

The total drift is the resulting vector of dRA and dDEC, see picture on the left side.

It can be calculated easily with the following formula:

If we take the values from above and use the given formula, the total drift of Lovejoy = 525.3751 "/h = 8.7563 "/min.

**= 0.1459 "/s**

In this case, there is no big difference between dRA = 525.1 "/h and tD = 525.3751 "/h resulting from the formula. But it is generally a good idea to calculate the total drift, because it can be a huge difference for other objects.

Now we have to find out the weakest point in terms of resolution. We need to know, what is the limiting factor. Is it our equipment or is it the expected seeing distortion?

We start with equipment. Here we have to compare the angular resolution of one pixel of the CCD to the theoretical resolution of the telescope and the expected seeing distortion (the weakest result counts):

Angular resolution of the CCD:

First step is to know the edge length of one pixel of our CCD camera. If it is not known, you can easily calculate it from the parameters a) total number of pixels and b) size of the CCD sensor.

As an example: My Canon EOS 450D measures 22.2 mm × 14.8 mm and on this area are 4272 × 2848 pixels. This means that one pixel = 5.197 µm.

Next thing is to calculate the angular resolution of one pixel. This can be easily calculated with the following formula:

α = Angular resolution of one pixel

206265 = Factor to convert radians into arc sec [arc sec]

d _{Pixel} = Edge size of one pixel [µm]

F = Focal length of the imaging system

For my setup, I get the following results: α = (206.265 × 5.197) / 605 = **1.7718"**

This result must be compared to the theoretical resolution of the telescope or the seeing distortion.

The theoretical resolution of the telescope can be calculated with the following formula:

α = Theoretical resolution of the telescope

D = Aperture of the telescope [mm]

For my setup: α = 138 / 102 = **1.3529"**

Lastly, we have to estimate the expected seeing distortion of the observing night. As an example, the average seeing distortion of the Paranal Observatory is 0.66" and for the Large Binocular Telescope, it's 0.58". The average seeing for my location is around 1.1". As a result, I would expect circa **0.80"** at its best.

Now we have to compare those figures:

If the theoretical resolution of the telescope or the expected seeing distortion is better than the angular resolution of one pixel of the CCD, I allow a drift of 2x the size of the angular resolution of one pixel:

The picture on the left illustrates the situation, that the theoretical resolution of the telescope or the expected seeing distortion is better (smaller value) than the angular resolution of one pixel of the CCD. A motion of the comet starts to be detectable, if the comet drifts across 2 pixels and more.

On the other hand: If the angular resolution of the telescope or expected seeing distortion is worse than the angular resolution of one pixel of the CCD, I allow a drift 2x the size of the theoretical resolution of the telescope or the expected seeing distortion (the worst value counts).

The picture on the left side illustrates the situation, that the theoretical resolution of the telescope or the expected seeing distortion is worse than the angular resolution of one pixel of the CCD. In this case, a motion of the comet will become detectable if it moves across a distance of 2x the size of the resolution of the telescope or the expected seeing distortion (worst value counts).

Let's take a look at the values of our example:

Angular resolution of one pixel = 1.7718"

Theoretical resolution of the telescope = 1.3529"

Expected seeing = 0.80"

Both values are smaller than 1.7718", this means that I will allow a drift across 2 pixels for further calculations.

Remark: 1.3529", means less resolution than 0.80" here the bigger value counts.

The final calculation is now a simple one:

t = Max. time span

α = Worst resolution power of the system (see 3.2)

tD = Total drift ["/s]

In my example t = (2 × 1.7718) / 0.1459 = **24.29 s**

This is a very low time frame!

I have created a small spreadsheet that does the calculation for you. You can download the file from here by clicking the link below. Before you do so, please be aware that you use the spreadsheet on your own risk. I don't take any responsibility for false calculation, damage, or loss of data this spreadsheet could cause.

`SHA-1 Checksum: d204d9b77d8bd6cd5f52fd1f2fd7b0d4089d3dc4`

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