Jupiter
Determining the rotation speed with the help of photographs

On February 14th, 2003, there was beautiful weather in Stuttgart. No danger that clouds would move in and hide the view of the night sky. Since Jupiter had its opposition on February 2nd, 2003, I expected very good observing conditions of this giant gas planet. To add to the good fortune, the Great Red Spot (GRS) was predicted to transit at 11:50pm UT (0:50 local time MEZ).

Already early in the evening I set up my 155mm Astro-Physics StarFire EDFS refractor (around 6 inch aperture), to allow the scope to reach the thermal equilibrium. The first view of Jupiter through the scope showed a very steady image and confirmed that I had picked a night with very good seeing conditions. I immediately decided to photograph the upcoming transit of the GRS and to create a small amination – provided I could produce enough high quality single images. Therefore, I mounted my „Philips ToUCam 740k“ to the refractor. To increase the focal length, I used the well proven “Fluorit Flatfield Converter” from Baader-Planetarium. This extended the effective focal length to 4650mm with a focal ratio of 1:30. The employed WebCam has a pixel size of 5.4mm, i.e. a „sampling rate“ of 0.24“ per pixel. This was enough to illustrate even the finest details that could be attained with this optic under the prevailing circumstances.

Starting at 9:30pm UT until 2:00am UT I took two video files every 10 minutes, each with a length of 20 seconds. The image rate was 25 frames per second so that each single AVI video file contained 500 individual frames. The resulting animation, a time-lapse animation of the 4 hours and 30 minutes transit of the GRS, can be viewed [here]. (Note: file size is about 500 megabyte!).

I am very satisfied with the image quality because one doesn’t always succeed in imaging the distinctly darker core region of the GRS with a 6 inch aperture scope. Therefore, I decided to try to determine the rotation duration of Jupiter on the basis of these images.

I decided to use a graphical and geometrical method of resolution.

The starting point are two sharp images of the planet where the GRS is clearly visible. These images must be taken some time apart and the exact recording time must be known. I chose one image from 11:00pm UT (12:00pm MEZ) and one from 0:50am UT (1:50am MEZ).

Using the first image from 11:00pm UT (12:00pm MEZ) as an example, I would like to demonstrate which auxiliary lines are necessary:


Image 1: The finished Jupiter image from 11:00pm UT (12:00pm MEZ)

If the images are aligned in such a way that the equator is located exactly horizontally and the rotation axis exactly vertically, you can draw the location of these axes into the image quite precisely, by counting the pixels in the image (width and height).


Image 2: Image of Jupiter with geometrical determined equator- and rotation axes

In the next step you draw a line from the point of intersection of the two axes through the center of the Great Red Spot.


Image 3: Line from the center of the planetary disk through the GRS

Now you add a vertical line which runs exactly through the center of the Great Red Spot.


Image 4: Vertical Line through the GRS

Now we have reached a critical point: A further diagonal line is needed, which begins in the middle of the planetary disk and crosses exactly that point, where the vertical line from the last step is cutting the edge of the Jupiter disk. This is not easy because the edge of the planet disk is not always clearly identifiable. In particular on darker images it is often unsharp and blurry. Therefore, it is better to use images with sufficient exposure time and/or a brighter processing that show a well defined edge of the disk. In this respect please keep in mind that Jupiter pictures that are captured 2 to 3 weeks before or after the opposition exhibit a phase defect on Jupiter’s East or West side. This prohibits precise measurements on this side of the planet. Fortunately, I didn´t have to worry about this, because in my case the opposition was only 12 days ago. After this step you receive the following image:


Image 5: Diagonal line (light blue) through the point of intersection, where the vertical line from the last step is cutting the edge of the disk.

All these steps should now be applied to the second image.


Image 6: Image of Jupiter from 0:50am UT (1:50am MEZ) including all auxiliary lines.

Now the two images are merged. In this new image you can neglect planet details since the following calculations are based on the lines in the image only. The next step is to determine the angle between the two blue diagonal lines, highlighted in the following image:


Image 7: After merging the two images you have to determine the highlighted angle

This can be done with the help of an image processing software, which can directly measure angels. Or with the help of the tangent trigonometric function, after you have divided the angle to be measured into two triangles with one right angle each. In the following image these triangles are highlighted in red and green respectively.


Image 8: Dividing the angle to be measured, into two right-angled triangles (red and green).

The angles are calculated as follows:

a = arctan(B/A) and b = arctan(B'/A)

The advantage of this method is, that you can measure the length of the lines A, B and B´ precisely, because they are exact vertical and horizontal lines that you can measure in pixel.

To increase the accuracy of the measurement, I doubled the size of the image before the actual pixel count. In the present example I received the following values:

A = 481 pixel, B = 278 pixel, B’ = 349 pixel

Using the above-mentioned formula the values of the angles are:

a = 30°, b = 36°

Therefore, the missing angle amounts to a + b = 66°

In order to determine the rotation time, a rule of three must now be applied.

In one hour and 50 minutes (=110 minutes; this is the time period between the two images) the GRS is rotating 66°. How many minutes does it need for one full rotation (360°)?

66° = 110 minutes
360° = 600 minutes

The calculated rotation duration for the GRS is 600 minutes, which equals exactly 10 hours.

The GRS is located in the System II of the Jupiter atmosphere. According to scientific literature the value of the rotation speed of the System II lies at 9h55m40.6s

Therefore, the deviation of the measured value from the literature value is only 0.726%!

This is not bad at all for one single measurement with possible inaccuracies, when drawing the auxiliary lines and determining the angles. Even higher accuracy can be attained by repeating the measurement and averaging the results.
 
Stefan Seip, Nov 23, 2003
 
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