The tumble period of the UNHA-3 upper stage from the recent North Korean launch is slowly changing

click image to enlarge

The image above, taken in the evening of 5 March 2016,  is a 10-second exposure showing several flashes of the tumbling UNHA-3/Kwangmyŏngsŏng rb 2016-009B, the upper stage from North Korea's recent Kwangmyŏngsŏng-4 launch. It was taken during a very favourable 67-degree elevation pass, using my Canon EOS 60D and a SamYang 1.4/85mm lens (set at F2.0). The sky had cleared just in time for this pass (a last wisp of clouds is still visible in the image).

The flashes had a brightness of about mag. +3.5 and were visible by the naked eye. The resulting brightness variation curve is this one:

click diagram to enlarge

I have briefly mentioned the tumbling behaviour of this rocket stage in an earlier post. Over the past week I have been following this rocket when weather allowed, obtaining observations in the evenings of Feb 28, Feb 29, March  3 and March 5. This now allows a first look at how the tumble rate is (very) slowly changing.

The theory behind tumbling rocket stages and why their tumble rate changes over time, is briefly discussed here on the satobs.org site. After the payload and the upper stage separate, usually by means of exploding bolts, the upper stage gets a momentum from this separation.

Over time, the resulting tumble is influenced by interaction of the rocket stage body with the earth's magnetic field. Spent upper stages are basically hollow metal tubes, and the Earth's magnetic field causes induction in it, leading to the tube getting an electric charge. Basically, the rocket stage becomes a dynamo. The Earth's magnetic field then further interacts with this electrically charged rocket stage, by means of the Lorentz force exerting a magnetic torque on the rocket stage's spinning motion. It is the latter effect which by "tugging" on the tumbling stage, changes its momentum, with a changing tumble period as a result. The resulting change is one towards a slower tumble rate, and eventually the stage might stop tumbling altogether.

I earlier established a peak-to-peak period of 2.39 seconds for 2009-009B from observations on Feb 28 and 29. Analysis of the new data obtained on March 3 and 5 show that the period is changing: I get 2.43 seconds for March 3 and 2.45 seconds for March 5.

I re-analyzed the Feb 28 and 29 data as well, this time using a fit to a running 5-point average on the raw data, which leads to somewhat better refined peaks. I also found that the initial autofit made by PAST is actually not the best fit, based on the r-square values of the fit. re-analysis leads to a 0.01 second revision to 2.38 seconds of the Feb 28 period, while the Feb 29 period stays at 2.39 seconds as initially established.

So the sequence is:

peak-to-peak periods
-----------------------------------
Date        TLE date    Period(sec)
-----------------------------------
Feb 28.81   16059.81    2.38 ± 0.01
Feb 29.79   16060.79    2.39 ± 0.01
Mar 03.79   16063.79    2.43 ± 0.01
Mar 05.82   16065.82    2.45 ± 0.01
-----------------------------------

(NB: the listed uncertainty is an estimate)

Even though the differences are very small, there appears to be an increasing trend to the periodicity, at the rate of about 0.01 second per day. As the difference is systematic, it is probably real and not just scatter due to measuring uncertainty (time will tell if this indeed holds).

[edit 7 March 2016 19:55]
One caveat: the synodic effect. As the viewing angle changes over the pass, this has some influence on the determined period. For fast tumblers this effect is small, but as we are talking about differences in the order of a few 0.01 seconds, the synodic effect comes into play.
The observations of Feb 28, 29 and March 3 were all made some 30 degrees beyond culmination, so the synodic effect should be about the same. The March 5 observation was done at culmination (I actually have a second image post-culmination as well but have not analysed it yet)
[end of edit]

Below are the brightness curves on which these values are based (click diagrams to enlarge):

click diagrams to enlarge

Appendix: on the construction of these brightness curves

I got a number of questions on how, and with what software, I produce these brightness curves. I will briefly explain below.

(a) calibrate exposure duration
What is first necessary, is that the real duration of the exposure is carefully calibrated. A "10-second" exposure set on your camera is not exactly 10.000 seconds: with my Canon EOS 60D for example, it is 10.05 seconds in reality (this deviation seems to increase exponentially with exposure time: a "15-second" exposure for example is in reality closer to 16 seconds!).

(b) measure pixel values with IRIS
The pixel brightness over the trail on the photograph is measured using the free astrophoto software IRIS.  Load the image, and chose "slice" from the menu option "view". Put the cursor at the start of the trail, and draw a line over the trail to the end of the trail. A window pops up with a diagram. You can save the data behind this diagram as a .txt table.
NB: be aware that Iris always measures from left to right (no matter how you draw the line), so if the satellite moved from right to left, you will later have to invert the obtained data series.

(c) Excel manipulation
The resulting .txt data file is read into excel. There, if necessary I first invert the series (see remark above). The result is a table with a column with pixel brightness values,  to which I ad an increasing pixel count. I then ad another column, representing the time for each pixel measurement. The value of the first cell is the start time of the image in seconds (I usually take the number of seconds after a whole minute, e.g. if the image started at 19:43:32.25 UT the value in this cell is "32.25". If I have a total number of pixels of say 430 (with 430 corresponding pixel brightness values), and an exposure time of 10.05 seconds, then I type this in the cell below it: "=[cell above it]+(exposure/number of pixels - 1)". In our example: "[cell above it] +(10.05/429)".
Then drag this down to the end of the column: the last value now should correspond to the end time of the exposure (in our example, it should be "42.30", i.e. 32.25 + 10.05).

If the raw data graph shows a lot of scatter, it can be useful to apply a running average to the data.

(Note: this approach assumes that the angular motion of the tumbling satellite or rocket stage was fixed over the exposure time in question. In reality, this is not the case. But for short time spans of a few seconds, this can usually be ignored, certainly if the image was taken near culmination of the object. It does introduce some deviation in the result though. Compensating for this makes the exercise a hell of a lot more complicated).

(d) read into PAST and analyse
I then copy the columns with the times and pixel brightness values, and paste them into PAST v.3 (very neat and free statistical software developed by paleontologists. I like it because it is versatile and able to create publication quality vector-format diagrams - the latter ability is something often lacking in such packages).
Press "shift" and select the two columns. Next, under "model" chose "sum-of-sinusoids". Next, a pop-up screen with a diagram appears.
Select "points" under "graph style". I leave "Phase" on "free". You then check the checkbox "fit periods" and click the "compute" button. It will fit a period.
However, I have noted that for some odd reason, the fitted period is not always the best fitting period! Check this by unchecking the "fit period" box, and in the box with the period result, varying the value from the initial fit slightly, after which you press the button "compute" again (leave the "fit period" box unchecked). Look at the R^2 values, and by trial and error find the best R^2 value. This is your actual period.
If your graph shows clearly skewed rather than sinusoidal peaks, than there is a second period interacting with the main period (for example, complex spin motion over two axis, or weaker secondary peaks present). You can try to model this by chosing "2" under "partials".

If you want a nice publishable diagram, press "graph settings" after you are done and adjust the diagram to your liking. Save it as .svg if you want to edit it further in for example Illustrator (as I do), otherwise use one of the other image formats available.