Brian Skiff and Joe Pollock have sent me their photometric data for 2011
GP59 that they took from Lowell and with PROMPT on April 11,
respectively. I am going to comment on my analysis of the Brian's data.

Using the method of Pravec et al. (Tumbling asteroids, Icarus 173,
108-131, 2005), I found it to be a tumbler rated PAR = -3 on the scale
we defined in the paper. Two main frequencies present in the data are:
f_1 = 1/(0.122501 +/- 0.000007 h)
f_2 = 1/(0.17096 +/- 0.00005 h)
corresponding to periods 7.350 min and 10.258 min. (The errors are
formal.) The tumbler's lightcurve contains harmonics of the two
frequencies as well as their linear combinations (n_1*f_1 + n_2*f_2),
where n_1, n_2 are integers. The strongest signal is in the 2nd
harmonic of f_1 (causing the prominent brightness variation that you
have easily identified), the next strongest signal is in the 1st
harmonic of f_1 and in the 2nd harmonic of f_2, and then there is
significant signal in several linear combinations of the two
frequencies, e.g., (f_1 - f_2), (2 f_1 + f_2), (2 f_2 - f_1), (2 f_2 +
f_1), (2 f_1 + 2 f_2). For those of you who wish to plot the fit to the
lightcurve themselves, I copy below the coefficients of the Fourier
series. (Note: The fit is in flux units, not in magnitudes.)

The fitted curve and residuals of the Brian's data:
http://www.asu.cas.cz/~asteroid/2011gp59_110411_lowell.png
You can notice that the tumbling rotation causes each maximum having a
different level and each minimum having a different depth.

The two frequencies apparent in the data can be interpreted in terms of
rotation and precession with a dynamical and shape modeling, like we did
in the case of 2008 TC3 (Scheirich et al., Meteoritics & Planetary
Science 45, 1804-1811, 2010). This has to wait until the apparition of
the asteroid is over and reductions of photometric observations are
completed.

In this case, the detection of tumbling was easy; the amplitudes of the
harmonics of the two frequencies were much larger than the photometric
errors of the Brian's observations and the data covered tens of
rotation/tumbling cycles. In most cases, however, it is more demanding
to reveal tumbling uniquely. More typically it occurs that tumbling
amplitudes are burried in observational noise or not many
rotation/tumbling cycles are covered (especially for more slowly
rotating tumblers), hampering a unique resolving of tumbling. Those
biases against tumblers cause that our knowledge of the tumbling
asteroid population is rather limited. To get a better understanding,
we prepare a survey for tumblers among small super-fast rotating
asteroids that is designed properly to suppress the observational
selection effects. The observations will have to be done with
photometric errors not greater than a few 0.01 mag so that we can detect
also a low-amplitude tumbling, and a full-night coverage will be needed
so that we can resolve tumbling also in slower rotating asteroids (with
periods of several tens minutes to a couple hours). With such
low-biased data, we shall be able to constrain theories of rotational
excitation of small asteroids.

Cheers,

Petr Pravec

Fourier coefficients of the Lowell 2011 April 11 data:
C_0 = 1 (i.e., the flux was normalized)
n1 n2 Ampl.*1000 C-coef S-coef |1/(n1/P1+n2/P2)|, hours
1 0 130.7 -0.1281 0.0258 0.122501
2 0 620.1 -0.6162 0.0697 0.061250
3 0 16.7 -0.0079 -0.0147 0.040834
4 0 16.8 -0.0119 0.0118 0.030625
-4 1 0.2 0.0002 0.0000 0.037309
-3 1 20.2 0.0042 0.0198 0.053647
-2 1 45.5 -0.0029 -0.0454 0.095446
-1 1 63.9 0.0070 -0.0635 0.432175
0 1 31.6 -0.0016 0.0316 0.170960
1 1 27.5 0.0087 -0.0261 0.071365
2 1 71.7 0.0221 0.0683 0.045094
3 1 14.9 -0.0087 -0.0121 0.032961
4 1 10.0 -0.0073 -0.0068 0.025973
-4 2 5.7 -0.0011 -0.0056 0.047723
-3 2 9.3 0.0078 -0.0051 0.078180
-2 2 48.4 0.0421 0.0238 0.216088
-1 2 79.3 0.0192 0.0770 0.282850
0 2 130.4 -0.1026 -0.0805 0.085480
1 2 76.5 -0.0628 -0.0437 0.050348
2 2 80.1 0.0590 0.0542 0.035682
3 2 9.9 0.0099 -0.0011 0.027633
4 2 13.3 -0.0102 -0.0085 0.022547
-4 3 21.3 -0.0192 -0.0092 0.066204
-3 3 1.2 0.0011 0.0004 0.144058
-2 3 36.0 0.0259 -0.0249 0.818621
-1 3 32.5 0.0324 0.0023 0.106556
0 3 45.2 0.0449 0.0055 0.056987
1 3 34.3 -0.0338 -0.0055 0.038894
2 3 32.1 -0.0312 0.0076 0.029521
3 3 4.6 0.0042 -0.0019 0.023788
4 3 13.1 -0.0082 -0.0103 0.019920
-4 4 8.0 -0.0075 -0.0028 0.108044
-3 4 33.4 -0.0046 -0.0331 0.915491
-2 4 1.3 0.0006 0.0012 0.141425
-1 4 10.1 0.0086 0.0054 0.065642
0 4 9.5 -0.0039 -0.0086 0.042740
1 4 8.5 -0.0047 -0.0071 0.031685
2 4 18.9 0.0155 0.0110 0.025174
3 4 20.9 -0.0207 0.0027 0.020883
4 4 4.5 -0.0038 -0.0025 0.017841

> Properly Petr Pravec's, not mine. Nick James' video clip is
> indeed a good representation of what the object was doing
> photometrically, and what we had to deal with in the reductions.
> I sent my data to Pravec, who is the wizard of the tumbling asteroids.
> As Brian W intimated, Petr finds two very clear non-commensurate
> periods of about 7.3 and 10.3 minutes. Phasing with the
> double-period significantly reduces the scatter in the plot posted
> previously (rms scatter now 0.085 mag) --- so my dismissing the
> scatter as just crappy data was overstated. Petr also has similar
> data with better time-resolution from Joe Pollack using the PROMPT
> telescope in Chile.
> Since this is not my analysis, I won't spill the beans
> altogether without Petr's permission, partly since some refinement
> in the analysis may be called for in merging the PROMPT and Lowell
> series.
> I didn't look at the observing circumstances after April 11 UT,
> but additional runs with the different viewing geometry (and the
> asteroid much brighter but moving faster and approaching the Moon...)
> would be useful in defining the shape and pole-orientation.
>
>
> \Brian
>