Attempt to measure (1036) Ganymed rotation axis
The following message was posted on AUDEL Yahoo group (14 June 2011) by J.Caron, in French:
« The asteroid 1036 Ganymed looks like an interesting target for the coming months :it will
pass close to Earth, at 0.360 AU in next October.It will also always stay at high declinations
(from +10 to +65 deg) with a low magnitude (8 to 11). Besides, MPC is encouraging
astrometry on it.
According to JPL Horizons, the rotation period is 10.31 hours which can be reasonnably
measured while being not too fast. In the first part of its orbit (JuneJuly) the motion is from 1.8
to 1.3 arcsec per minute in azimuth 30deg. Then in the second part (end of Sept, October,
beginning Nov) the motion is from 2 to 3 arcsec per minute, in the opposite direction (azimuth
170 deg) !!
When we measure a rotation period with photometry, I think that the measurement also
contains a contribution from the asteroid motion over the sky. The information that we are
looking for is the inertial rotation period, while we are measuring the rotation with respect to
the line of sight from Earth to asteroid. When we neglect the difference, we neglect the rotation
of this line of sight in the inertial frame (which is exactly the asteroid motion over the sky).For
1036 Ganymed, if I am not wrong, the combination of a slow rotation (T=10.31h gives 0.6094
rad/h) and a large change of motion over the sky (+/2 arcsec per minute gives +/ 0.0006
rad/h) gives a relative importance of 10^3.
If all this is correct, and if the accuracies over the rotation periods derived by Raoul Behrend
are realist (we often get better than 10^5) we could try to measure the term linked to the
asteroid motion. Then, taking the difference between the periods measured in JuneJuly and
October, we could determinate whether the rotation is prograde or retrograde.What do you
think ?
If this is confirmed, it would be nice to involve a few people to do it (to be sure to have the right
coverage in time due to weather, and to improve the measurements). »
The mathematics behind the method are described further
in this document.
A few persones expressed their interest. The contributors are, in order of importance:
 Maurice Audejean (mpc B92)
 Jerome Caron (mpc C70)
 Francois Kugel (mpc A77)
 Emmanuel Conseil
After a few months the lightcurve of Ganymed was accurately measured. We could see a clear drift of the rotation period with time.
Unfortunately it was not possible to obtain the rotation axis of Ganymed. Our results were suggesting a rotation period of 20.6h, but we received a confirmation from Marina Brozovic (NASAJPL) who was observing Ganymed from Arecibo and Goldstone that Ganymed period was about 10 hours.
We suspect the reason for the above proposed model being not right is that it does not include Ganymed's phase. The fraction of the Ganymed disc which is illuminated changes with time and this also influences the apparent rotation period.
Results for the rotation period
Preliminary determination of rotation period has been done by Raoul Behrend with
Courbrot software, see
here.
Measurement in 
JuneJuly 2011 
August 2011 
September 2011 
from / to 
21 June / 13 July 
09 Aug / 01 Sept 
08 Sept / 01 Oct 
nb of nights 
13 
7 
10 
nb of Fourier harmonics 
6 
6 
6 
period (days) 
0.428976 
0.429330 
0.429756 
3sigma uncertainty (days) 
0.000025 
0.000035 
0.000024 
Assumptions:
Results obtained with a Python equivalent of
Courbrot software:
 simple script (~150 lines of code) thanks to the routine scipy.optimize.curve_fit which completely manages the least square fitting
 fit with a Fourier series at order 6. Fitting parameters are: period + 12 Fourier coefficients (cos and sin) + magnitude offsets to shift each measurement individually
 copypaste of MPC ephemeris in a text file to get the distance asteroid / Earth. The propagating time of light is subtracted from the measuring times (small impact)
 scipy.optimize.curve_fit also gives the covariance matrix, the period uncertainty is thus also known (sigma=sqrt(variance))
The uncertainty on the period directly depends on the uncertainty of the magnitude measurements.
The magnitude error is estimated as the quadratic sum of two terms: (1) the rms noise of the obtained magnitude curve,
(2) a value of 0.02 that represents all other errors
(atmospheric effects, color of reference stars, interfering star when measuring flux, etc). The resulting period uncertainty is then multiplied by 3 to get a 3sigma value.
JuneJuly 2011


Measurements (magnitude vs Julian date)

Lightcurve (magnitude vs rotation phase)

August 2011


Measurements (magnitude vs Julian date)

Lightcurve (magnitude vs rotation phase)

September 2011


Measurements (magnitude vs Julian date)

Lightcurve (magnitude vs rotation phase)

Dephasing (added 05 sept 2011)
If we keep the fit calculated with the JuneJuly measurements, and put the AugustSeptember data on it we obtain the figure below. The shift is evident
despite the slight change of shape of the lightcurve: as the period gets longer, the same magnitudes occur a bit later than initially expected.