hartley 2 - nucleus rotation and coma profiles

8/7/2019 Hartley 2 - nucleus rotation and coma profiles

http://slidepdf.com/reader/full/hartley-2-nucleus-rotation-and-coma-profiles 1/10

Astronomy & Astrophysics manuscript no. Report˙Tom˙Daan c ESO 2011

January 17, 2011

Comet 103P/Hartley 2: nucleus rotation and coma profiles

T. Hendrix1

and D. Camps1

Instituut voor Sterrenkunde, K.U.Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

Preprint online version: January 17, 2011

ABSTRACT

Aims. To determine the profile in the coma of 103P/ Hartley 2 and derive constraints on the outgassing. To probe the rotation periodof the cometary nucleus.Methods. 103P/ Hartley 2 was observed for 6 consecutive nights in October 2010 (October 12-17) with the Mercator telescope at LaPalma, Spain. CCD photometry in Cousins I and Geneva V, B and U was performed with the Merope CCD camera. 103P/ Hartley2 was close to perihelion passage when observed and near its closest approach to Earth (0.121 AU on October 21). Small aperturephotometry is used to filter the nucleus signal from the flux of the light reflected in the coma.Results. A rotation period of P = 16.6 ± 0.7 is derived from our measurements in the IC band, which is in good agreement with

other publications, however a period of P = 25.3h cannot be excluded. From a comparison between I band observations and theorydeviations from simple behavior of the coma surface brightness profile are found. A brightness gradient m1 = -1.14 ± 0.08 for theinner coma and m2 = -1.45 ± 0.07 for the outer coma is found. The terminal outflow velocity of the grains is determined to bevgr = 191 ± 25 m/ s. We also note a possible detection of jetslike structures in the IC band.

Key words. Comets: nucleus - Comets: rotation- Comets: coma - Comets: individual: 103P / Hartley 2

1. Introduction

In the field of solar system astronomy the understanding of the origin and nature of comets is of particular interest. Thisbecause they are tracers of our past, formed in the protosolarnebula at 10-35AU. During the migration of the outer gas giants

to their present location mean motion resonances occurred. Thiscaused these objects to be catapulted outwards, forming what isknown as the Kuiper belt. Some of these objects may eventuallylose their stable position and start to make a comeback intothe inner solar system, reborn as a comet. In the meanwhiletheir chemical composition has not changed since its formation.The knowledge of their physical properties is essential in thispicture, especially the nucleus is interesting. Cometary nucleiare of limited size, typically 500m to 40km, which is why theyare irregular of shape. They are chemically complex bodiesconsisting of ices and frozen grains. These objects are amongthe darkest in the solar system, having a very low albedoand thus reflecting just a small fraction of in falling sunlight.But even though their nucleus is small, the ion tail and coma

can extend over very large distances near perihelion passage(lion ≈ 1AU , lcoma ≈ 106km) making them both the largest andthe smallest celestial bodies in our solar system.The object of this paper, 103P/ Hartley 2, is a small (Rn ≈ 0.6km)periodic comet with a period of 6.47 years. Because its periodis shorter than 200 years, it is called a short period comet. Ithas been forced into the present orbit with a aphelion distanceof 5.87 AU by perturbations from Jupiter and has becomea Jupiter family comet. 103P/ Hartley was observed with theMercator telescope at La Palma, Spain, for 6 consecutive nights,

Send off print requests to: [email protected] Based on observations made with the Mercator Telescope, operated

on the island of La Palma by the Flemish Community, at the Spanish

Observatorio del Roque de los Muchachos of the Instituto de Astroficade Canarias.

Table 1. Basic properties of Comet 103P/ Hartley 2

Comet 103P/ Hartley 2

Discovery March 15 1986 by Hartley, M.Closest Earth passage ∆ = 0.121 AU on October 20, 2010Perihelion passage rh = 1.059 AU on October 28, 2010

Inclination i = 13.62◦

Orbital period 6.47 yearsRadius nucleus Rnucleus ∼ 0.6 km

starting from the night of October 12 until the night of October17. It was close to perihelion passage at the time, which took place on October 28. Its closest approach to Earth was onOctober 20 when ∆ = 0.121 AU (the distance Earth-cometis typically denoted as ∆). This close encounter provides anexcellent chance to get a close look at the internal structureof the coma and the surroundings of the nucleus. This means

that 103P/ Hartley 2 was easily detectable when observed, butalso highly active with a pronounced coma. Table 1 lists somegeneral characteristic properties of 103P/ Hartley 2.

This work covers two parts of cometary physics, first the in-ner coma of 103P/ Hartley 2 is studied. Studying brightness pro-files in the coma is useful because these give a clue about thestructure of the coma when compared with theoretical coma pro-files and the terminal velocity of grains ejected from the nucleuscan be constrained in this way (Jewitt 1991). The second partof this work studied the very smallest part of the comet, beingthe nucleus with a largest diameter of slightly more than 1 km(Lisse et al. 2009) this entity covers less than 1 pixel on the CCDof Mercator’s Cassegrain camera, MEROPE. Even though its

limited dimensions it is possible to study brightness variationscaused by the rotation of the nucleus by aperture photometry



2 T. Hendrix and D. Camps: Comet 103P/ Hartley 2: nucleus rotation and coma profiles

with small apertures around the nucleus. The fact that it is possi-ble to measure the rotational period of the nucleus is due to theirirregular shape. During the rotation the projected surface of thenucleus on our line of sight changes continuously, changing theamount of light reflected in our direction. It is easy to show thata limited aperture is necessary to increase the amplitude of thebrightness variation (Licandro et al. 2000). The brightness mea-sured in an aperture around the cometary optocenter is given by

BT (ρ) = BN + BC (ρ) (1)

where BC (ρ) is the brightness contribution of the coma within theaperture of radius ρ and BN is the brightness contribution of thenucleus. The latter is independent of the aperture radius as longas that radius is larger than the radius of the nucleus, which isalways the case as mentioned earlier. In this case, the amplitudeof the light curve is

∆m(ρ) = −2.5logBN 2 + BC (ρ)

BN 1 + BC (ρ). (2)

In this equation BN 2 is the maximal brightness of the nucleus

and BN 1 the minimal brightness. Also note that it is assumed inthis equation that the brightness contribution of the coma is con-stant over the entire time span of the observations. This is an as-sumption which also favors a small aperture around the nucleusbecause a smaller aperture minimizes the eff ect of possible tem-poral changes in activity which cause brightness variations in thecoma. Another eff ect that has an impact on the measured magni-tude in the aperature is the seeing eff ect (Licandro et al. 2000).The variations in the seeing during the night and over diff erentnights causes the flux to be smeared out over a larger surfacewhen the seeing is high and be more concentrated when the see-ing is low. This eff ect is most prominent when small aperturesare used, as required for a maximal amplitude eff ect, and thus a

correction on the images is necessary. This correction implies areduction of the seeing of all images to the seeing value of theimage with the worst seeing. The complete process is explainedlater in this paper.

2. Observations

The observations were done with the 1.2m Mercator telescopeon La Palma, Spain, starting from the night of October 12 untilthe night of October 17. The complete scientific dataset availableafter the observing run consists of 46 images which accounts forapproximately 3 hours of observing time. The Cousins I (IC)and Geneva V, B and U (VG, BG and UG) photometric filters

available on the Merope camera (Davignon et al. 2004) wereused. The Merope camera has a plate scale of 0.19” / pixel andthe frame-transfer Eddington CCD (Østensen 2010) has an im-age format of 2048 × 3074 pixels, resulting in a field of viewof 6.39’ × 9.73’. Our main interest was in IC band observa-tions because these are most optimal to detect brightness vari-ations of the nucleus and to determine the brightness profile inthe cometary coma (Reyniers et al. 2009). Furthermore, the lowextinction in the IC band filter enables us to do the observationswithout observing standard stars (see 3.4). The observing strat-egy was thus focused on observations in the IC band resultingin a dataset consisting of 29 IC band, 3 VG band, 6 BG bandand 8 UG band images. The observations in the bands other thanIC were done to have a interruption between IC observations

during the available blocks of observation time, as contentiousmonitoring in a small amount of time would not have improved

Table 2. Overview of the observations. ∆ and rh are the geocen-tric and heliocentric distances of the comet, respectively. Bothare given in AU.

Date n(U) n(B) n(V) n(I) ∆ rh

10-12-2010 3 1 2 5 0.132 1.07910-13-2010 0 1 1 2 0.129 1.07610-14-2010 0 2 0 3 0.127 1.074

10-15-2010 0 0 0 3 0.125 1.07210-16-2010 5 0 0 13 0.123 1.07010-17-2010 0 1 0 3 0.122 1.068Total 8 6 3 29

the result. Not every IC band image was useful because some-times the pointing of the telescope went wrong or a very brightstar crossed the comet in the field of view, which finally gaveus 24 usable IC images, 13 of which were made in the night of October 16. A log with an overview of all the observations isgiven in table 2. However, this obtained number of observationsin the IC band is considerably smaller then our aim of 100 obser-

vations. Even though obtaining 100 observations in the limitedtimeframe that was available might have been quiet ambitious, aconsiderable amount of observing time was lost due to weatherconditions.

3. Science Extraction

The images were reduced with a bias correction and the correc-tion with a master flatfield. For four out of six nights skyflatswere available, and for these nights the skyflats were used. Nocorrection for dark current needs to be made as the Merope CCDis cooled to 160K. Because the coma is much bigger than theregion captured on the CCD (6.39’ × 9.73’), the background

can not directly be read from the CCD values. However, a back-ground correction was done by calculating the median value of aregion in a corner of the field (see figure 1). The shape of theregion was chosen in a way that excludes the interference of smeared out background stars. The position of the region waschosen to minimize the coma contribution (i.e. in the directionopposite to that of the coma).To determine the periodicity of the nucleus rotation the magni-tude of the nucleus needs to be calculated first. The operationswe performed to do so are discussed in the following paragraphs.In the last paragraph of this section the methods we used to ex-tract a coma brightness profile from the images is described.

3.1. The Seeing Effect

As mentioned before, the seeing plays an important role in theinterpretation of the observations, as bad seeing smears out theflux from the nucleus. Because we want to work with aperturessmaller than the seeing disk to probe the variations in the nu-cleus, ignoring this eff ect might lead to a detection of the seeingin the periodicity if the seeing was variable during the observa-tions. The necessity of this correction can also be seen from asimple argument: if we represent the signal by a Gaussian withσ = 0.8 and integrate with an “aperture” window [-0.4,0.4] wefind a flux F ≈ 0.39. If the signal is broadened to σ = 2, the fluxdrops to F ≈ 0.16.Because no standard stars were observed together with the

comet, we deduced the value of the seeing from the field starsin the comet observations. To find the seeing we used a Markov



T. Hendrix and D. Camps: Comet 103P/ Hartley 2: nucleus rotation and coma profiles 3

Fig. 1. Example of an observation. Field stars are seen as stripes

due to diff erential tracking. The region in the lower left corneris used for the background correction. The bar in the lower righthas a length of one arcminute. The central condensation seemsto be saturated

Fig. 2. Seeing values obtained by fitting Gaussians perpendicularto the trails of field stars.

Chain Monte Carlo program that makes a Gaussian fit of theprofile of the field stars. The method seems to be stable as starsat diff erent positions in the frame and of diff erent brightnessgive similar values for the seeing. The seeing we adopted is themean of the obtained seeing in three diff erent stars on the frame.The resulting seeing values can be found in figure 2. This figureshows us that the seeing is in the range of 0.81” to 2”.All observations were then convolved with Gaussian kernels tocompensate for the seeing eff ect.

3.2. Flux Extraction

With the observations all brought to the same level, we can startwith the aperture photometry of the nucleus. Obviously the nu-

cleus in not resolved: the comet has a diameter of 1.14 ± 0.16km (Lisse et al. 2009), while one pixel of the Merope-EddingtonCCD (0.19”) would have a projected size of 17 km at the dis-tance of the closest approach during our observations. At first weapproximated the location of the nucleus with the brightest pixel.However, this approach gave some unexpected results (e.g. thecentral brightness sometimes increased after we convolved witha Gaussian kernel). We then adopted the weighted mean of the

pixel positions near the brightest pixel as our central value. Eventhough the eff ect of this correction is small (the center is nevermore than a fraction of a pixel from the brightest pixels center),this approach seemed to work much better as it solved the appar-ent conflicts encountered when using the brightest pixel.Most programs that extract fluxes work by dividing pixels insmaller entities (subpixels), and then calculate the flux by sum-ming the fluxes in all the subpixels with centers within the ra-dius of the aperture. To increase the accuracy, especially as wewanted to use very small aperture radii, we wrote a procedurethat calculates the exact analytic fraction of the area of a pixelthat falls within the aperture radius. The formulae used to cal-culate the intersection of a circle and a pixel can be found in

(Degroote 2008).

3.3. Distance Correction

During the five days of the observations, the comet moved rel-ative to the Earth and the sun. This has two distance eff ects onour observations for which we need to correct, namely the he-liocentric and the geocentric distance, r h and ∆ respectively. Themagnitude of a cometary nucleus can be approximated in the fol-lowing way:

m = m0 + 2.5log

∆2r nh

Φ(α)

. (3)

Here m0 represents the comets magnitude at r = ∆ = 1 AU, α =0, with α the sun-comet-Earth angle. Φ(α) is the phase functionthat gives the fraction of light scattered at phase angle α to thatat α = 0. It is often approximated by the empirical relation

Φ(α) = 10−αβ/2.5, (4)

β is know as the linear phase coefficient, and needs to be deter-mined empirically. Most comets have values close to β = 0.035mag/ deg (Lamy et al. 2004), so we adopted this value in our cor-rection.It can be seen from equation 3 that the dependence on the geo-centric distance follows the usual inverse square law: F ∝ ∆−2.The nucleus usually follows the same relation for the heliocen-

tric distance, giving n = 2. If one would want to correct forthe coma, a diff erent value of n needs to be adopted, because thecomet increases activity as it nears the sun, generating more dustto reflect the light. For the coma, a typical value would be n = 4,however values for in the range of -1 to 11 have been reported(Brandt & Chapman 2004).

3.4. Conversion to Magnitudes

To convert the found flux values F to magnitudes the relation

M = ZP + 2.5 log t − 2.5 log F − k λA (5)

is used, with A the airmass, ZP the zeropoint of the instrument

and k λ the extinction coefficient at wavelength λ and t the ex-posure time. This means we still need the values of two more




parameters to calculate the magnitude, namely ZP and k λ.However, as can be seen from this equation, the zeropoint onlyprovides an off set value. Since it does not change during theobservations, its true value will not influence our results, andthe value ZP I = 22.89 is adopted. The actual value of k λdoes matter somewhat more as it is more variable. Becausewe did not observe standard stars we do not have values of the extinction. Luckily, the extinction in the I band is rather

low, which also makes the variations smaller. Furthermore, theCarlsberg Meridian Telescope (CMT) (Evans 2001; Garcıa-Gilet al. 2010) at the Roque de los Muchachos provides nightlymeasurements of the extinction in the r’ band (e ff ective wave-length λ = 625nm). The CMT takes 40 observations of standardframes during the night, with 30-40 calibration stars on eachframe. For four out of six of our nights (October 13-16) extinc-tion values are available. During these nights the extinction val-ues were the same (taking the errors into account), having a valueof k r = 0.091. One of the other two nights was reported to beentirely unphotometric, and as the two observations in this nightshowed a clear shift these observations were no longer consid-ered when fitting the periodicity. The remaining night (October

12) was only unphotometric for a limited amount of time and forthese observations we assume that the extinction had the samevalue as the one reported during the other nights. Using valuesfrom the La Palma technical note no. 31 (King, 1985) we con-verted the extinction to the value in the I band, find a value of k I = 0.032.

3.5. Extraction of Brightness Profiles

The extended field of view of the Merope ccd camera allows usto study more than just the very inner regions around the nu-cleus. The more exterior regions of the coma also fall onto theccd up to just beyond 25000 km away from the nucleus. This

means that coma brightness profiles can be determined from thedata.The first method we used to extract a brightness profile from animage used the position of the nucleus and a given direction andthen scans all pixelvalues in a 1 pixel narrow row between thosepoints. This method exhibits one major problem, when a back-ground star crosses the pixel row a lump in the profile appears. Itwas tried to construct diff erent profiles in diff erent directions andaverage these, but in this manner the results are not that good.Although more or less in agreement with the next method whenthe profile was averaged over 8 directions.The second method we used to extract a coma profile scans thewhole image up to a certain pixel distance. This profile is then

binned in logarithmic steps to get a nice smooth curve. We binin logarithmic steps because the profiles are expected to have atypical shape, which is most easy visualized when plotted log-arithmic (see section 4.1.1). The process of binning has the ad-vantage that it removes the lumps due to stars or cosmics, whilekeeping the typical decline of the coma profile. It has to be notedthat in this manner we construct a coma profile averaged over alldirections away from the nucleus. This isn’t a problem, the direc-tional dependence of the profiles is limited. This was checked bydividing an image in 4 diff erent regions of equal size (top right,top left, bottom left and bottom right) and analyzing the binnedcoma profile of each region. The diff erences are fairly limitedand the major problem here is that the number of datapoints ineach region is to limited to completely remove all stellar lumps.

The standard method for our analysis is the second one, whichis explained in all details in section 4.1.2.

4. Results

4.1. Coma profiles

For our study of the coma of 103P/ Hartley 2 we used the methoddescribed in section 3.5. All I band images were used and alsosome of the B band images. This was done to investigate thediff erences between diff erent filters. The U band images wererejected from the sample because they did not reach a sufficientsignal to noise ratio to perform a qualitative analysis. Our studyof the V band images was very concise because their numberis very limited and the wavelength range of the V band is moreor less situated in between the B and I band. Meaning that thediff erence between B and I will be larger than between V and I.The V band images are not mentioned any more in this paper.

4.1.1. Theoretical profile

During the development of solar system astronomy, which goesback to ancient times but really flourishes in the past 100 years,comets have been of interest for research and diff erent modelshave been constructed to describe the structure of the coma in

terms of its surface brightness profile. This because the surfacebrightness profile provides a useful tool from which certain as-pects of the 3D structure of the coma can be deduced. A simple,but relevant model for the surface brightness profile was derivedby Jewitt (1991). This model is valid for a spherically symmetriccoma in a steady state case. Steady state meaning that the out-flow rate of particles is constant, no changes in activity occur.It is easy to show that in this case the surface brightness profilefollows the simple law

Bλ(ρ) =K

ρ, (6)

where K is a constant and ρ is the projected distance from thenucleus on the sky plane of the nucleus. Note here also that theprofile is expected to be wavelength dependent.To check the agreement between this model and observations itis most easy to introduce the parameter m, defined as

m =d log B(ρ)

d ρ. (7)

This means that an agreement with the model is reached whenm = −1, making this the most logic value for the steepness of the profile and it is expected to be valid for the inner regions of cometary comae under the standard assumptions, this region iscalled the inner coma. Deviations of this regime occur at large

ρ because the eff ect of the solar radiation pressure is neglectedthus far. At large distances where this eff ect becomes importantm ≈ −1.5 as shown by Jewitt & Meech (1987), this region isthe outer coma. The distance where the transition from outer toinner coma occurs is of the order of

X R ≈v2

gr r 2h

2βgsun(r h), (8)

as shown by Jewitt & Meech (1987). In this formula vgr the ter-minal velocity of the ejected grains, r h the heliocentric distanceof the comet, β is the ratio of the acceleration of a grain due toradiation pressure to the local solar gravity and gsun is the solargravity at r h. This means we can estimate the outflow velocity of

the grains if we determine the transition point from the observa-tional profiles.




Another useful relation is the empirical Bobrovmikoff -Delsemme relation (Delsemme 1982)

vgr ∼ 580 r −0.5h [m/s] (9)

which gives an estimate of the expected grain velocity. It has tobe noted that deviations from the simple m = −1/− 1.5 behavioris possible, not all coma brightness profiles behave exactly the

same and diff erences can occur because of changes in cometaryactivity, strong deviations from spherical symmetry or other ef-fects.

4.1.2. Coma profiles of 103P/Hartley 2

The coma brightness profiles of all IC images were extracted bythe second method described in section 3.5. This method expe-rienced some minor problems at the borders of all images. Thiswas because of three diff erent reasons. First the code bins thevalues over concentric circles but because the ccd has a rectan-gular shape the circles far away from the nucleus do not fall com-pletely on the ccd reducing the number of values to be binned.This eff ect will be rather small. Secondly the images show some

vignetting in the corners, which causes the profile to fall off rapidly but this eff ect is not physical, it is due to the optics. Thethird and most important eff ect is that the background has to besubtracted to retrieve the true brightness profile. But the comaspans the whole image, which makes it hard to find the truebackground value. As an approximation the median value over asmall bar in the left bottom of the image is used (see also figure1). In the outer part of the profile the intensity can be very closeto or even smaller than this background value causing the bright-ness profile to fall down very rapidly in loglog scale. Because of all these reasons the very outer part of all profiles was omittedand only the points up to 15000 km from the nucleus were used.The final set of extracted coma surface brightness profiles were

fitted to the expected brightness distributions in both the innerand outer coma. This was done with a Markov Chain MonteCarlo (MCMC) procedure which fits a power law brightness pro-file,

B(ρ) = 10C r m (10)

to the data. When plotted in loglog scale m is the slope of theline and C is the point of intersection with the y axis.

4.1.3. Comparison between theory and observations

All IC band profiles were analyzed and fitted to theoretical pro-files at the manner described in previous section. A point of in-

flection appears to be present in all the profiles. This point of in-flection, X R, is then defined as the point of intersection betweenthe fit to the data in both regions, this point is calculated fromthe m and C value in the inner and outer coma. From formula 8it is than possible to calculate the grain velocity of all profiles.The resulting values for X R, |m1|, |m2| and vgr are presented inFigure 4. Here |m1| is the absolute value of the slope of the in-ner coma and |m2| of the outer coma. This results in a slope of m1 = −1.14 ± 0.08 for the inner coma profile and a slope of m2 = −1.45 ± 0.07. This means that we find a significant devia-tion from the m = -1 value in the inner coma, while the outercoma satisfies a m = −1.5 behavior. The slope that is foundfor the inner coma is steeper than a 1 / ρ dependence and thusthe brightness falls of more rapidly. An example of such a fit is

given in the bottom half of figure 3. This is the profile that bestagrees with the average m1, m2 and X R values. The value found

for the point of inflection is X R = 3600 ± 1000 km which givesa terminal outflow velocity of the grains of vgr = 191 ± 25 m/ s.These values are rather low, in the literature the inflection pointlies typically at about 1−3×104km. And the grain velocity is lessthan half of what is expected from the Bobrovnikoff -Delsemmerelation (Equation 9).Figure 3 also shows a profile in the B band. This has a less steepfall of m1 = −0.90, and thus deviating from the simple m = −1

trend of the model. This value is consistent for all B band im-ages. The deviation from an m = −1 value in the B band canprobably be contributed to strong emission lines in this wave-length range. There is typically more emission in the wavelengthrange of the B band, eg. CN emission is detected in other comets(Reyniers et al. 2009), which makes that both continuum andemission features may contribute to the brightness profile. Thepresence of emission breaks the assumption of conservation of mass of a certain species, making the used model invalid. Thishypothesis to explain the disagreement between theory and ob-servations was not tested for the specific case of 103P / Hartley2 because a spectrum in the propper wavelength range was notavailable1

The significant deviation from m = −1 in the I band can be dueto diff erent reasons. It could be possible that some strong emis-sion features are present in the spectrum of 103P / Hartley 2 inthe wavelength range of the I filter, making the model to break down because of the same reason as before. Another possibilityis that strong changes in activity occur making the steady stateassumption invalid. Another, last possibility is that the field of view of the images at the distance of 103P/ Hartley 2 is to lim-ited, that there is a true other point of inflection from inner toouter coma beyond the edges of our images. This is possible be-cause the coma is much more extended than what we see on theimages, where the field of view is fairly limited because of theclose distance to earth of 103P/ Hartley 2, but it is very unlikelybecause we do find a point of inflection in all profiles in a con-

sistent way and the outer coma found in this manner satisfies theexpected m ≈ -1.5 behavior.Note that the very inner part of the coma (ρ ≤ 250 km)deviatesstrongly from the rest of the profile. Here, other eff ects becomeimportant: the acceleration zone close to the nucleus has someinfluence at this small scale, but also the seeing distorts a clearbrightness profile. This region was never considered for the fit-ting procedure.

4.2. Rotation Frequency

Using the technique explained in section 3 we were able tomake a time series of the inner regions of the observations. In

figure 5 the result is show for an aperture of 1 pixel. In figure 6we see the same figure, but for aperture radii of 5 and 15 pixels.The pattern of the points for an aperture of one pixel is almostthe same as the one for an aperture of 5 pixels. However, thetime series with an aperture radius of 15 pixels shows a di ff erentpattern. This diff erence can most easily be seen in the first andthe last night. This eff ect can be expected from equation 2. Todetermine the rotation frequency of the nucleus we used fourdistinct techniques: a Markov Chain Monte Carlo (MCMC) fit,a Lomb-Scargle periodogram analysis (Lomb 1976),(Scargle1982), a Fourier transform using Period04 (Lenz & Breger

1 During the observing run with the Mercator telescope one attemptwas made to make an observation of 103P/ Hartley 2 with the HERMES

spectrograph in order to obtain a propper spectrum. But this failed be-cause a correct pointing appeared to be impossible.




Fig. 3. Coma profile in B and I band.

Fig. 4. XR, | m1| , | m2|. and vgr of all I band coma profiles.

2005) and the Jurkevich-Stellingwerf PDM (Phase DispersionMinimalisation) method (Jurkevich 1971), (Stellingwerf 1978).The first one can be considered as the most straight foreward:we wrote a MCMC algorithm that fits a sine to the time series,fitting for the amplitude, frequency, phase and off set at the sametime. The Lomb-Scargle periodogram analysis fits sine wavesy = a cosωt + b sinωt using a least square fitting procedure.The frequency analysis routine used by Period04 works in three

steps: first it uses a fourier transform to calculate the frequen-cies. The second step does a least squares fitting using the same

Fig. 5. Time series of the central region of the coma. An apertureof 1 pixel was used.

Fig. 6. Time series of the central region of the coma. The resultis shown for aperture radii of 5 and 15 pixels.

parameters as our MCMC model to improve the frequenciesfound by the Fourier transform. The final step which comparesdiff erent periods is a calculation of uncertainties, using theerror-matrix of the least-square fitting. For a description of theJurkevich-Stellingwerf PDM method see (Aerts et al. 2010).When looking for the frequency of the rotation, we have to knowin what range of frequencies we have to look. The frequencycan not be too high: comets are very fragile balls of dust andice, and too high angular velocities will result in the breakupof comets. If the nucleus of a comet is taken to be spherical,with a density of ρ = 1000 kg m−3 a minimum period of 3.3h

is found (Jewitt & Meech 1988). However, comets tend to haveelongated shapes (especially smaller ones like 103P/ Hartley),




Fig. 7. The most reliable frequencies, as calculated by theMCMC fitting procedure. The program performed 107 fitting it-

erations to achieve the distribution in this figure.

and most cometary densities are believed to be around 600 kgm−3 (Britt et al. 2006). Together this gives us a rough maximumfrequency estimate of about 4.3 cycles per day (c d−1), or aminimal period of 5.6h. This is supported by a list of knowrotation periods (Samarasinha et al. 2004).For the frequency analysis we used aperture radii of one pixel.Both the MCMC method and Period04 give comparable for the

frequency identification, as can be seen in figures 7 and 8. Bothof these methods identify frequencies around 1.9 c d −1 and 2.9c d−1 as the most reliable frequencies (P ≈ 13h and P ≈ 8.3h).The two other methods also find these two peaks, however theyfavor them in the opposite order (see figure 9). Phase diagramsfor both frequencies are shown in figures 10 and 11. All fourmethods seem to favor their first and second candidate withapproximately the same prefence ratio. However, the remark can be made that the MCMC algorithm is somewhat lessreliable as it is not designed specially for frequency analysis,contrary to the three other algorithms, which might lead to aslight preference of 2.9 c d−1 as the most reliable frequency.Also, on the phase diagrams the frequency of 2.9 c d −1 seems

to give a much better fit with the individual datapoints, wherethe frequency of 1.9 c d−1 seems more like a fit that follows theaverage tendens of both good datapoints and outliers. Both of these arguments incline us to believe that 2.9 c d −1 is the mostreliable frequency, however the frequency of 1.9 c d−1 cannotbe ruled out.Normally we can assume that the frequencies we find will onlybe half of the rotation frequency: if the brightness diff erenceof the central region is caused by the diff erence in reflectionsurface of an elongated nucleus, there will be two maxima inone rotation. The first one when we see the frontside of thecomet, the other one when we see the backside of the comet.This means the actual frequency of the nuclear rotation is half of the frequency found before, giving us the final periods P1 =

16.6 ± 0.7 h and P2 = 25.3 ± 0.9 h.

Fig. 8. Reliability distribution for frequencies between 1 and 3 cd−1 as obtained by Period04

Fig. 9. Reliability distribution for frequencies between 1 and 4c d−1. The upper graph represents the result of the Jurkevich-Stellingwerf PDM method. As one might suspect, lower val-ues mean higher reliability for this method contrary to allother methods used. The lower graph is the Lomb-Scargle pe-riodogram.

Fig. 10. Phase diagram for frequency f =1.91 c d−1. The lowerpart of the graph shows the residuals of the fit.




Fig. 11. Phase diagram for frequency f =2.90 c d−1. The lowerpart of the graph shows the residuals of the fit.

Frequencies of the nuclear rotation

MCMC 1.87 c d−1 2.87 c d−1

Period04 1.89 c d

−1

2.93 c d

−1

Lomb-Scargle 2.90 cd−1 1.91 c d−1

Jurkevich-Stellingwerf 2.91 c d−1 1.91 c d−1

Table 3. List of the strongest frequencies, obtained by usingfour diff erent frequency analysis algorithms. The most reliablefrequencies are given for each method, ranked in order of im-portance from left to right.

4.3. Literature Values of the Rotation

Because 103P/ Hartley was the focus of the a study with

the NASA Deep Impact space probe extended mission calledEPOXI it allready gained much attention long before it becameclose to Earth. As a preparation for the EPOXI mission, Meechet al. (Meech et al. 2009) did a quite extensive campaign to deter-mine the rotation period between March and July of 2009, whenthe comet was still at a distance of r = 5.7 AU. At this distancecomets are normally inactive, as they are no longer su fficientlyheated. This makes it easier to see the nuclear brightness varia-tion, however at this distance the nucleus is very faint: nuclearmagnitudes near m ∼ 25 were expected. Observations consistedof 10 hours on the LBT (only one telescope was operative atthat time), 28 hours of time on the Gemini N and S telescopes,2 nights on the VLT 8m, 7 hrs on the GTC 10.4m and 20 hrs on

the SALT telescope in S. Africa, and in addition they received 12orbits of HST time, spanning a clock time of 1 day. Both fromthe data of the Gemini telescopes and the HST data they found arotation period of 16.6h.A diff erent approach was used by Knight et al. (Knight et al.2010): they used a CN filter (λe f f 387 nm with a width of 6.2nm) to locate jets emerging from the nucleus. These jets rotatetogether with the nucleus, allowing to determine a nuclear rota-tion period. A value of 16.6 ± 0.5h was derived.The Arecibo radio telescope made 20 observations of 103P/ Hartley on October 24-27, resulting in a rotation period of 18.1 ± 0.3h, however they could not exclude a less likely periodof 13.2h. These observations also showed the comet is highlyelongated.

Our proposed period of P = 16.6 ± 0.7 is in very good agree-ment with the findings of both Meech et al. and Knight et al.,

and we therefore come to the conclusion that this must indeedbe the period of nuclear rotation.

4.4. Nucleus Axis Ratio

In equation 3 we saw how the magnitude of the nucleus shows along term variation as the comet travels trough the solar system.As described above there is also a signal superponated on thislong term variation due to the rotation of the nucleus. The firstterm in the right hand side of equation 3 is the magnitude termrelated to the physical properties of the nucleus itself, as itquantizes the amount of solar light reflected from the sun at adistance rh = ∆ = 1AU and α = 0. We can therefore understandthat the amount of light reflected must be proportional to thereflective surface of the comet and its albedo:

F comet ∝ F R2N pv (11)

⇒ m0 = C − 5log RN − log pv (12)

With F the solar flux, RN the radius of the nucleus, pv the geo-

metric albedo and C a constant. In section 4.2 we found the mostconvincing period to be P = 16.6 ± 0.7. In figure 11 a wave withthis period was comared with the observations. Fitting the phaseplot allows us to predict the variations in magnitude during a cy-cle. Doing so gives us a value of a value of ∆m = 0.46 ± 0.06.If we use equation 12 together with this value, and assume thealbedo of the comet is constant and has the shape of a prolatespheroid we find

e0.4∆m = 1.20 ± 0.03 ≤ a/b (13)

This value is a lower boundary for the axis ratio because we onlyobserve the projection of the nucleus.

4.5. Jetlike Structure

On the night of the 16th of October we were able to obtain thelargest number of observations: 13 of our 29 observations inthe IC filter were done on this night. We combined all our ob-servations in the IC filer to create a median frame of the inner38” × 38” around the nucleus. We then subtracted that medianframe from all our observations to enhance small diff erences. Acomposite image of 12 of these observations is shown in fig-ure 12. The center of each frame corresponds to the pixel as-sociated with the nucleus. Starting at 03h12 a structure seemsto emanate from the nucleus, seemingly rotating anti-clockwiseduring the rest of the night. In several of the following frames

two-lobed structures are observed, consisting of a small and abig lobe. These structures might be jets as they seem to have anarrow collimation and there is a certain morphological agree-ment with other observations of jets. However, jets are normallylooked for in narrowband filters positioned on gas emission fea-tures. Normally the light in the IC filter comes from the reflectionof light on dust particles. Knight et al. observed 103P/ Hartleywhen it was at r = 1.74 AU and detected jets in the CN filter, butno structures were detected in their R filter (Knight et al. 2010).Another possibility is that the structure is an enhanced amountof dust originating from a minor breakup or sudden eruption,which moves into the dusttail during the observations. Howeverwe cannot exclude the possibility that these structures are an ar-tifact of the process used to obtain these images, however we see

no clue for this. Because we cannot confirm the origin of thesestructures and we only discovered them during the last stages of




this project, we did not perform a deeper investigation on thesejetlike structures.

5. Conclusions

From our limited dataset obtained from our observing run wewere able to investigate the structure of the coma, the rotation of

the nucleus and possibly discovered some jetlike structures.The analysis of the coma surface brightness profiles in the ICfilter lead us to a m value of −1.14 ± 0.07 for the inner coma,and m = −1.45 ± 0.07 for the outer coma. Thus the structure of the inner coma clearly deviates from the expected m = −1 valuethat is obtained from theoretical considerations. Further obser-vations, such as spectroscopy and photometric observations inwavelength ranges where only continuum emission is important,could help to clear out this discrepancy. The outer coma struc-ture is, within the margin of error, in agreement with what isexpected from the literature. The point of inflection from innerto outer nucleus, X R, is found to be situated at 3600 ± 1000 km,which is particularly close to the nucleus. This finally gives aterminal grain ejection velocity of 191 ± 25 m/ s, which is sig-

nificantly lower than expected from an empirical law. Possiblyindicating that the activity of 103P/ Hartley 2 is less than whatis normal for a comet close to perihelion or otherwise indicatingthat the structure in the inner coma of 103P / Hartley 2 is diff erentfrom a spherically symmetric, steady state model and the theorydoes not make sense anymore. Further observation could per-haps exclude one of the possibilities.Using small aperture photometry of the region around the nu-cleus and correcting for the seeing we were able to detect ampli-tude variations due to the rotation of the nucleus. Our frequencyanalysis resulted intwo possible rotation periods, of which P1 =

16.6 ± 0.7 h was found to be the most reliable. This value is invery good agreement with other publications. No other observa-tions seem to come close to our second period of P

2= 25.3 ±

0.9 h. This period might have been caused by outliers due to ourlimited amount of observations. A larger amount of observationsmight have made it possible to rule out this period.

Acknowledgements. We would like to thank Maarten Reyniers for providing usthe idea for this research project, and Pieter Degroote for his master’s thesiswhich provided us a solid background on cometary physics. We are grateful toProfessor Van Winckel for his support and usefull remarks during both research-project II and III, and to Peter Papics for helping us out during the observations.

References

Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology,ed. Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W., 348–350

Brandt, J.C. & Chapman, R.D. 2004, Introduction toComets, ed. Brandt, J.C. &Chapman, R. D., p35

Britt, D. T., Consolmagno, G. J., & Merline, W. J. 2006, in Lunar and PlanetaryInstitute Science Conference Abstracts, Vol. 37, 37th Annual Lunar andPlanetary Science Conference, ed. S. Mackwell and E. Stansbery, 2214– +

Davignon, G., Blecha, A., Burki, G., et al. 2004, in Presented at the Societyof Photo-Optical Instrumentation Engineers (SPIE) Conference, Vol. 5492,Society of Photo-Optical Instrumentation Engineers (SPIE) ConferenceSeries, ed. A. F. M. Moorwood & M. Iye, 871–879

Degroote, P. 2008, K.U.Leuven Master Thesis, Faculty of ScienceDelsemme, A. H. 1982, Icarus, 49, 438Evans, D. W. 2001, Astronomische Nachrichten, 322, 347Garcıa-Gil, A., Munoz-Tunon, C., & Varela, A. M. 2010, PASP, 122, 1109Jewitt, D. 1991, in Astrophysics and Space Science Library, Vol. 167, IAU

Colloq. 116: Comets in the post-Halley era, ed. R. L. Newburn Jr.,M. Neugebauer, & J. Rahe, 19–65

Jewitt, D. C. & Meech, K. J. 1987, ApJ, 317, 992Jewitt, D. C. & Meech, K. J. 1988, ApJ, 328, 974Jurkevich, I. 1971, Ap&SS, 13, 154

Knight, M., Schwieterman, E., & Schleicher, D. 2010, IAU Circ., 9163, 2Lamy, P. L., Toth, I., Fernandez, Y. R., & Weaver, H. A. 2004, The sizes, shapes,

albedos, and colors of cometary nuclei, ed. Festou, M. C., Keller, H. U., &Weaver, H. A., 223–264

Lenz, P. & Breger, M. 2005, Communications in Asteroseismology, 146, 53Licandro, J., Serra-Ricart, M., Oscoz, A., Casas, R., & Osip, D. 2000, AJ, 119,

3133Lisse, C. M., Fernandez, Y. R., Reach, W. T., et al. 2009, PASP, 121, 968Lomb, N. R. 1976, Ap&SS, 39, 447Meech, K. J., Hainaut, O., Weaver, H. A., et al. 2009, in AAS / Division for

Planetary Sciences Meeting Abstracts, Vol. 41, AAS/ Division for PlanetarySciences Meeting Abstracts, 20.07–+

Østensen, R. H. 2010, ArXiv e-printsReyniers, M., Degroote, P., Bodewits, D., Cuypers, J., & Waelkens, C. 2009,

A&A, 494, 379Samarasinha, N. H., Mueller, B. E. A., Belton, M. J. S., & Jorda, L. 2004,

Rotation of cometary nuclei, ed. Festou, M. C., Keller, H. U., & Weaver,H. A., 281–299

Scargle, J. D. 1982, ApJ, 263, 835Stellingwerf, R. F. 1978, ApJ, 224, 953




Fig. 12. Observations of the 16th of October in the IC filter after extracting a median of these observations. Observation time isgiven in the bottom right corner of every frame. After 03h12 a structure seems to emerge from the nucleus. Each frame has a field

of view of 38” × 38”.

hartley 2 - nucleus rotation and coma profiles

Documents