Quantizing comets: semiclassical methods in action

One key aspect of theoretical physics is that the (rather few!) basic equations are not tied to a narrow field of applications, but that insights from celestial mechanics are equally relevant for the dynamics of quantum objects. I have written before how to interpret the Coulomb Green’s function of hydrogen or Rydberg molecules in terms of Lamberts theorem of cometary orbit determination.

Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.
Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.

Equally interesting is the dynamics of small celestials bodies in the vicinity of a parent body (the “restricted three body problem“). Near the Lagrange points the attraction of the sun and the effective potential in a coordinate system moving with the parent body around the sun cancel and the small objects can be trapped.
The comet Shoemaker-Levy 9 (SL9) is a prime example: SL9 was captured by Jupiter, broke apart at a close approach, and finally the string of fragments collided with Jupiter in 1994.

What would have happened with SL9 if Jupiter was contracted to a point mass?

Since SL9 was once captured, it should also have been released again. Indeed, in 2014 SL9 would have left Jupiter, as shown in the numerically integrated JPL orbit. To illustrate and simplify the transient dynamics, I have assumed in a recent publication a circular orbit of Jupiter and that SL9, Jupiter, and the sun are located in a plane. In reality the changing distance from the sun can open and close the entry points and in conjunction with precise location of the comet an escape or trapping becomes feasible:

Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped).
Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped). (C) Tobias Kramer.

The Shoemaker-Levy 9 case has been studied extensively in the astronomical literature (see for instance the orbital analysis by Benner et al). So what new insights are there for quantum objects? The goal is not to claim that comets have to be treated as quantum mechanical objects, but to realize that exactly the same dynamics seen in celestial mechanics guides electrons in magnetic fields through wave guides. I refer you for the details to my article Transient capture of electrons in magnetic fields, or: comets in the restricted three-body problem, but want to close by showing the electronic eigenfunctions, which show a real quantum feature absent in the classical case: electrons can tunnel through the forbidden area and thus will always escape from the parent body:

Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field.
Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field. (C) Tobias Kramer.

Added on March 1, 2020: another case of a transient capture, this time around Erath: small body 2020 CD3:

Visualization of the orbit of small body 2020 CD3, ephemeris taken from the JPL Horizons system (the uncertainties of the 1 year backward integration are several hours), geocentric ecliptical coordinates. The inset shows the close encounters with Earth (blue disk) on April 4, 2019 and February 13, 2020.

Visitors from other stars

Star chart (open source stellarium program) showing the location of comet 2I/Borisov on Dec 1st, 2019, 5:36 UTC for comparison with the corresponding image in the gallery of telescope.org (C) The Open University.

This year has revealed the first two visitors coming from another stellar system far away. The giveaway signature of these visits is the very hyperbolic orbit, which means that the objects are not gravitationally bound by our sun. They must have undertaken the long journey from another star to us. Many fascinating questions arise: are these distant worlds we will never get to similar to our solar system? Which elements are formed, how much water is around?

While the first visitor 1I/’Oumuamua (I=interstellar) did not show much activity (besides a surprisingly large change in its orbital acceleration presumable caused by sublimating ices), the second one 2I/Borisov is more active and can be analyzed for its chemical composition.
I submitted a request for an image to the fine online observatory operated by The Open University on the Teide (Tenerife, Spain), and tonight the COAST 14″ telescope took a 120 seconds image of 2I/Borisov. You can find the
corresponding image in the newest image gallery of telescope.org, all images there (C) The Open University. Can you spot the comet? It is the tiny elongated speck in the lower right quadrant (the image covers about 1/2 degree of the sky), or have a look at the finder chart shown here.

In addition to the hyperbolic orbit, both objects showed a non-gravitational acceleration (discussed here before for the solar-system comet 67P/Churyumov-Gerasimenko). Both interstellar comets show a rather large extra acceleration if compared to Jupiter family comets.

Non-graviational acceleration of Jupiter family comets (taken from the JPL Small-Body database), compared with the two visiting comets (preliminary orbit data from JPL).

Random scattering: the small scale structure of the universe

Branched flow
Emergence of branched electron flow from weak scattering across a random potential. (C) Tobias Kramer

Our universe displays various mass concentrations of matter and is not a homogeneous density soup of particles as assumed in the simplest cosmological models where isotropy is assumed (cosmological principle). Interestingly recent supernova data (see Colin et al Evidence of anisotropy of cosmic acceleration A&A 631 L13, also on the arxiv) shows deviations of the angular distribution of matter as seen from Earth. One possibility (also not taken into account in the standard model) is the topology of the cosmos, discussed in this blog by Peter Kramer before. But back to the “small structures” (< 100 Mega parsec): How do mass concentrations arise and how get small fluctuations amplified?

One universal mechanism at work across many domains of physics is structure formation and concentration into branches by random weak scattering. The key-point is: despite randomness this type of scattering does not smear out the density as expected by diffusion processes. This important mechanism has been independently discovered in various domains of physics, but has been rarely discussed and further explored within a consistent framework. Interestingly the fundamental branching behavior can be seen in both quantum mechanics and classical physics. Together with Rick Heller and Ragnar Fleischmann we have written a short overview  article (available on the arxiv: “Branched flow”) which provides some background information and serves as a starting point for further exploration. We also provide a short python script to generate intricate patterns out of random deformations (script available on github).

Sunlight creating a web of caustics at the bottom of a swimming poll. Picture (C) Tobias Kramer.

One example discussed in this blog is the formation of “dust concentrations” in the near nucleus coma of comets from a homogeneously emitting cometary surface. The structure formation of the universe (the cosmic web) is discussed by Y. Zeldovich and reviewed by P.J.E. Peebles in his monograph The Large-Scale Structure of the Universe (Princeton University Press, 1980) in the chapter “caustics and pancakes”. A more accessible incarnation of the caustics are the ripples of sunlight at a pool bottom.

Accelerating comets: the non-gravitational force

Orbit of comet 67P/Churyumov-Gerasimenko seen from the North pole of the rotation axis of the nucleus. Perihelion, aphelion, and the equinox positions are shown. (C) Tobias Kramer 2019

Celestial mechanics is firmly rooted in Newton’s (and Einsteins) laws of gravity and enables us to compute the positions of planets, moons, and comets with high accuracy. Besides the solar attraction, the major gravitational perturbation of comets are caused by Jupiter. Non-gravitational forces are important to understand the dynamics and composition of our interstellar visitor ʻOumuamua, which not only followed a hyperbolic trajectory but additionally accelerated.

How does a “non-gravitational force” arise?

For asteroids and comets, the term “non-gravitational force” refers to all effects besides the standards law of gravitation. For comets the most obvious one is the force induced by the sublimation of ice. The gas molecules fly into space and carry along momentum transferred from the cometary nucleus. The momentum transfer affects cometary motion considerably:

Fortunately, Rosetta was a faithful companion of 67P/Churyumov-Gerasimenko and provided the required positions in space (accuracy better than 10 km). The accurate determination of the spacecrafts is an art by itself, as described in this report. Only this data allowed us to retrieve the non-gravitational acceleration and to deduce how much water ice sublimated during the 2015 apparition.

Magnitude of the non-gravitational acceleration of 67P/Churyumov-Gerasimenko retrieved from the orbital evolution. The red line is an independent determination of the gas production based on ROSINA data. Adapted from Kramer & Läuter 2019.

Flying close to the nucleus provided us with spectacular views of the surface from the OSIRIS camera, but made it more difficult to assess the overall gas production of the comet, since Rosetta moved actually across the escaping molecules and probed the density “in-situ” using the ROSINA devices. Together with the ROSINA principal scientists Kathrin Altwegg and Martin Rubin, we (Matthias Läuter and Tobias Kramer) reconstructed the surface emission rate from the in-situ data (“Surface localization of gas sources on comet 67P/Churyumov-Gerasimenko based on DFMS/COPS data“, free arxiv version). This completely independent determination of the water gas production of 67P agrees matches up nicely with the acceleration data as shown in the figure.

You can run your own experiments with various parameters for the outgassing induced acceleration by using the NASA Horizons web interface. You need to input the 6 osculating elements (an equivalent to specifying the position and velocity vector of the comet) and you can enter values for the non-gravitational parameters A1,A2,A3. Usually these parameters are determined by a careful analysis of several apparitions of a comet, since one apparition as seen and measured from Earth does not allow us to retrieve the values with enough confidence.

For 67P a good orbit representation is given by these values from our article:

Osculating elements of 67P Churyumov Gerasimenko:
Epoch                    2456897.7196990740
Eccentricity             0.6410114978
Perihelion distance      1.24317813856152504
Perihelion Julian date   2454893.7138435622
Longitude Ascending Node 50.1459466115
Argument of perihelion   12.7813547059
Inclination              7.0405649967

Non-gravitational parameters:
A1 +1.066669896245E-9 au/d^2
A2 −3.689152188599E-11 au/d^2
A3 +2.483436092734E-10 au/d^2
∆T 35.07142 d

Comet 21P/Giacobini-Zinner from Berlin Mitte

cof comet21p_fc

Observing comet 21P/Giacobini-Zinner from the city center of Berlin proved to be challenging. I put up my 130 mm Newton telescope in the living room to catch a glimpse of Capella above the roof of the adjacent building around 3 am on Sept. 5. Not far from Capella, 21P/Giacobini-Zinner was barely visible, but a 10s exposure with the smartphone camera revealed the fuzzy coma of the comet. Since the comet is close to perihelion it moves swiftly across the stars within 1 hour, as seen in the 3 frames animation from my observations at 02:39, 03:04 and 03:14 local time.

It would be nice to knock out the street light or to block the city lights, but it turns out the new LED street lights are emitting with a broad spectrum across to the entire visible range.
I analyzed the spectra with a hand-build spectrometer from Astromedia to evaluate if a UHC filter might help, but unfortunately this is not the case.

The hopes are high for Christmas comet 46P/Wirtanen to be much brighter and better visible at the end of the year.
Wirtanen gets this time very close to Earth (0.071 AU, no danger of collision though) and will be extensively studied. For an overview of our professional work about comet 67P/Churyumov-Gerasimenko have a look at this ZIB feature.
Current comet light curves are compiled on the excellent COBS.SI webpage.


Octahedral gravity – the case of asteroid (162173) Ryugu

Gravitational potential of an octahedron (rough estimate: 1.4 km diameter, density 1500 kg/m3), with added centrifugal term (7 h rotation). Credit: Tobias Kramer, 2018

Asteroid Ryugu (June 24, 2018). Credit: JAXA, University of Tokyo, Kochi University, Rikkyo University, Nagoya University, Chiba Institute of Technology, Meiji University, Aizu University, AIST

Asteroid Ryugu

The Hayabusa 2 target asteroid (162173) Ryugu displays a strange diamond like shape, at least from the sunny side explored so far.
Time for me to reuse the code developed for comet 67P/Churyumov-Gerasimenko and to compute the gravitational potential (with added centrifugal term, assuming a 7 h rotation period). The rotation weakens the gravitational potential in the equator regions. Otherwise on the triangular faces the potential minima are close to the centers of the triangle.

Update: new JAXA images, an animation of the rotation, and a shape model extracted by Doug Ellison. This allows me to recompute the effective gravitational potential (assuming a uniform density):


Cometary activity: the case of 67P/Churyumov-Gerasimenko

Visualization of dust trajectories emerging from comet 67P/C-G from sun-lit (reddish) and shadowed areas (bluish lines). Taken from Advances in Physics: X, 3(1), 1404436, 2018

The Rosetta spacecraft has come to rest on comet 67P/Churyumov-Gerasimenko. The comet is retreating from the sun, but the analysis of the scientific data is an ongoing endeavour with many discoveries yet to be made. As described before, the coma structure of 67P/C-G followed a surprisingly predictable pattern: dust is emitted from the entire sunlit surface and later in space forms intricate dust bundles and rays, directly reflecting the surface topography. The rotation of the nucleus leads to a bending of the dust trajectories which allows us to read of the velocity of the particles: around 3 m/s at distances 2-3 km from the surface. Our detailed prediction of the dust coma from May 2015 recently appeared in Advances in Physics: X, 3(1), 1404436, 2018 (open access), where we compare the model with Rosetta images such as these ones: (1, 2, 3).

Distribution of gas emitting regions on 67P/Churyumov-Gerasimenko (blue to red color scale: increasing gas emission). The black needles mark the reported locations of short-lived dust outbursts. Adapted from MNRAS 469, S20, 2017

The dust is propelled by the gas emitted from the surface by sublimation processes. It is a non-trivial task to back-out the surface ice distribution from the measured gas densities at Rosetta’s orbit via the COmetary Pressure Sensor (COPS), built by the ROSINA team (PI: Prof. Kathrin Altwegg, University of Bern, Switzerland). Using an analytical ansatz for the gas distribution, we managed to retrieve the “best-fit” distribution of the gas sources on the surface (T. Kramer, M. Läuter, M. Rubin, K. Altwegg: Seasonal changes of the volatile density in the coma and on the surface of comet 67P/Churyumov-Gerasimenko Monthly Notices of the Royal Astronomical Society, 469, S20, 2017).
Interestingly, the sources of higher gas emission are linked to reported short-lived outburst locations. A unified picture and modelling of gas and dust emission holds important clues about the composition of the cometary surface and we continue our investigation in that direction.

Metadata analysis of 80,000 arxiv:physics/astro-ph articles reveals biased moderation

Have you ever thought of arXiv moderation in astro-ph being a problem? Did you experience a >5 months delay from submission of your pre-print to the arXiv  to being publicly visible? Did this happen without any explanation or reaction from the arXiv moderators despite the same article being published after peer review in the Astrophysical Journal Letters?

Chances are high that your answer is no, to be precise the odds are 81404/81440=99.9558 percent that this did not happen to you.  Lucky you! Now let me tell about the other 36/81440=0.0442043 percent. My computer based analysis of the last 80,000 deposited arxiv:astro-ph articles shows interesting results about the moderation patterns in astrophysics. To repeat the analysis

  • get the arXiv metadata, which is available (good!) from the arxiv itself. I used the excellent metha tools from Martin Czygan to download all metadata from the astro-ph and quant-ph sections since 5/2014.
  • parse the resulting 200 MB XML file, for instance with Mathematica. To get the delay from submission to arXiv publication, I  took the time difference between the submission date stamp (oldest XMLElement[{http://purl.org/dc/elements/1.1/, date}) and the arXiv identifier, which encodes the year and month of public visibility.
  • Example: the article  arxiv:1604.00876 went public in April 2016, 5 months after submission to the arXiv (November 5, 2015) and publication in the Astrophysical Journal Letters (there total processing time from submission to online publication, including peer review 1.5 months).

The analysis shows different patterns of moderation for the two sections I considered, quant-ph and astro-ph. It reveals problematic moderation effects in the arXiv astro-ph section:

  1. Completely suitable articles are blocked, mostly peer reviewed and published for instance in the Astrophysical Journal, Astrophysical Journal Letters, Monthly Notices of the Royal Astronomical Society.
  2. This might indicate a biased moderation toward specific persons and subjects. In contrast to scientific journals with their named editors, the arXiv moderation is opaque and anonymous. The metadata analysis shows that the moderation of the physics:astro-ph and physics:quant-ph use very different guidelines, with astro-ph having a strong bias to block valid contributions.
  3. It makes the astro-ph arXiv less usable as a medium for rapid dissemination of cutting edge research via preprints.
  4. This hurts careers, citation histories, and encourages plagiarism. New scientific findings are more easily plagiarized by other groups, since no arXiv time-stamped preprint establishes the precedence.
  5. If we, the scientists, want a publicly funded arXiv we must ensure that it is operated according to scientific standards which serve the public. This excludes biased blocking of valid and publicly funded research.
  6. Finally, the arXiv was not put in place to be a backup server for all journals, but rather to provide a space to share upcoming scientific publications without months of delay.

I will be happy to share comments I receive about similar cases. I am not talking about dubious articles or non-scientific theories, but about standard peer-reviewed contributions published in established physics journals, which should be on the astrophysical preprint arXiv.

Here follows the list of all articles which were delayed by more than 3 months from arxiv:physics/astro-ph (out of a total of 81,440 deposited articles) and if known where the peer reviewed article got published. I cannot exclude other factors besides moderation for the delay, but can definitely confirm incorrect moderation being the cause for the 2 cases I have experienced. Interestingly the same analysis on arxiv:physics/quant-ph did not reveal such a moderation bias of peer reviewed articles. This gives hope that the astrophysical section could recover and return to 100 percent flawless operation. Then the arXiv fulfils its own pledge on accountability and on good scientific practices (principles of the arXiv’s operation).

http://arxiv.org/abs/1009.0420 The Astrophysical Journal
http://arxiv.org/abs/1101.3547 Publications of Astronomical Society of Japan
http://arxiv.org/abs/1506.01700 Journal of Astrophysics and Astronomy
http://arxiv.org/abs/1509.08311 EPJ Web of Conferences
http://arxiv.org/abs/1501.03478 Astrophysics and Space Sciences
http://arxiv.org/abs/1602.07334 The Astrophysical Journal
http://arxiv.org/abs/1602.07511 Physical Review C
http://arxiv.org/abs/1603.09573 Monthly Notices of the Royal Astronomical Society
http://arxiv.org/abs/1604.00876 The Astrophysical Journal Letters
http://arxiv.org/abs/1604.00291 Journal of Statistical Mechanics: Theory and Experiment
http://arxiv.org/abs/1604.07760 The Astrophysical Journal Letters

Predicting comets: a matter of perspective

Contrast stretched NAVCAM image of the nucleus of comet 67P/Churyumov-Gerasimenko to highlight the “jets” of dust emitted from all over the surface. CC BY-SA IGO 3.0

In general, any cometary activity is difficult to predict and many comets are known for sudden changes in brightness, break ups and simple disappearances. Fortunately, the Rosetta target comet 67P/Churyumov-Gerasiminko (67P/C-G) is much more amendable to theoretical predictions. The OSIRIS and NAVCAM images show light reflected from a highly structured dust coma within the space probe orbit (ca 20-150 km).

Is is possible to predict the dust coma and tail of comets?

Starting in 2014 we have been working on a dust forecast for 67P/C-G, see the previous blog entries. We had now the chance to check how well our predictions hold by comparing the model outcome to a image sequence from the OSIRIS camera during one rotation period of 67P/C-G on April 12, 2015, published by Vincent et al in A&A 587, A14 (2016) (arxiv version, there Fig. 13).

Comparison of Rosetta observations by Vincent et al A&A 2016 (left panels) with the homogeneous model (right panels). Taken from Kramer&Noack (ApJL 2016) Credit for (a, c): ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA

Our results appeared in Kramer & Noack, Astrophysical Journal Letters, 823, L11 (preprint, images). We obtain a surprisingly high correlation coefficient (average 80%, max 90%) between theory and observation, if we stick to the following minimal assumption model:

  1. dust is emitted from the entire sunlit nucleus, not only from localized active areas. We refer to this as the “homogeneous activity model”
  2. dust is entering space with a finite velocity (on average) along the surface normal. This implies that close to the surface a rapid acceleration takes place.
  3. photographed “jets” are highly depending on the observing geometry:
    rotateif multiple concave areas align along the line of sight, a high imaged intensity results, but is not necessarily the result of a single main emission source. As an exemplary case, we analysed the brightest points in the Rosetta image taken on April 12, 2015, 12:12 and look at all contributing factors along the line of sight (yellow line) from the camera to the comet. The observed jet is actually resulting from multiple sources and in addition from contributions from all sunlit surface areas.

What are the implications of the theoretical model?

If dust is emitted from all sunlit areas of 67P/C-G, this implies a more homogeneous surface erosion of the illuminated nucleus and leaves less room for compositional heterogeneities. And finally: it makes the dust coma much more predictable, but still allows for additional (but unpredictable) spontaneous, 20-40 min outbreak events. Interestingly, a re-analysis of the comet Halley flyby by Crifo et al (Earth, Moon, and Planets 90 227-238 (2002)) also points to a more homogeneous emission pattern as compared to localized sources.

Weathering the dust around comet 67P/Churyumov–Gerasimenko

Bradford robotic telescope image of comet 67P (30th Oct 2015)
Bradford robotic telescope image of comet 67P/Churyumov–Gerasimenko (180s exposure time, 5:43 UTC, 30-10-2015). © 2015 University of Bradford

Comet 67P/Churyumov–Gerasimenko is past its perihelion and is currently visible in telescopes in the morning hours. The picture is taken from Tenerife by the Bradford robotic telescope, where I submitted the request. The tail is extending hundred thousands kilometers into space and consists of dust particles emitted from the cometary nucleus, which measures just a few kilometers. In a recent work just published in the Astrophysical Journal Letters (arxiv version), we have explored how dust, which does not make it into space, is whirling around the cometary nucleus. The model assumes that dust particles are emitted from the porous mantle and hover over the cometary surface for some time (<6h) and then fall back on the surface, delayed by the gas drag of gas molecules moving away from the nucleus. As in the predictions for the cometary coma discussed previously, we are sticking to a minimal-assumption model with a homogeneous surface activity of gas and dust emission.

Dust trajectories reaching the Philae descent area computed from a homogeneous dust emission model. Figure from Kramer/Noack Prevailing dust-transport directions on comet 67P/Churyumov-Gerasimenko, Astrophysical Journal Letters, 813, L33 (2015)
Dust trajectories reaching the Philae descent area computed from a homogeneous dust emission model. From Kramer/Noack “Prevailing dust-transport directions on comet 67P/Churyumov-Gerasimenko”, Astrophysical Journal Letters, 813, L33 (2015).

The movements of 40,000 dust particles are tracked and the average dust transport within a volumetric grid with 300 m sized boxes is computed. Besides the gas-dust interaction, we do also incorporate the rotation of the comet, which leads to a directional transport.
The Rosetta mission dropped Philae over the small lobe of 67P/C-G and Philae took a sequence of approach images which reveal structures resembling wind-tails behind boulders on the comet. This allowed Mottola et al (Science 349.6247 (2015): aab0232) to derive information about the direction of impinging particles which hit the surface unless sheltered by the boulder. Our model predicts a dust-transport inline with the observed directions in the descent region, it will be interesting to see how wind-tails at other locations match with the prediction. We put an interactive 3d dust-stream model online to visualize the dust-flux predicted from the homogeneous surface model.

Day and night at comet 67P/Churyumov–Gerasimenko

Comet 67P/Churyumov–Gerasimenko has passed its nearest distance to the sun and its tail has been observed from earth. The comet emits dust and displays spectacular but short-lived outbreaks of localized jet activity. Very detailed OSIRIS pictures of the near-surface dust emission ready for stereo viewing have been posted by Brian May. The pictures also allow one to have a look at the prediction from the homogeneous dust emission model discussed previously. When you direct your attention in Brian May’s pictures to the background activity, you find very similar patterns as expected from the homogenous emission model. This activity is dimmer but steadily blowing off dust from the nucleus. Matthias Noack and I have generated and uploaded a visualization of the dust data obtained from the homogeneous activity model. In contrast to a localized activity models, collimated jets arise from a bundle of co-propagating dust trajectories emanating from concave surface areas. The underlying topographical shape model is a uniform triangle remesh of Mattias Malmer’s excellent work based on the release of Rosetta’s NAVCAM images via the Rosetta blog. The following video takes you on a flight around 67P/C-G, with 16 hours condensed into 90 sec.

The video is a side-by-side stereoscopic 3d rendering of 67P/Churyumov–Gerasimenko and the dust cloud, which can be viewed in 3d with  a simple cardboard viewer. While the observer is encircling the nucleus, day and night passes and different parts of the comet are illuminated.

Gas flow around 67P/C-G computed from a homogeneous activity model.
Gas flow around 67P/C-G computed from a homogeneous activity model. arxiv:1505.08041

In the homogeneous activity model each sunlit triangle emits dust with an initial velocity component along the surface normal. Then dust is additionally dragged along within the outwards streaming gas, which is also incorporated in the model. In contrast to compact dust particles, the gas molecules are diffusing also in lateral directions and thus gas is not helping to collimate jets by itself. The Rosetta mission with its long term observation program offers fascinating ways to perform a reality check on various models of cometary activity, which differ considerably in the underlying physics and assumptions about the original distribution and lift-off conditions of the dust eventually forming the beautiful tails of comets.

Homogeneous dust emission and jet structure near active cometary nuclei: the case of 67P/Churyumov-Gerasimenko by Tobias Kramer, Matthias Noack, Daniel Baum, Hans-Christian Hege, Eric J. Heller.

For red-cyan glasses try our 3d video on youtube (flash player required, watch out for the settings and 3d options, 1080p HD recommended).

Dusting off cometary surfaces: collimated jets despite a homogeneous emission pattern.

Effective Gravitational potential of the comet (including the centrifugal contribution), the maximal value of the potential (red) is about 0.46 N/m, the minimal value (blue) 0.31 N/m computed with the methods described in this post.
Effective Gravitational potential of the comet (including the centrifugal contribution), the maximal value of the potential (red) is about 0.46 N/m, the minimal value (blue) 0.31 N/m computed with the methods described in this post. The rotation period is taken to be 12.4043 h. Image computed with the OpenCL cosim code. Image (C) Tobias Kramer (CC-BY SA 3.0 IGO).

Knowledge of GPGPU techniques is helpful for rapid model building and testing of scientific ideas. For example, the beautiful pictures taken by the ESA/Rosetta spacecraft of comet 67P/Churyumov–Gerasimenko reveal jets of dust particles emitted from the comet. Wouldn’t it be nice to have a fast method to simulate thousands of dust particles around the comet and to find out if already the peculiar shape of this space-potato influences the dust-trajectories by its gravitational potential? At the Zuse-Institut in Berlin we joined forces between the distributed algorithm and visual data analysis groups to test this idea. But first an accurate shape model of the comet 67P C-G is required. As published in his blog, Mattias Malmer has done amazing work to extract a shape-model from the published navigation camera images.

  1. Starting from the shape model by Mattias Malmer, we obtain a re-meshed model with fewer triangles on the surface (we use about 20,000 triangles). The key-property of the new mesh is a homogeneous coverage of the cometary surface with almost equally sized triangle meshes. We don’t want better resolution and adaptive mesh sizes at areas with more complex features. Rather we are considering a homogeneous emission pattern without isolated activity regions. This is best modeled by mesh cells of equal area. Will this prescription yield nevertheless collimated dust jets? We’ll see…
  2. To compute the gravitational potential of such a surface we follow this nice article by JT Conway. The calculation later on stays in the rotating frame anchored to the comet, thus in addition the centrifugal and Coriolis forces need to be included.
  3. To accelerate the method, OpenCL comes to the rescue and lets one compute many trajectories in parallel. What is required are physical conditions for the starting positions of the dust as it flies off the surface. We put one dust-particle on the center of each triangle on the surface and set the initial velocity along the normal direction to typically 2 or 4 m/s. This ensures that most particles are able to escape and not fall back on the comet.
  4. To visualize the resulting point clouds of dust particles we have programmed an OpenGL visualization tool. We compute the rotation and sunlight direction on the comet to cast shadows and add activity profiles to the comet surface to mask out dust originating from the dark side of the comet.

This is what we get for May 3, 2015. The ESA/NAVCAM image is taken verbatim from the Rosetta/blog.

Comparison of homogeneous dust model with ESA/NAVCAM Rosetta images.
Comparison of homogeneous dust mode (left panel)l with ESA/NAVCAM Rosetta images. (C) Left panel: Tobias Kramer and Matthias Noack 2015. Right panel: (C) ESA/NAVCAM team CC BY-SA 3.0 IGO, link see text.

Read more about the physics and results in our arxiv article T. Kramer et al.: Homogeneous Dust Emission and Jet Structure near Active Cometary Nuclei: The Case of 67P/Churyumov-Gerasimenko (submitted for publication) and grab the code to compute your own dust trajectories with OpenCL at github.org/noma/covis

Trilobites revived: fragile Rydberg molecules, Coulomb Green’s function, Lambert’s theorem

The trilobite state
The trilobite Rydberg molecule can be modeled by the Coulomb Green’s function, which represents the quantized version of Lambert’s orbit determination problem.

The recent experimental realization observation of giant Rydberg molecules by Bendkowsky, Butscher, Nipper, Shaffer, Löw, Pfau [theoretically studied by Greene and coworkers, see for example Phys. Rev. Lett. 85, 2458 (2000)] shows Coulombic forces at work at large atomic distances to form a fragile molecule. The simplest approach to Rydberg molecules employs the Fermi contact potential (also called zero range potential), where the Coulomb Green’s function plays a central role. The quantum mechanical expression for the Coulomb Green’s function was derived in position space by Hostler and in momentum space by Schwinger. The quantum mechanical expression does not provide immediate insights into the peculiar nodal structure shown on the left side and thus it is time again to look for a semiclassical interpretation, which requires to translate an astronomical theorem into the Schrödinger world, one of my favorite topics.

Johann Heinrich Lambert was a true “Universalgelehrter”, exchanging letters with Kant about philosophy, devising a new color pyramid, proving that π is an irrational number, and doing physics. His career did not proceed without difficulties since he had to educate himself after working hours in his father’s tailor shop. After a long journey Lambert ended up in Berlin at the academy (and Euler choose to “escape” to St. Petersburg).

Lambert followed Kepler’s footsteps and tackled one of the most challenging problems of the time: the determination of celestial orbits from observations. In 1761 Lambert did solve the problem of orbit determination from two positions measurements. Lambert’s Theorem is a cornerstone of astronavigation (see for example the determination of Sputnik’s orbit using radar range measurements and Lambert’s theorem). Orbit determination from angular information alone (without known distances) is another problem and requires more observations.

Lambert poses the following question [Insigniores orbitae cometarum proprietates (Augsburg, 1761), p. 120, Lemma XXV, Problema XL]: Data longitudine axis maioris & situ foci F nec non situ punctorum N, M, construere ellipsin [Given the length of the semi-major axis, the location of one focal point, the points N,M, construct the two possible elliptical orbits connecting both points.]

Lambert's construction of two ellipses.
Lambert’s construction to find all possible trajectories from N to M and to map them to a ficticious 1D motion from n to m.

Lambert finds the two elliptic orbits [Fig. XXI] with an ingenious construction: he maps the rather complicated two-dimensional problem to the fictitious motion along a degenerate linear ellipse. Some physicists may know how to relate the three-dimensional Kepler problem to a four-dimensional oscillator via the Kustaanheimo–Stiefel transformation [see for example The harmonic oscillator in modern physics by Moshinsky and Smirnov]. But Lambert’s quite different procedure has its advantages for constructing the semiclassical Coulomb Green’s function, as we will see in a moment.

Shown are two ellipses with the same lengths of the semimajor axes 1/2 A1B1=1/2 A2 B2 and a common focus located at F. The centers of the two ellipses are denoted by C1 and C2. Lambert’s lemma allows to relate the motion from N to M on both ellipses to a common collinear motion on the degenerate linear ellipse Fb, where the points n and m are chosen such that the time of flight (TOF) along nm equals the TOF
along the elliptical arc NM on the first ellipse. On the second ellipse the TOF along the arc NB2M equals the TOF along nbm. The points n and m are found by marking the point G halfway between N and M. Then the major axis Fb=A1 B1=A2 B2 of the linear ellipse is drawn starting at F and running through G. On this line the point g is placed at the distance Fg=1/2(FN+FM). Finally n and m are given by the intersection points of a circle around g with radius GN=GM. This construction shows that the sum of the lengths of the shaded triangle α±=FN + FM ± NM is equal to α±=fn+ fm ± nm. The travel time depends only on the distances entering α±, and all calculations of the travel times etc. are given by one-dimensional integrations along the ficticious linear ellipse.

Lambert did find all the four possible trajectories from N to M which have the same energy (=semimajor axis a), regardless of their eccentricity (=angular momentum). The elimination of the angular momentum from Kepler’s equation is a tremendous achievement and the expression for the action is converted from Kepler’s form

  • [Kepler] W(r,r‘;E)=√μ a Kc [ξ + ε sin(ξ) – ξ’ – ε sin(ξ’)], with eccentricity ε, eccentric anomaly ξ to
  • [Lambert] W(r,r‘;E)=√μ a Kc[γ + sin(γ) – δ – sin(δ)], with
    sin2(γ/2)=(r+r’+ |r‘-r|)/(4a) and sin2(δ/2)=(r+r’- |r‘-r|)/(4a).

The derivation is also discussed in detail in our paper [Kanellopoulos, Kleber, Kramer: Use of Lambert’s Theorem for the n-Dimensional Coulomb Problem Phys. Rev. A, 80, 012101 (2009), free arxiv version here]. The Coulomb problem of the hydrogen atom is equivalent to the gravitational Kepler problem, since both are subject to a 1/r potential. Some readers might have seen the equation for the action in Gutzwiller’s nice book Chaos in classical and quantum mechanics, eq. (1.14). It is worthwhile to point out that the series solution given by Lambert (and Gutzwiller) for the time of flight can be summed up easily and is denoted today by an inverse sine function (for hyperbolic motion a hyperbolic sine, a function later introduced by Riccati and Lambert). Again, the key-point is the introduction of the linear ficticious ellipse by Lambert which avoids integrating along elliptical arcs.

The surprising conclusion: the nodal pattern of the hydrogen atom can be viewed as resulting from a double-slit interference along two principal ellipses. The interference determines the eigenenergies and the eigenstates. Even the notorious difficult-to-calculate van Vleck-Pauli-Morette (VVPM) determinant can be expressed in short closed form with the help of Lambert’s theorem and our result works even in higher dimensions. The analytic form of the action and the VVPM determinant becomes essential for our continuation of the classical action into the forbidden region, which corresponds to a tunneling process, see the last part of our paper.

Lambert is definitely a very fascinating person. Wouldn’t it be nice to discuss with him about philosophy, life, and science?