# Ceres gravity from spherical harmonics from Dawn, how to get the coefficients, definitions and potential?

update: I think I've found what I'm looking for. The file JGDWN_CER18C_SHA.TAB has what looks like spherical harmonics up to order 18, except I don't know the definitions. I still need the expression for the gravitational potential that uses this particular set of indices.

Each harmonic has four floats. Two could be the sine and cosine terms, but would the third and four float be their uncertainty? The low-order values show the last pair of floats to be much lower than the first pair, so this might be a reasonable interpretation.

For the first line of each group ((2, 0), (3, 0), (4, 0)...), the 2nd coefficient is always exactly zero. This suggests that the first line of each group represents the $\tilde{J}_n$ and the rest the $\tilde{C}_n^m$ terms in Eq. 10 in Wikipedia's Geopotential model for example, but I don't know.

For the top line, 0.470E+03 could be the normalizing radius $R_0$ since is close to the radius of Ceres of about 473 km, and 0.626E+02 is very close to the standard gravitational parameter $GM_{Ceres}$ of about 62.63 km^3/s^2. The next number I don't recognize (an uncertainty?) the next is 18 the maximum order, the rest is 1, 0, 0.

 0.4700000000000E+03, 0.6262735874220E+02, 0.3985003000000E-03,   18,   18,    1, 0.0000000000000000E+00, 0.0000000000000000E+00
1,  0,  0.0000000000000E+00, 0.0000000000000E+00, 0.0000000000000E+00, 0.0000000000000E+00
1,  1,  0.0000000000000E+00, 0.0000000000000E+00, 0.0000000000000E+00, 0.0000000000000E+00
2,  0, -0.1185081209724E-01, 0.0000000000000E+00, 0.4423323117973E-07, 0.0000000000000E+00
2,  1,  0.4584485474872E-08, 0.3639162051337E-08, 0.2155525472595E-07, 0.2153934044801E-07
2,  2,  0.2469729305807E-03,-0.2743726604078E-03, 0.5213051258454E-07, 0.5076931635838E-07
3,  0,  0.4152413352854E-04, 0.0000000000000E+00, 0.1520927120725E-07, 0.0000000000000E+00
3,  1,  0.2337951903625E-04, 0.6215005087427E-04, 0.1947755967293E-07, 0.1954951268792E-07
3,  2, -0.1917104114695E-04, 0.7075307362085E-04, 0.1863860600869E-07, 0.1818690157217E-07
3,  3, -0.4883368345013E-04,-0.9002646059395E-04, 0.2876655517205E-07, 0.2665508252055E-07
4,  0,  0.5729460133105E-03, 0.0000000000000E+00, 0.2259871225770E-07, 0.0000000000000E+00
4,  1, -0.2273796738721E-04,-0.2351160823946E-05, 0.2631912096005E-07, 0.2625579001914E-07
4,  2,  0.1237405960611E-04,-0.1722009659098E-04, 0.2902786583947E-07, 0.2915156210308E-07
4,  3, -0.2344736481214E-04,-0.3324468519353E-05, 0.2300918515991E-07, 0.2324047471504E-07
4,  4,  0.2661506582348E-04, 0.2753957135508E-04, 0.2313102127907E-07, 0.2316537801342E-07
5,  0, -0.3969538990586E-06, 0.0000000000000E+00, 0.3732369694857E-07, 0.0000000000000E+00
[...]
18, 16, -0.3285042287832E-05,-0.2204432934687E-05, 0.4782896096099E-05, 0.4777775535815E-05
18, 17, -0.3053493116667E-05,-0.2491134971988E-05, 0.4437864110763E-05, 0.4434679553983E-05
18, 18, -0.7169766777004E-06,-0.9266714099251E-05, 0.4488625034504E-05, 0.4493431031223E-05


above: GIF of Ceres image and surface gravity maps from here, originally from https://photojournal.jpl.nasa.gov/catalog/PIA22083 Credit: NASA/JPL-Caltech/UCLA/MPS/DLR/IDA

I've found and downloaded the contents from the Planetary Data Systems (PDS) page Dawn Ceres Gravity Models. The file AAREADME.TXT includes the following diagram and guide.

Reading that, ideally what I'd like to find is something like a SHA or Spherical Harmonics ASCII Data Record. The .DAT files are binaries (which I don't know how to read) and not small ~0.5 MB and so are probably more data than I need or know how to process, however they might contain what I'm looking for but I don't know how to open them.

The answer might be a guide how to find the data here, or an independently published list of spherical harmonics coefficients in an easy to read format. Either is fine.

 |===================================================================|
|                                                                   |
|             Derived Data Directory Structure for Ceres            |
|                                                                   |
|===================================================================|
|                                                                   |
|  root                                                             |
|   |                                                               |
|   |- VOLDESC.CAT                                                  |
|   |                                                               |
|   |- [CATALOG]                                                    |
|   |    |- CATINFO.TXT                                             |
|   |    |- DAWNMISSION.CAT                                         |
|   |    |- DAWNINSTHOST.CAT                                        |
|   |    |- INST.CAT                                                |
|   |    |- REF.CAT                                                 |
|   |    |- PERSON.CAT                                              |
|   |    - DATASET.CAT                                             |
|   |                                                               |
|   |- [DOCUMENT]                                                   |
|   |    |- DOCINFO.TXT                                             |
|   |    |- ...                                                     |
|   |                                                               |
|   |- [INDEX]                                                      |
|   |    |- INDXINFO.TXT                                            |
|   |    |- INDEX.LBL                                               |
|   |    - INDEX.TAB                                               |
|   |                                                               |
|   - [DATA]                                                       |
|        |- [SHBDR]                                                 |
|        - [RSDMAP]                                                |
|                                                                   |
|===================================================================|


and further down:

==============================================================================
ACRONYMS AND ABBREVIATIONS
==============================================================================

ASCII  American Standard Code for Information Interchange
DOY    Day of year
DSN    Deep Space Network
JPL    Jet Propulsion Laboratory
NASA   National Aeronautics and Space Administration
PDS    Planetary Data System
SHA    Spherical Harmonics ASCII Data Record
SIS    Software Interface Specification


What you're missing is the descriptions of the contents of that file. That's in another file, right next to the one you found. The data for the Ceres gravity model are in JGDWN_CER18C_SHA.TAB and the descriptions are in JGDWN_CER18C_SHA.LBL.

Each harmonic has four floats. Two could be the sine and cosine terms, but would the third and four float be their uncertainty?

Correct.

For the first line of each group ((2, 0), (3, 0), (4, 0)...), the 2nd coefficient is always exactly zero.

Those are the zonal harmonics. One way to represent them is they are quite different from the tesseral and sectoral harmonics. (The wikipedia page you referenced does exactly this.) Another way is they are no different from the tesseral and sectoral harmonics: They need a cosine and sine contribution. The two representations are the same because $\cos(m\phi)=1$ and $\sin(m\phi)=0$ when $m=0$.

That $\sin(0)=0$ means $\tilde S_{n0}$ could be anything because $\tilde S_{nm} \tilde P_{nm}(\cos\theta) \sin(m\phi) \equiv 0$ when $m=0$. So $\tilde S_{n0}$ might as well be zero.

For the top line, 0.470E+03 could be the normalizing radius $R_0$ since is close to the radius of Ceres of about 473 km, and 0.626E+02 is very close to the standard gravitational parameter $GM_\text{Ceres}$ of about 62.63 km^3/s^2. The next number I don't recognize (an uncertainty?) the next is 18 the maximum order, the rest is 1, 0, 0.

The label file once again describes those entries. You are correct on the first five. The 1 means the coefficients are normalized, and the final 0, 0 are the reference longitude and latitude for the model.

• Thanks! Okay, with this I'll recheck the Wikipedia page to better understand the specifics of how the terms "zonal", "tesseral" and "sectoral" are used. I've seen these a few times (probably in your other posts here and in other SE sites) so it's about time I learn what they mean. – uhoh Jun 7 '18 at 16:58
• @uhoh - The zonal harmonics terms are those for which $m=0$. These are of special interest because longitude does not contribute to these terms; the zonal harmonics divide the sphere into latitude zones. The sectoral (aka sectorial) harmonics terms are those for which $m=n$. These are of special interest because the only zeros of the associated Legendre functions $\tilde P_{nm}$ are at the poles when $m=n$; the sectoral harmonics divide the sphere into longitude sectors. The intermediate term ($1<m<n$) divide the sphere into a more complex mosaic; a set of tesserae. – David Hammen Jun 7 '18 at 17:20
• Thanks once more! fyi I've just asked How do I check my reconstructed gravity field of Ceres from spherical harmonics? – uhoh Jun 11 '18 at 8:11