Gravity modelling of asteroid 433 Eros. Any help will be appreciated.

[AdityaNanda]

Hello.
I am an engineering student and i am attempting to model the gravity around the asteroid Eros by expanding the potential as a spherical harmonic expansion. I got the coefficients (Clm and Slm) from the Planetary Data Model of Eros(NEAR Collected Shape and Gravity Models available at http://sbn.psi.edu/pds/resource/nearbrowse.html).

My potential function had 16 parts. Each part corresponds to a particular value of l. Now each part had again l+1 number of terms.Thus the l=6 part had 7 terms. The l=16 part had 17 terms and so on.

Now since the spherical harmonics are an infinite series, the contribution from each part should decrease in magnitude. Example:- contribution from l=6 part should be less than l=5 or l=4 part.

But that is not the case in my model. The contributions from terms follow the decreasing trend till l=3 but after that the contributions start to increase.

Can you please tell me what is wrong with my model?

 

[BobG]

Is there some physical reason the coefficients should decrease? Or is that something that seems like it should happen because your spherical harmonics are an infinite series?

The spherical harmonics are just a good mathematical way to 'draw' an object in ever and ever greater detail and I guess, at some point, the coefficients will be pretty flat since your ability to 'view' the object has a limited resolution. But, if you were clever enough, you could draw a person with a big nose using spherical harmonics.

Now, usually, your first coefficient is going to be a lot larger than the rest simply because most rotating celestial bodies will bulge around their equator due to their rotation. In fact, the first few will probably define the overall shape pretty well and the rest might be larger or smaller depending on the irregularities in the shape.

But there's no physical reasons for your coefficients other than that they happen to model the actual shape of the object, and asteroids are pretty irregularly shaped objects.

Here's what Earth's coefficients look like for comparison (in fact, they have the coefficients for a few other objects, as well). http://geophysics.ou.edu/solid_earth/notes/geoid/Earth's_geoid.htm

 

[D H]

Hello.
I am an engineering student and i am attempting to model the gravity around the asteroid Eros by expanding the potential as a spherical harmonic expansion. I got the coefficients (Clm and Slm) from the Planetary Data Model of Eros(NEAR Collected Shape and Gravity Models available at http://sbn.psi.edu/pds/resource/nearbrowse.html).

My potential function had 16 parts. Each part corresponds to a particular value of l. Now each part had again l+1 number of terms.Thus the l=6 part had 7 terms. The l=16 part had 17 terms and so on.

I see several different gravity models at that site:

Model name                    Size    Normalized    Origin=CoM
NEARMOD-NAVGRAV-200204        16x16      Yes            No  
NEARMOD-NAVSHCOEFF-200204     34x34      Yes            No  
NEARMOD-SHCOEFF-200204       180x180     Yes            No  
NEARMOD-N15ACOEFF-200204      15x15      Yes           Yes 
NEARMOD-N393COEFF-200204      16x16      Yes            No

About the columns:

• Model name - the name of the model.
• Size - The size of the model, degree×order of the highest degree/order terms.
• Normalized - Gravity models are almost always published using normalized coefficients nowadays. You need to use the fully normalized associated Legendre polynomials with these. The typical textbook description of spherical harmonics uses (unnormalized) associated Legendre polynomials. Make sure you are using the fully-normalized polynomials, typically designated as $\overline P_{nm}$ rather than the typical textbook (and wikipedia) $P_{nm}$. In particular, do not use a formulation directly based on
  $$P_{nm}(\cos\theta) = (\sin\theta)^m \frac{d^m}{d(\cos\theta)^m}P_n(\cos\theta)$$
• Origin=CoM - Is the origin of the model at the center of mass? If this is the case the dipole terms (degree=1) will be zero. If it is non-zero, the origin of the gravity model frame is offset from the center of mass. You will need to know the offset (some of those offset models supply the offset; some don't).

You mentioned that you have 16 parts, but that the 16th part has 17 terms. You appear to have an off by one error here. I suspect it is the number of parts. You are forgetting the dominant 0th degree term. So you appear to be using either the NAVGRAV or N393COEFF coefficients, both of which have an offset CoM.

I strongly recommend you use the N15ACOEFF coefficients.

Reason #1: This is the only model with the model origin at the center of mass. There are all kinds of contortions you need to go through when the model origin is offset from the CoM.

Reason #2: This is apparently the model described in the paper "A Global Solution for the Gravity Field, Rotation, Landmarks, and Ephemeris of Eros", Konopliv et al. 2001.

Now since the spherical harmonics are an infinite series, the contribution from each part should decrease in magnitude. Example:- contribution from l=6 part should be less than l=5 or l=4 part.

Why would you think that? Look at that lumpy potato:

It's not anything close to spherical. The coefficients (normalized coefficients particularly so) are not going to start dropping until you get to very high degree/order terms.Addendum
The paper "Fast gravity, gravity partials, normalized gravity, gravity gradient torque and magnetic field" (http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19940025085_1994025085.pdf) discusses the mathematics of and a software implementation of a spherical harmonics gravity model. It carefully distinguishes between normalized and unnormalized coefficients and polynomials. There are a few bugs in the code related to the computation of the gravity gradient (the gradient of the force field, a second order tensor), and besides, the code is in Ada.

 

[AdityaNanda]

Thanks a lot BobG and DH.

I realized my mistake. I had been using the unnormalized Legendre polynomials. At high degrees it was leading to unnaturally high values of potential. For example, the potential(per unit mass) at a distance of 32kms from the centre of eros , according to my model, was in the order of e13. G*M/R predicts a value of -13 odd joules.

@DH yes it is an extremely lumpy potato.

Also, I am using NEARMOD-NAVGRAV-200204. Do you think that will create inaccuracies? It is mentioned in the accompanying label that the center of mass is actually offset. You say the the l=1 terms will not be zero and they are not zero in my case. What changes do I have to make (if any) if I plan to continue using NEARMOD-NAVGRAV-200204?

Thanks again!

 

[D H]

Yep. Using unnormalized Legendre polynomials with normalized coefficients will result in garbage. Regarding your use of NEARMOD-NAVGRAV-200204: I guess it should be okay. However, that the origin of the gravity frame is not the center of mass means that all of the coefficients except the degree=order=0 coefficient are artificially inflated by this offset. There is going to be a lot of cancellation and loss of precision because of this offset. I suggest you iterate down to zero, starting with the highest degree/order terms and adding the contribution from the 0,0 term last.