AE-8/AP-8 Questions

[Bob B.]

I've been looking at the AE-8/AP-8 Radiation Belt Models and I have some questions I'm hoping somebody can answer.

► My first question is rather straightforward but I just want to verify that I'm correctly understanding what I'm seeing.  The fluxes appear to for the number of particles with an energy equal to or greater than a given value.  For example, let's say the flux for protons => 10 MeV is 10⁴ protons/cm²-s and the flux for protons => 50 MeV is 10³ protons/cm²-s.  Then am I correct in saying that the number of protons with energies between 10-50 MeV is equal to 10⁴ - 10³ = 9000 /cm²-s?

► The coordinates for particles within the belts are L and B/B_o.  I understand the meaning of L and B/B_o but don't know to transform a known position in polar or rectangular coordinates to L and B/B_o coordinates.

L is pretty straightforward as it is simply radial distance from the center of Earth in Earth radii, which I assume is measured in the geomagnetic equatorial plane and not on a direct line to my known position.  In other words, if I'm 3 radii out from the center of Earth along the equatorial plane and 1 radii above the equatorial plane, then my position is SQRT(3²+1²) = 3.16 radii from Earth's center.  I assume my L coordinate in this case is 3 and not 3.16.  Is this correct?

B/B_o is a much bigger problem for me.  I understand that B/B_o is magnetic field strength normalized to the equatorial value, and I know how to calculate the value of B_o.  What I haven't been able to figure out is how to find the value of B for a given physical location within the magnetic field.  Say, for instance, we're at the previously given location of 3 radii along the equator and 1 radii above the equator.  Surely there is a way to find the field strength, B, at that location.  Is there a formula or some other means by which I can determine this?  Any help provided is greatly appreciated.

[Bob B.]

The following might be what I need regarding the value of B:

http://en.wikipedia.org/wiki/Dipole_model_of_the_Earth's_magnetic_field

I still welcome additional answers/comments.

[JayUtah]

To answer your first question, yes.

[pause for laughter]

Which is to say, the model computes fluxes for the given energy and greater, so if you specify some low energy E₁ and some high energy E₂ in a single run, the flux values you get back for each energy is for that energy and greater.  So yes, the flux for the energy range (E₁,E₂) is found by the subtraction you derived.

The B-L coordinates system makes grown men either cry, drink, or both.  And for any practical purpose, no one uses the web interface or computes the McIlwain coordinates by hand.  In fact, I'm pretty sure I would have to spend a morning with McIlwain's paper again before I could even remember how to do it.  The practical way is to use additional computer programs from NSSDC at Goddard that automate converting from geodetic coordinates or orbital elements to B-L coordinates, and further automate stuffing long sequences of B-L coordinates into the AX8 models, which they embed.

I think Jim Vette wrote most of that code.  NASA/Goddard repackages their software approximately every eight minutes, so I don't know off the top of my head what package it's in these days, or if you have to pay for it.  Let me check.

[Bob B.]

Thanks for your answers, Jay.  If you can find out anything more about that NASA/Goddard software I would be very grateful.

I also happened across the following, which gives equations I might be able to use:

http://onlinelibrary.wiley.com/doi/10.1029/JZ069i023p05089/abstract

Although not defined in the abstract, the previously referenced Wikipedia article defines R as the radius and lambda as the magnetic latitude.  One problem is that the image quality is so poor that I'm having trouble making out some of the text.  I think I'm seeing:

(1)  B = B_o (4 - 3 cos²(lambda))^1/2 / cos^?(lambda)

(2)  R = L cos²(lambda)

(3)  B_o = M/L³

The only number that I really can't make out at all is the exponent for cos(lambda) in the denominator of equation (1), which I show above as a question mark.  Any idea what that is?

[Mag40]

Thanks for your answers, Jay.  If you can find out anything more about that NASA/Goddard software I would be very grateful.

I also happened across the following, which gives equations I might be able to use:

http://onlinelibrary.wiley.com/doi/10.1029/JZ069i023p05089/abstract

Although not defined in the abstract, the previously referenced Wikipedia article defines R as the radius and lambda as the magnetic latitude.  One problem is that the image quality is so poor that I'm having trouble making out some of the text.  I think I'm seeing:

(1)  B = B_o (4 - 3 cos²(lambda))^1/2 / cos^?(lambda)

(2)  R = L cos²(lambda)

(3)  B_o = M/L³

The only number that I really can't make out at all is the exponent for cos(lambda) in the denominator of equation (1), which I show above as a question mark.  Any idea what that is?

I wonder whether this helps?

http://www.spacewx.com/Docs/AIAA-655-543.pdf

[JayUtah]

Thanks for your answers, Jay.  If you can find out anything more about that NASA/Goddard software I would be very grateful.

I also happened across the following, which gives equations I might be able to use:

http://onlinelibrary.wiley.com/doi/10.1029/JZ069i023p05089/abstract

Although not defined in the abstract, the previously referenced Wikipedia article defines R as the radius and lambda as the magnetic latitude.  One problem is that the image quality is so poor that I'm having trouble making out some of the text.  I think I'm seeing:

(1)  B = B_o (4 - 3 cos²(lambda))^1/2 / cos^?(lambda)

(2)  R = L cos²(lambda)

(3)  B_o = M/L³

The only number that I really can't make out at all is the exponent for cos(lambda) in the denominator of equation (1), which I show above as a question mark.  Any idea what that is?

It's almost certainly a 2.  I know NSSDC has code to convert R-λ to B-L, but I can't remember for sure what the core equations are so I can't confirm that yours are they.  However, the subexpression cos² λ is reasonably ubiquitous as a normalization divisor.

This source https://www.spenvis.oma.be/help/background/magfield/rlambda.html gives Eqn (3) as

B = M R - 3 (1+3 sin²λ)^1/2

[JayUtah]

http://www.spacewx.com/Docs/AIAA-655-543.pdf

Eqn 8 seems to be stabbing at the uniform-dipole problem, so ... maybe?

[Bob B.]

I wonder whether this helps?

http://www.spacewx.com/Docs/AIAA-655-543.pdf

It appears that equation 24 in that document is the same as the previously posted equation 1.  Equation 24 shows the exponent that I couldn't read as 6.  It's looks that might be the equation I need to do what I want to do, though I have some more reading to do before I'm convinced.

What I'm wanting to do is to calculate a better estimate of the radiation dose received along the trajectory of Apollo 11 for my web page,

http://www.braeunig.us/apollo/apollo11-TLI.htm

I already have all the R and lambda coordinates from my previous calculations, so all I need are equations to convert them to L and B.

[ajv]

It appears that equation 24 in that document is the same as the previously posted equation 1.  Equation 24 shows the exponent that I couldn't read as 6.

Equation 9 on Jay's linked page is your equation 1 (with a 6 exponent as above), isn't it?

[Bob B.]

This source https://www.spenvis.oma.be/help/background/magfield/rlambda.html gives Eqn (3) as

B = M R^-3 (1+3 sin²λ)^1/2

(above equation edited by me, changing R - 3 to R^-3)

That is almost the same equation given in the Wikipedia article, in which M is replaced by B_o

B = B_o R^-3 (1+3 sin²λ)^1/2

After doing a little rearranging, I currently have two equations,

B/B_o = R^-3 (1+3 sin²λ)^1/2

B/B_o = (4 - 3 cos²λ)^1/2 / cos⁶λ

Of these I'm more inclined to believe that latter.  The reason is because the online AE-8/AP-8 model allows a range of B/B_o values of 1 to 1000.  The first equation doesn't yield values anywhere close to that.  On the other hand, the second equation gives B/B_o = 1000 when λ = 69.3855^o.

[Bob B.]

It appears that equation 24 in that document is the same as the previously posted equation 1.  Equation 24 shows the exponent that I couldn't read as 6.

Equation 9 on Jay's linked page is your equation 1 (with a 6 exponent as above), isn't it?

Yes it is.  Rearranging equation (7) we have,

RL^-1 = cos²λ

Substituting cos²λ for RL^-1 in equation (9) gives,

(BB_o^-1)² (cos²λ)⁶ - (4 - 3 cos²λ) = 0

Which when rearranged gives,

B = B_o (4 - 3 cos²λ)^1/2 / cos⁶λ

I'm starting to feel pretty confident that this is the equation I want to use.

[ajv]

I'm trying to reconcile Bob B's two equations:
B/B_o = R^-3 (1+3 sin²λ)^1/2
B/B_o = (4 - 3 cos²λ)^1/2 / cos⁶λ

Using equations 6 and 8 from Jay's linked page, the first one seems to be missing an L³ factor. If you add it in and rearrange you get the second equation.

[Bob B.]

I'm trying to reconcile Bob B's two equations:
B/B_o = R^-3 (1+3 sin²λ)^1/2
B/B_o = (4 - 3 cos²λ)^1/2 / cos⁶λ

Using equations 6 and 8 from Jay's linked page, the first one seems to be missing an L³ factor. If you add it in and rearrange you get the second equation.

That first equation comes from this Wikipedia article.  Their definition of B₀ is,

"First, define B₀ as the mean value of the magnetic field at the magnetic equator on the Earth's surface. Typically B₀ = 3.12x10^-5 T."

which is not the same definition used in other sources.  What Wikipedia is calling B₀ is defined in other sources as M.  These other sources define B₀ as the value of B along the magnetic equator at a distance L (in radii) from the center of Earth, calculated by B₀ = M/L³.  At Earth's surface, B₀ = M.

Therefore, the first equation is incorrect if we are to be consistent with our definitions.  It should be rewritten as,

B = MR^-3 (1+3 sin²λ)^1/2

which is equation (1) in Jay's link.  Substituting M = B₀L³ we get,

B/B₀ = L³ R^-3 (1+3 sin²λ)^1/2

(edited to add)

Eureka!  That does it.  Although it is not apparent at first observation, the following two equations are indeed equivalent and yield the same answers:

B/B₀ = L³ R^-3 (1+3 sin²λ)^1/2, where  R = L cos²λ

B/B_o = (4 - 3 cos²λ)^1/2 / cos⁶λ

[Bob B.]

Based on our conversations here, below is how I think it's done.  Comments and criticisms are welcomed.  Thanks to everybody who helped.

[JayUtah]

That seems graphically right, Bob.

As for the quantitative stuff, yeah I transcribed those equations completely wrong.  I was late for Comic Con and didn't really have time to delve much further.