THE SDP8 MODEL

The NORAD mean element sets can be used for prediction with SDP8. All symbols not defined below are defined in the list of symbols in Section Twelve. The original mean motion (n''o) and semimajor axis (a''o) are first recovered from the input elements by the equations

#math317#

a1 = #tex2html_wrap_indisplay4547##tex2html_wrap_indisplay4548##tex2html_wrap_indisplay4549#

#math318#

δ1 = #tex2html_wrap_indisplay4551##tex2html_wrap_indisplay4552##tex2html_wrap_indisplay4553#

#math319#

ao = a1#tex2html_wrap_indisplay4555#1 - #tex2html_wrap_indisplay4556#δ1 - δ12 - #tex2html_wrap_indisplay4559#δ13#tex2html_wrap_indisplay4562#

#math320#

δo = #tex2html_wrap_indisplay4564##tex2html_wrap_indisplay4565##tex2html_wrap_indisplay4566#

#math321#

n''o = #tex2html_wrap_indisplay4568#

#math322#

a''o = #tex2html_wrap_indisplay4570#.

The ballistic coefficient (B term) is then calculated from the B* drag term by

#math323#

B = 2B*/ρo

where

#math324#

ρo = (2.461×10-5)#tex2html_wrap_indisplay4575#

is a reference value of atmospheric density.

Then calculate the constants

#math325#

β2 = 1 - e2

#math326#

θ = cos i

#math327#

#tex2html_wrap_indisplay4579# = - #tex2html_wrap_indisplay4580##tex2html_wrap_indisplay4581#(1 - 3θ2)

#math328#

#tex2html_wrap_indisplay4583# = - #tex2html_wrap_indisplay4584##tex2html_wrap_indisplay4585#(1 - 5θ2)

#math329#

#tex2html_wrap_indisplay4587# = - 3#tex2html_wrap_indisplay4588#θ

#math330#

#tex2html_wrap_indisplay4590# = #tex2html_wrap_indisplay4591##tex2html_wrap_indisplay4592#(13 - 78θ2 +137θ4)

#math331#

#tex2html_wrap_indisplay4594# = #tex2html_wrap_indisplay4595##tex2html_wrap_indisplay4596#(7 - 114θ2 +395θ4) + #tex2html_wrap_indisplay4597##tex2html_wrap_indisplay4598#(3 - 36θ2 +49θ4)

#math332#

#tex2html_wrap_indisplay4600# = #tex2html_wrap_indisplay4601##tex2html_wrap_indisplay4602#θ(4 - 19θ2) + #tex2html_wrap_indisplay4603##tex2html_wrap_indisplay4604#θ(3 - 7θ2)

#math333#

#tex2html_wrap_indisplay4606# = n''o + #tex2html_wrap_indisplay4607# + #tex2html_wrap_indisplay4608#

#math334#

#tex2html_wrap_indisplay4610# = #tex2html_wrap_indisplay4611# + #tex2html_wrap_indisplay4612#

#math335#

#tex2html_wrap_indisplay4614# = #tex2html_wrap_indisplay4615# + #tex2html_wrap_indisplay4616#

#math336#

ξ = #tex2html_wrap_indisplay4618#

#math337#

η = esξ

#math338#

ψ = #tex2html_wrap_indisplay4621#

#math339#

α2 = 1 + e2

#math340#

Co = #tex2html_wrap_indisplay4624#Bρo(qo - s)4n''a''ξ4α-1ψ-7

#math341#

C1 = #tex2html_wrap_indisplay4626#n''α4Co

#math342#

D1 = ξψ-2/a''β2

#math343#

D2 = 12 + 36η2 + #tex2html_wrap_indisplay4629#η4

#math344#

D3 = 15η2 + #tex2html_wrap_indisplay4631#η4

#math345#

D4 = 5η + #tex2html_wrap_indisplay4633#η3

#math346#

D5 = ξψ-2

#math347#

B1 = - k2(1 - 3θ2)

#math348#

B2 = - k2(1 - θ2)

#math349#

B3 = #tex2html_wrap_indisplay4638#sin i

#math350#

C2 = D1D3B2

#math351#

C3 = D4D5B3

#math352#

#tex2html_wrap_indisplay4642# = C1#tex2html_wrap_indisplay4643#2 + 3η2 +20eη +5eη3 + #tex2html_wrap_indisplay4644#e2 +34e2η2 + D1D2B1 + C2cos 2ω + C3sinω#tex2html_wrap_indisplay4645#

#math353#

#tex2html_wrap_indisplay4647# = - #tex2html_wrap_indisplay4648##tex2html_wrap_indisplay4649#(1 - e)

where all quantities are epoch values.

At this point SDP8 calls the initialization section of DEEP which calculates all initialized quantities needed for the deep-space perturbations (see Section Ten).

The secular effect of gravity is included in mean anomaly by

#math354#

MDF = Mo + #tex2html_wrap_indisplay4651#(t - to)

and the secular effects of gravity and atmospheric drag are included in argument of perigee and longitude of ascending node by

#math355#

ω = ωo + #tex2html_wrap_indisplay4653#(t - to) + #tex2html_wrap_indisplay4654#Z7

#math356#

Ω = Ωo + #tex2html_wrap_indisplay4656#(t - to) + #tex2html_wrap_indisplay4657#Z7

where

#math357#

Z7 = #tex2html_wrap_indisplay4659#Z1/n''o

with

#math358#

Z1 = #tex2html_wrap_indisplay4661##tex2html_wrap_indisplay4662#(t - to)2.

Next, SDP8 calls the secular section of DEEP which adds the deep-space secular effects and long-period resonance effects to the six classical orbital elements (see Section Ten).

The secular effects of drag are included in the remaining elements by

#math359#

n = nDS + #tex2html_wrap_indisplay4664#(t - to)

#math360#

e = eDS + #tex2html_wrap_indisplay4666#(t - to)

#math361#

M = MDS + Z1 + #tex2html_wrap_indisplay4668#Z7

where nDS, eDS, MDS are the values of no, eo, MDF after deep-space secular and resonance perturbations have been applied.

Here, SDP8 calls the periodics section of DEEP which adds the deep-space lunar and solar periodics to the orbital elements (see Section Ten). From this point on, it will be assumed that n, e, I, ω, Ω, and M are the mean motion, eccentricity, inclination, argument of perigee, longitude of ascending node, and mean anomaly after lunar-solar periodics have been added.

Solve Kepler's equation for E by using the iteration equation

#math362#

Ei+1 = Ei + ΔEi

with

#math363#

ΔEi = #tex2html_wrap_indisplay4684#

and

#math364#

E1 = M + e sin M + #tex2html_wrap_indisplay4686#e2sin 2M.

The following equations are used to calculate preliminary quantities needed for the short-period periodics.

#math365#

a = #tex2html_wrap_indisplay4688##tex2html_wrap_indisplay4689##tex2html_wrap_indisplay4690#

#math366#

β = (1 - e2)#tex2html_wrap_indisplay4692#

#math367#

sin f = #tex2html_wrap_indisplay4694#

#math368#

cos f = #tex2html_wrap_indisplay4696#

#math369#

u = f + ω

#math370#

r'' = #tex2html_wrap_indisplay4699#

#math371#

#tex2html_wrap_indisplay4701# = #tex2html_wrap_indisplay4702#sin f

#math372#

(r#tex2html_wrap_indisplay4704#)'' = #tex2html_wrap_indisplay4705#

#math373#

δr = #tex2html_wrap_indisplay4707##tex2html_wrap_indisplay4708#[(1 - θ2)cos 2u + 3(1 - 3θ2)] - #tex2html_wrap_indisplay4709##tex2html_wrap_indisplay4710#sin iosin u

#math374#

δ#tex2html_wrap_indisplay4712# = - n#tex2html_wrap_indisplay4713##tex2html_wrap_indisplay4714##tex2html_wrap_indisplay4715##tex2html_wrap_indisplay4716##tex2html_wrap_indisplay4717#(1 - θ2)sin 2u + #tex2html_wrap_indisplay4718##tex2html_wrap_indisplay4719#sin iocos u#tex2html_wrap_indisplay4720#

#math375#

δI = θ#tex2html_wrap_indisplay4722##tex2html_wrap_indisplay4723##tex2html_wrap_indisplay4724#sin iocos 2u - #tex2html_wrap_indisplay4725##tex2html_wrap_indisplay4726#e sinω#tex2html_wrap_indisplay4727#

#math376#

δ(r#tex2html_wrap_indisplay4729#) = - n#tex2html_wrap_indisplay4730##tex2html_wrap_indisplay4731##tex2html_wrap_indisplay4732#δr + na#tex2html_wrap_indisplay4733##tex2html_wrap_indisplay4734##tex2html_wrap_indisplay4735##tex2html_wrap_indisplay4736#δI

#math377#

δu = #tex2html_wrap_indisplay4738#

#math378#

δλ = #tex2html_wrap_indisplay4740#

The short-period periodics are added to give the osculating quantities

#math379#

r = r'' + δr

#math380#

#tex2html_wrap_indisplay4743# = #tex2html_wrap_indisplay4744# + δ#tex2html_wrap_indisplay4745#

#math381#

r#tex2html_wrap_indisplay4747# = (r#tex2html_wrap_indisplay4748#)'' + δ(r#tex2html_wrap_indisplay4749#)

#math382#

y4 = sin#tex2html_wrap_indisplay4751#sin u + cos u sin#tex2html_wrap_indisplay4752#δu + #tex2html_wrap_indisplay4753#sin u cos#tex2html_wrap_indisplay4754#δI

#math383#

y5 = sin#tex2html_wrap_indisplay4756#cos u - sin u sin#tex2html_wrap_indisplay4757#δu + #tex2html_wrap_indisplay4758#cos u cos#tex2html_wrap_indisplay4759#δI

#math384#

λ = u + Ω + δλ.

Unit orientation vectors are calculated by

#math385#

Ux = 2y4(y5sinλ - y4cosλ) + cosλ

#math386#

Uy = - 2y4(y5cosλ + y4sinλ) + sinλ

#math387#

Uz = 2y4cos#tex2html_wrap_indisplay4764#

#math388#

Vx = 2y5(y5sinλ - y4cosλ) - sinλ

#math389#

Vy = - 2y5(y5cosλ + y4sinλ) + cosλ

#math390#

Vz = 2y5cos#tex2html_wrap_indisplay4768#

where

#math391#

cos#tex2html_wrap_indisplay4770# = #tex2html_wrap_indisplay4771#.

Position and velocity are given by

#math392#

#tex2html_wrap_indisplay4773# = r#tex2html_wrap_indisplay4774#

#math393#

#tex2html_wrap_indisplay4776# = #tex2html_wrap_indisplay4777##tex2html_wrap_indisplay4778# + r#tex2html_wrap_indisplay4779##tex2html_wrap_indisplay4780#.

A FORTRAN IV computer code listing of the subroutine SDP8 is given below. #center4781#