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fly-by.fs
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\ fly-by anomaly
\ Bahngleichung im Gravitationsfeld
\ Die antriebslose Bahn eines
\ Körpers mit Masse m im Schwerefeld eines Planeten mit Masse M ist
\ allgemein gegeben durch die Gleichung
\ r(θ)=k/(1 + ε cosθ).
\ r=Abstand zum Planetenzentrum
\ θ=Bahnwinkel
\ k=L²/Gm²M=U²b²/GM (Drehimpulskonstante)
\ L=Bahndrehimpuls des Körpers
\ Bahndrehimpuls: L=mUb, b ist der Abstand zwischen Planet und
\ einlaufender Asymptote der Flugbahn.
\ G=Gravitationskonstante
\ ε=√(1+(2EL²/G²M²m³)) (Exzentrizität) = √(1+(U²b/GM)²)
\ E=0.5mU², kinetische Energie des Körpers,
\ Der Bahntyp wird durch ε festgelegt:
\ ε=0 -> Kreis
\ 0 < ε < 1 -> Ellipse
\ ε=1 -> Parabel
\ ε > 1 -> Hyperbel
include galaxy.fs
332946e FConstant sun-mass \ relative to earth
1.496e11 FConstant sun-earth
sun-mass sun-earth f**2 f/ FConstant bg-scaler
\ all units are SI units, m kg s
FVariable U
FVariable b
FVariable GM 5.9736e24 6.67428e-11 f* GM f! \ earth mass
FVariable epsilon
FVariable k
FVariable phi0 0e phi0 f!
FVariable rho 0e rho f!
12.756e6 f2/ FConstant rearth
: >epsilon ( -- ) U f@ f**2 b f@ f* GM f@ f/ !1 f+ fsqrt
epsilon f! ;
: >k ( -- ) U f@ b f@ f* f**2 GM f@ f/ k f! ;
: orbit ( U b -- ) b f! U f! >k >epsilon ;
: orbit2 ( peri epsilon -- ) fdup epsilon f!
!1 f+ fswap rearth f+ f* k f!
k f@ epsilon f@ f**2 !1 f- f/ b f!
k f@ GM f@ f* fsqrt b f@ f/ U f! ;
: orbit3 180e f/ pi f* rho f! orbit2 ;
: orbit4 180e f/ pi f* phi0 f! orbit3
phi0 f@ rho f@ fsin f/ phi0 f! ;
: r ( phi -- r ) fcos epsilon f@ f* !1 f+ k f@ fswap f/ ;
: range ( -- maxphi ) epsilon f@ 1/f fnegate facos ;
: to-sun ( r -- scale ) f**2
f>r fr@ 1/f bg-scaler f+ fr> f* 1/f ;
\ : to-sun ( r -- scale ) f**2 bg-scaler f* 1/f ;
: diff-a ( r -- a )
sun-earth f- f**2 sun-mass fswap f/
bg-scaler f- GM f@ f* ;
: delta-a ( alpha r -- a ) fdup to-sun f>r
fswap fsin f* diff-a fr> f* ;
: v ( r -- ) GM f@ fswap f/ f2* U f@ f**2 f+ fsqrt ;
\ integration
: phi>pos ( phi -- x y ) fdup r fswap r,phi>xy ;
: phi>pos' ( phi -- x y ) fdup r fswap phi0 f@ f+ r,phi>xy ;
: delta-step ( maxphi stephpi -- delta-a )
fswap fdup r fover phi0 f@ f+ fover delta-a { f: a |
v { f: v |
fswap fover fover
f- phi>pos { f: x1 f: y1 |
f+ phi>pos { f: x2 f: y2 |
x1 x2 f- f**2 y1 y2 f- f**2 f+ fsqrt
v f/ a f*
x2 x1 f- y2 y1 f- fatan2 phi0 f@ f+ fcos f* } } } } ;
: integrate ( n -- result )
range fdup dup 2* fm/ { f: maxphi f: stepphi } 0e
dup negate 1+ DO
I abs I' 1999 2000 */ < IF
maxphi I I' fm*/ stepphi delta-step f+
THEN
LOOP ;
: phis ( n m -- ) \ print mm/s
dup negate 1+ ?DO
pi I I' 1- fm*/ phi0 f!
dup integrate phi0 f@ pi f/ 180e f* f.
( phi0 f@ fcos f* ) 1e3 f* f. cr
LOOP drop ;
\ Examples:
: galileo 959.9e3 2.47e 142.9e 23.35e orbit4 ;
: NEAR 538.8e3 1.81e 108.8e 46.4e orbit4 ;
: cassini 1173e3 5.8e 25.4e 8.95e orbit4 ;
: rosetta 1954e3 1.327e 144.9e 18.55e orbit4 ;
\ considder earth rotation
: set-disc ( -- )
s" " test-disc $!
disc# 1+ elements test-disc $!len test-disc $@ erase
range fdup fnegate fswap disc# 2/ 1+ fm/ { f: maxphi f: stepphi }
disc# 0 ?DO
maxphi stepphi I 1+ fm* f+ phi>pos' fswap
I disc dup element x df!
fdup rho f@ fcos f* dup element y df!
rho f@ fsin f* element z df!
LOOP ;
: .element ( addr -- ) ." ("
dup element x df@ fx.
dup element y df@ fx.
dup element z df@ fx. ." ) <"
dup element ax df@ fx.
dup element ay df@ fx.
dup element az df@ fx. ." > ["
dup element ax+ df@ fx.
dup element ay+ df@ fx.
dup element az+ df@ fx. ." ]" drop cr ;
: disc-a+'xyz ( x y z -- x' y' z' )
0 star star# elements bounds ?DO
I dup a+@ dxyz@abs 1/f vscale v+
I dup -a+@ -dxyz@abs 1/f vscale v+
sizeof element +LOOP ;
: disc-a+' ( -- )
disc# 0 ?DO I disc >xyz
>x df@ >y df@ >z df@ vdup-abs fdup fsqrt f* 1/f
central# fm* vscale
disc-a+'xyz
I disc !1 star# 2* central# + fm/
dup element msum df@ msum+ f@ f+ f/ vscale
dup element az+ df!
dup element ay+ df!
element ax+ df!
pause LOOP ;
bg-scaler msum+ f!
: setups set-disc disc-msum disc-a+' ;
$4000 to star# init-stars set-earth
\ star# 20e fm* f>s to central# set-earth-msum+
\ integrate over precalculated positions
: norm { f: dx f: dy f: dz }
dx f**2 dy f**2 dz f**2 f+ f+ fsqrt 1/f
dx fover f* fswap dy fover f* fswap dz f* ;
: v* { f: x1 f: x2 f: x3 }
x3 f* fswap x2 f* f+ fswap x1 f* f+ ;
[IFUNDEF] vscale
: vscale { f: x f: y f: z f: scale }
x scale f* y scale f* z scale f* ;
[THEN]
: (integrate' ( x end start -- x' ) ?DO
I 1- disc >xyz
I 1+ disc dxyz@
I disc xyz@ fsqsum fsqrt v 1/f vscale
I disc a+@ v* f+
LOOP ;
: integrate' ( -- result )
0e disc# 1- 2 (integrate' ;
: phis' ( m -- )
dup negate 1+ ?DO
pi I I' 1- fm*/ phi0 f! setups
integrate' phi0 f@ pi f/ 180e f* f. f. cr
LOOP ;
: frac-earth ( n -- )
star# swap ?DO
I star
!0 dup element ax df!
!0 dup element ay df!
!0 element az df!
LOOP ;
: run-all cr
." Galileo:" galileo setups integrate' f. cr
." Near: " near setups integrate' f. cr
." Cassini:" cassini setups integrate' f. cr
." Rosetta:" rosetta setups integrate' f. cr ;