Inhaltsverzeichnis
Bahnelemente
Die Bahnelemente der Planeten beziehen sich auf die Epoche $J2000$, d.h. den 1.1.2000 12:00 $UT$.
Es gilt:
$$d = JD - 2451545.0\tag{1}$$
$d$ = Tage vor/nach der Epoche $J2000$, in Dezimaldarstellung.
Erläuterung
$a$ = große Halbachse = $\overline{ZP} = \overline{ZA}$
$e$ = lineare Exzentrizität = $\overline{ZS}$
$i$ = Bahnneigung gegen die Ekliptik (Inklination)
$\Omega$ = Länge des aufsteigenden Knotens (= Winkel in der Ekliptikebene vom Frühlingspunkt 
an gemessen)
$u = \omega + \nu$ = Argument der Breite (= Winkel in der Bahnebene vom aufsteigenden Knoten bis zum Objekt)
$\omega$ = Argument des Perihels (= Winkel in der Bahnebene vom aufsteigenden Knoten an gemessen)
$\nu$ = wahre Anomalie (= Winkel vom Perihel bis zum Objekt)
$\epsilon = \frac{e}{a}$ = numerische Exzentrizität
$P$ = Perihel (sonnennächster Punkt)
$A$ = Aphel (sonnenfernster Punkt)
$S$ = Sonne (Brennpunkt, Fokus)
$O$ = Objekt (Planet, Komet, etc.)
$r$ = Radiusvektor = $\overline{SO}$
$q$ = Periheldistanz = $\overline{SP}$
= Richtung zum Frühlingspunkt
☊ = aufsteigender Knoten (Übergang Süd → Nord)
☋ = absteigender Knoten (Übergang Nord → Süd)
= Frühlingspunkt
$N$ = ☊ = Aufsteigender Knoten
$P$ = Perihel (sonnennächster Punkt)
$O$ = mittlere Position des Objekts
$O'$ = wahre Position des Objekts
$O''$ = Projektion des wahren Objekts in die Ekliptikebene
$\Omega$ = Länge des aufsteigenden Knotens = Winkel → ☊
$\omega$ = Argument des Perihels, Winkel ☊ → $P$
$\varpi = \Omega + \omega$ = Länge des Perihels, Winkel → $P$ (in verschiedenen Ebenen)
$L = \Omega + \omega + M$ = mittlere Länge des Objekts
$M$ = Winkel $P$ → $O$, mittlere Anomalie des Objekts
$C$ = Winkel $O$ → $O'$, Mittelpunktsgleichung
$\nu = M + C$ = wahre Anomalie des Objekts
$i$ = Inklination, Bahnneigung gegen die Ekliptik
Die mittleren Bahnelemente I (IMCCE)
Die Bahnelemente beziehen sich auf eine mittlere dynamische Ekliptik und ein mittleres Äquinoktium des Datums. Ferner bezieht sich die Epoche auf den 01. Januar 2000 um $12^{h}$ dynamischer Weltzeit ($J2000.0 = JDE 2451545.0$). Sie sind im Zeitraum von 1000 n.Chr. bis 3000 n.Chr. gültig. Die mittleren Bahnelemente wurden J.L. Simon, P. Bretagnon & al. entnommen und stammen vom Institut de mécanique céleste et de calcul des éphémérides.
| Tabelle 1 | |
|---|---|
| Merkur | |
| $a_1=$ | $0.3870983098\text{ AE}$ |
| $\epsilon_1=$ | $0.2056317526 + 2.040653\cdot 10^{-5}\cdot T - 2.8349\cdot 10^{-8}\cdot T^2$ |
| $- 1.805\cdot 10^{-10}\cdot T^3 + 2.3\cdot 10^{-13}\cdot T^4 - 2.0\cdot 10^{-15}\cdot T^5$ | |
| $i_1=$ | $7\overset{\circ}{.}00498625 + (+ 6\overset{''}{.}557301\cdot T - 6\overset{''}{.}51516\cdot 10^{-2}\cdot T^2$ |
| $+ 2\overset{''}{.}0113\cdot 10^{-4}\cdot T^3 + 1\overset{''}{.}9\cdot 10^{-8}\cdot T^4 - 1\overset{''}{.}9\cdot 10^{-9}\cdot T^5)/3600''$ | |
| $\Omega_1=$ | $48\overset{\circ}{.}33089304 + (+ 4270\overset{''}{.}001444\cdot T + 6\overset{''}{.}314994\cdot 10^{-1}\cdot T^2 + 7\overset{''}{.}7259\cdot 10^{-4}\cdot T^3$ |
| $- 2\overset{''}{.}0893\cdot 10^{-5}\cdot T^4 - 2\overset{''}{.}19\cdot 10^{-8}\cdot T^5 + 1\overset{''}{.}6\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $\omega_1=$ | $29.125226 + (+ 1333\overset{''}{.}041201\cdot T + 4\overset{''}{.}320722\cdot 10^{-1}\cdot T^2 - 7\overset{''}{.}3841\cdot 10^{-4}\cdot T^3$ |
| $- 2\overset{''}{.}623\cdot 10^{-6}\cdot T^4 + 4\overset{''}{.}3\cdot 10^{-9}\cdot T^5 + 4\overset{''}{.}0\cdot 10^{-11}\cdot T^6)/3600''$ | |
| $\varpi_1=$ | $77\overset{\circ}{.}45611904 + 1\overset{\circ}{.}556400735\cdot T +(+ 1\overset{''}{.}0635716\cdot T^2 + 3\overset{''}{.}418\cdot 10^{-5}\cdot T^3$ |
| $- 2\overset{''}{.}3516\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}76\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $M_1=$ | $174\overset{\circ}{.}79478648 + 149400\overset{\circ}{.}0\cdot T + 72\overset{\circ}{.}51577147\cdot T +(+ 2\overset{''}{.}90227\cdot 10^{-2}\cdot T^2$ |
| $+ 3\overset{''}{.}104\cdot 10^{-5}\cdot T^3 + 1\overset{''}{.}6\cdot 10^{-8}\cdot T^4 - 3\overset{''}{.}0\cdot 10^{-10}\cdot T^5)/3600''$ | |
| $L_1=$ | $252\overset{\circ}{.}25090552 + 149400\overset{\circ}{.}0\cdot T + 74\overset{\circ}{.}07217222\cdot T +(+ 1\overset{''}{.}0925943\cdot T^2$ |
| $+ 6\overset{''}{.}522\cdot 10^{-5}\cdot T^3 - 2\overset{''}{.}35\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}79\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $n_{1.m}=$ | $4\overset{\circ}{.}092334449596106 + \text{höhere Terme von } M_1$ |
| $n_{1.l}=$ | $4\overset{\circ}{.}092377061525872 + \text{höhere Terme von } L_1$ |
| Venus | |
| $a_2=$ | $0.7233298200\text{ AE}$ |
| $\epsilon_2=$ | $0.0067719164 - 4.776521\cdot 10^{-5}\cdot T + 9.8127\cdot 10^{-8}\cdot T^2$ |
| $+ 4.639\cdot 10^{-10}\cdot T^3 + 1.23\cdot 10^{-12}\cdot T^4 - 3.0\cdot 10^{-15}\cdot T^5$ | |
| $i_2=$ | $3\overset{\circ}{.}39466189 + (+ 3\overset{''}{.}613261\cdot T - 3\overset{''}{.}1523\cdot 10^{-3}\cdot T^2$ |
| $- 2\overset{''}{.}525\cdot 10^{-5}\cdot T^3 + 8\overset{''}{.}5\cdot 10^{-8}\cdot T^4 - 8\overset{''}{.}0\cdot 10^{-10}\cdot T^5)/3600''$ | |
| $\Omega_2=$ | $76\overset{\circ}{.}67992019 + (+ 3243\overset{''}{.}757636\cdot T + 1\overset{''}{.}4622586\cdot T^2 - 3\overset{''}{.}3446\cdot 10^{-4}\cdot T^3$ |
| $- 2\overset{''}{.}3007\cdot 10^{-5}\cdot T^4 - 8\overset{''}{.}8\cdot 10^{-9}\cdot T^5 + 9\overset{''}{.}0\cdot 10^{-11}\cdot T^6)/3600''$ | |
| $\omega_2=$ | $54\overset{\circ}{.}88378281 + (+ 1803\overset{''}{.}989445\cdot T - 5\overset{''}{.}3365131\cdot T^2 - 2\overset{''}{.}010602\cdot 10^{-2}\cdot T^3$ |
| $- 7\overset{''}{.}2941\cdot 10^{-5}\cdot T^4 + 1\overset{''}{.}32\cdot 10^{-8}\cdot T^5 + 1\overset{''}{.}1\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $\varpi_2=$ | $131\overset{\circ}{.}56370300 + 1\overset{\circ}{.}402151967\cdot T +(- 3\overset{''}{.}8742545\cdot T^2 - 2\overset{''}{.}044048\cdot 10^{-2}\cdot T^3$ |
| $- 9\overset{''}{.}5948\cdot 10^{-5}\cdot T^4 + 4\overset{''}{.}4\cdot 10^{-9}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $M_2=$ | $50\overset{\circ}{.}41609785 + 58320\overset{\circ}{.}0\cdot T + 197\overset{\circ}{.}8108014\cdot T +(+ 4\overset{''}{.}9907566\cdot T^2$ |
| $+ 2\overset{''}{.}049416\cdot 10^{-2}\cdot T^3 + 7\overset{''}{.}2432\cdot 10^{-5}\cdot T^4 - 2\overset{''}{.}23\cdot 10^{-8}\cdot T^5)/3600''$ | |
| $L_2=$ | $181\overset{\circ}{.}97980085 + 58320\overset{\circ}{.}0\cdot T + 199\overset{\circ}{.}2129533\cdot T +(+ 1\overset{''}{.}1165021\cdot T^2$ |
| $+ 5\overset{''}{.}368\cdot 10^{-5}\cdot T^3 - 2\overset{''}{.}3516\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}79\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $n_{2.m}=$ | $1\overset{\circ}{.}6021303436376 + \text{höhere Terme von } M_2$ |
| $n_{2.l}=$ | $1\overset{\circ}{.}602168732466264 + \text{höhere Terme von } L_2$ |
| Erde | |
| $a_3=$ | $1.0000010178\text{ AE}$ |
| $\epsilon_3=$ | $0.0167086342 - 4.203654\cdot 10^{-5}\cdot T - 1.26734\cdot 10^{-7}\cdot T^2$ |
| $+ 1.444\cdot 10^{-10}\cdot T^3 - 2.0\cdot 10^{-14}\cdot T^4 + 3.0\cdot 10^{-15}\cdot T^5$ | |
| $i_3=$ | $\Omega_3 = 0\overset{\circ}{.}0$ |
| $\omega_3=$ | $L_3$ |
| $\varpi_3=$ | $102\overset{\circ}{.}93734808 + 1\overset{\circ}{.}719459803\cdot T +(- 1\overset{''}{.}6447797\cdot T^2 - 6\overset{''}{.}365\cdot 10^{-5}\cdot T^3$ |
| $- 1\overset{''}{.}209\cdot 10^{-5}\cdot T^4 + 2\overset{''}{.}98\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $M_3=$ | $357\overset{\circ}{.}52920875 + 35640\overset{\circ}{.}0\cdot T + 359\overset{\circ}{.}0502911\cdot T +(- 5\overset{''}{.}531988\cdot 10^{-1}\cdot T^2$ |
| $+ 1\overset{''}{.}3572\cdot 10^{-4}\cdot T^3 - 1\overset{''}{.}144\cdot 10^{-5}\cdot T^4 - 4\overset{''}{.}78\cdot 10^{-8}\cdot T^5)/3600''$ | |
| $L_3=$ | $100\overset{\circ}{.}46645683 + 36000\overset{\circ}{.}0\cdot T + 0\overset{\circ}{.}769750952\cdot T +(+ 1\overset{''}{.}0915809\cdot T^2$ |
| $+ 7\overset{''}{.}207\cdot 10^{-5}\cdot T^3 - 2\overset{''}{.}353\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}8\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $n_{3.m}=$ | $0\overset{\circ}{.}9856002817563237 + \text{höhere Terme von } M_3$ |
| $n_{3.l}=$ | $0\overset{\circ}{.}985647358 + \text{höhere Terme von } L_3$ |
| Mars | |
| $a_4=$ | $1.5236793419\text{ AE} + 3.0\cdot 10^{-11}\text{ AE}\cdot T$ |
| $\epsilon_4=$ | $0.0934006477 + 9.048438\cdot 10^{-5}\cdot T - 8.0641\cdot 10^{-8}\cdot T^2$ |
| $- 2.519\cdot 10^{-10}\cdot T^3 + 1.24\cdot 10^{-12}\cdot T^4 - 1.0\cdot 10^{-14}\cdot T^5$ | |
| $i_4=$ | $1\overset{\circ}{.}84972648 + (- 2\overset{''}{.}163885\cdot T + 4\overset{''}{.}5935\cdot 10^{-2}\cdot T^2 - 2\overset{''}{.}376\cdot 10^{-5}\cdot T^3$ |
| $- 1\overset{''}{.}708\cdot 10^{-6}\cdot T^4 + 6\overset{''}{.}5\cdot 10^{-9}\cdot T^5 + 5\overset{''}{.}0\cdot 10^{-11}\cdot T^6)/3600''$ | |
| $\Omega_4=$ | $49\overset{\circ}{.}55809321 + (+ 2779\overset{''}{.}268736\cdot T + 5\overset{''}{.}60611\cdot 10^{-2}\cdot T^2 + 8\overset{''}{.}16222\cdot 10^{-3}\cdot T^3$ |
| $- 4\overset{''}{.}5709\cdot 10^{-5}\cdot T^4 - 4\overset{''}{.}722\cdot 10^{-7}\cdot T^5 + 4\overset{''}{.}35\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $\omega_4=$ | $286\overset{\circ}{.}50214074 + (+ 3848\overset{''}{.}216254\cdot T + 4\overset{''}{.}290999\cdot 10^{-1}\cdot T^2 - 6\overset{''}{.}23091\cdot 10^{-3}\cdot T^3$ |
| $+ 1\overset{''}{.}7591\cdot 10^{-5}\cdot T^4 + 4\overset{''}{.}378\cdot 10^{-7}\cdot T^5 - 4\overset{''}{.}15\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $\varpi_4=$ | $336\overset{\circ}{.}06023395 + 1\overset{\circ}{.}840968053\cdot T +(+ 4\overset{''}{.}85161\cdot 10^{-1}\cdot T^2 + 1\overset{''}{.}93131\cdot 10^{-3}\cdot T^3$ |
| $- 2\overset{''}{.}8118\cdot 10^{-5}\cdot T^4 - 3\overset{''}{.}44\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $M_4=$ | $19\overset{\circ}{.}373041 + 19080\overset{\circ}{.}0\cdot T + 59\overset{\circ}{.}85540224\cdot T +(+ 0\overset{''}{.}6327064\cdot T^2$ |
| $- 1\overset{''}{.}87507\cdot 10^{-3}\cdot T^3 + 4\overset{''}{.}602\cdot 10^{-6}\cdot T^4 + 1\overset{''}{.}64\cdot 10^{-8}\cdot T^5)/3600''$ | |
| $L_4=$ | $355\overset{\circ}{.}43299958 + 19080\overset{\circ}{.}0\cdot T + 61\overset{\circ}{.}6963703\cdot T +(+ 1\overset{''}{.}1178674\cdot T^2$ |
| $+ 5\overset{''}{.}624\cdot 10^{-5}\cdot T^3 - 2\overset{''}{.}3516\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}8\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $n_{4.m}=$ | $0\overset{\circ}{.}5240206817862879 + \text{höhere Terme von } M_4$ |
| $n_{4.l}=$ | $0\overset{\circ}{.}5240710847446117 + \text{höhere Terme von } L_4$ |
| Jupiter | |
| $a_5=$ | $5.2026032092\text{ AE} + 1.9132\cdot 10^{-7}\text{ AE}\cdot T - 3.9\cdot 10^{-11}\text{ AE}\cdot T^2$ |
| $- 6.0\cdot 10^{-12}\text{ AE}\cdot T^3 - 1.0\cdot 10^{-13}\text{ AE}\cdot T^4 + 1.0\cdot 10^{-15}\text{ AE}\cdot T^5$ | |
| $\epsilon_5=$ | $0.0484979255 + 1.6322542\cdot 10^{-4}\cdot T - 4.71366\cdot 10^{-7}\cdot T^2 - 2.0063\cdot 10^{-9}\cdot T^3$ |
| $+ 1.018\cdot 10^{-11}\cdot T^4 - 2.1\cdot 10^{-14}\cdot T^5 + 1.0\cdot 10^{-16}\cdot T^6$ | |
| $i_5=$ | $1\overset{\circ}{.}30326698 + (- 19\overset{''}{.}787442\cdot T + 1\overset{''}{.}67744\cdot 10^{-2}\cdot T^2 - 8\overset{''}{.}38\cdot 10^{-6}\cdot T^3$ |
| $- 7\overset{''}{.}35\cdot 10^{-7}\cdot T^4 + 8\overset{''}{.}5\cdot 10^{-9}\cdot T^5 + 4\overset{''}{.}0\cdot 10^{-11}\cdot T^6)/3600''$ | |
| $\Omega_5=$ | $100\overset{\circ}{.}46440702 + (+ 3675\overset{''}{.}518747\cdot T + 1\overset{''}{.}4513295\cdot T^2 + 1\overset{''}{.}45556\cdot 10^{-3}\cdot T^3$ |
| $- 5\overset{''}{.}9609\cdot 10^{-5}\cdot T^4 - 4\overset{''}{.}324\cdot 10^{-7}\cdot T^5 + 1\overset{''}{.}75\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $\omega_5=$ | $273\overset{\circ}{.}86679985 + (+ 2129\overset{''}{.}967878\cdot T + 2\overset{''}{.}2581721\cdot T^2 - 1\overset{''}{.}752666\cdot 10^{-2}\cdot T^3$ |
| $+ 1\overset{''}{.}.10795\cdot 10^{-4}\cdot T^4 + 2\overset{''}{.}056\cdot 10^{-7}\cdot T^5 - 1\overset{''}{.}71\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $\varpi_5=$ | $14\overset{\circ}{.}33120687 + 1\overset{\circ}{.}612635174\cdot T +(+ 3\overset{''}{.}7095016\cdot T^2 - 1\overset{''}{.}60711\cdot 10^{-2}\cdot T^3$ |
| $+ 5\overset{''}{.}1186\cdot 10^{-5}\cdot T^4 -2\overset{''}{.}268\cdot 10^{-7}\cdot T^5 + 4\overset{''}{.}0\cdot 10^{-11}\cdot T^6)/3600''$ | |
| $M_5=$ | $20\overset{\circ}{.}02031187 + 2880\overset{\circ}{.}0\cdot T + 154\overset{\circ}{.}6901397\cdot T +(- 2\overset{''}{.}9056316\cdot T^2$ |
| $+ 1\overset{''}{.}620437\cdot 10^{-2}\cdot T^3 - 7\overset{''}{.}0036\cdot 10^{-5}\cdot T^4 + 2\overset{''}{.}679\cdot 10^{-7}\cdot T^5 - 1\overset{''}{.}8\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $L_5=$ | $34\overset{\circ}{.}35151874 + 2880\overset{\circ}{.}0\cdot T + 156\overset{\circ}{.}3027748\cdot T +(+ 8\overset{''}{.}0387\cdot 10^{-1}\cdot T^2$ |
| $+ 1\overset{''}{.}3327\cdot 10^{-4}\cdot T^3 - 1\overset{''}{.}885\cdot 10^{-5}\cdot T^4 + 4\overset{''}{.}11\cdot 10^{-8}\cdot T^5 - 1\overset{''}{.}4\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $n_{5.m}=$ | $0\overset{\circ}{.}08308528787609704 + \text{höhere Terme von } M_5$ |
| $n_{5.l}=$ | $0\overset{\circ}{.}08312943942089132 + \text{höhere Terme von } L_5$ |
| Saturn | |
| $a_6=$ | $9.5549091915\text{ AE} - 2.13896\cdot 10^{-6}\text{ AE}\cdot T + 4.44\cdot 10^{-10}\text{ AE}\cdot T^2 + 6.7\cdot 10^{-11}\text{ AE}\cdot T^3$ |
| $+ 1.1\cdot 10^{-12}\text{ AE}\cdot T^4 - 7.0\cdot 10^{-15}\text{ AE}\cdot T^5 - 1.0\cdot 10^{-16}\text{ AE}\cdot T^6$ | |
| $\epsilon_6=$ | $0.0555481426 - 3.4664062\cdot 10^{-4}\cdot T - 6.43639\cdot 10^{-7}\cdot T^2 + 3.3956\cdot 10^{-9}\cdot T^3$ |
| $- 2.19\cdot 10^{-12}\cdot T^4 - 3.0\cdot 10^{-15}\cdot T^5 + 6.0\cdot 10^{-16}\cdot T^6$ | |
| $i_6=$ | $2\overset{\circ}{.}48887878 + (- 1\overset{''}{.}3450388\cdot T - 5\overset{''}{.}468\cdot 10^{-2}\cdot T^2 + 3\overset{''}{.}1168\cdot 10^{-4}\cdot T^3$ |
| $+ 3\overset{''}{.}207\cdot 10^{-6}\cdot T^4 - 2\overset{''}{.}37\cdot 10^{-8}\cdot T^5 - 2\overset{''}{.}3\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $\Omega_6=$ | $113\overset{\circ}{.}66550252 + (+ 3157\overset{''}{.}516875\cdot T - 4\overset{''}{.}383321\cdot 10^{-1}\cdot T^2 - 8\overset{''}{.}09520\cdot 10^{-3}\cdot T^3$ |
| $+ 1\overset{''}{.}8433\cdot 10^{-5}\cdot T^4 + 6\overset{''}{.}867\cdot 10^{-7}\cdot T^5 - 2\overset{''}{.}76\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $\omega_6=$ | $339\overset{\circ}{.}39173496 + (+ 3912\overset{''}{.}02387\cdot T + 3\overset{''}{.}4534476\cdot T^2 + 2\overset{''}{.}583694\cdot 10^{-2}\cdot T^3$ |
| $+ 8\overset{''}{.}1195\cdot 10^{-5}\cdot T^4 + 3\overset{''}{.}265\cdot 10^{-7}\cdot T^5 + 9\overset{''}{.}98\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $\varpi_6=$ | $93\overset{\circ}{.}05723748 + 1\overset{\circ}{.}963761318\cdot T +(+ 3\overset{''}{.}0151155\cdot T^2 + 1\overset{''}{.}774174\cdot 10^{-2}\cdot T^3$ |
| $+ 9\overset{''}{.}9628\cdot 10^{-5}\cdot T^4 + 1\overset{''}{.}0132\cdot 10^{-6}\cdot T^5 + 7\overset{''}{.}22\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $M_6=$ | $317\overset{\circ}{.}02020682 + 1080\overset{\circ}{.}0\cdot T + 141\overset{\circ}{.}5473073\cdot T +(- 1\overset{''}{.}1464338\cdot T^2$ |
| $- 1\overset{''}{.}784922\cdot 10^{-2}\cdot T^3 - 1\overset{''}{.}34632\cdot 10^{-4}\cdot T^4 - 1\overset{''}{.}1762\cdot 10^{-6}\cdot T^5 - 6\overset{''}{.}19\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $L_6=$ | $50\overset{\circ}{.}07744430 + 1080\overset{\circ}{.}0\cdot T + 143\overset{\circ}{.}5110686\cdot T +(+ 1\overset{''}{.}8686817\cdot T^2$ |
| $- 1\overset{''}{.}0748\cdot 10^{-4}\cdot T^3 - 3\overset{''}{.}5004\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}63\cdot 10^{-7}\cdot T^5 + 1\overset{''}{.}03\cdot 10^{-9}\cdot T^6)/3600''$ | |
| $n_{6.m}=$ | $0\overset{\circ}{.}03344429467102442 + \text{höhere Terme von } M_6$ |
| $n_{6.l}=$ | $0\overset{\circ}{.}03349790742290669 + \text{höhere Terme von } L_6$ |
| Uranus | |
| $a_7=$ | $19.2184460618\text{ AE} - 3.716\cdot 10^{-8}\text{ AE}\cdot T + 9.79\cdot 10^{-10}\text{ AE}\cdot T^2$ |
| $\epsilon_7=$ | $0.0463812221 - 2.729293\cdot 10^{-5}\cdot T + 7.8913\cdot 10^{-8}\cdot T^2 + 2.447\cdot 10^{-10}\cdot T^3$ |
| $- 1.71\cdot 10^{-12}\cdot T^4$ | |
| $i_7=$ | $0\overset{\circ}{.}77319689 + (+ 2\overset{''}{.}787845\cdot T + 0\overset{''}{.}1349529\cdot T^2 - 3\overset{''}{.}3095\cdot 10^{-4}\cdot T^3$ |
| $- 3\overset{''}{.}444\cdot 10^{-6}\cdot T^4 + 1\overset{''}{.}71\cdot 10^{-8}\cdot T^5 + 1\overset{''}{.}2\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $\Omega_7=$ | $74\overset{\circ}{.}00595701 + (+ 1876\overset{''}{.}059902\cdot T - 4\overset{''}{.}8221068\cdot T^2 + 6\overset{''}{.}654269\cdot 10^{-2}\cdot T^3$ |
| $- 3\overset{''}{.}5249\cdot 10^{-4}\cdot T^4 - 3\overset{''}{.}2819\cdot 10^{-6}\cdot T^5 + 3\overset{''}{.}056\cdot 10^{-8}\cdot T^6)/3600''$ | |
| $\omega_7=$ | $98\overset{\circ}{.}99933405 + (+ 3474\overset{''}{.}904364\cdot T - 4\overset{''}{.}0515\cdot T^2 - 6\overset{''}{.}497992\cdot 10^{-2}\cdot T^3$ |
| $+ 3\overset{''}{.}35569\cdot 10^{-4}\cdot T^4 + 3\overset{''}{.}2639\cdot 10^{-6}\cdot T^5 - 3\overset{''}{.}036\cdot 10^{-8}\cdot T^6)/3600''$ | |
| $\varpi_7=$ | $173\overset{\circ}{.}00529106 + 1\overset{\circ}{.}486378963\cdot T +(+ 0\overset{''}{.}7706068\cdot T^2 + 1\overset{''}{.}56227\cdot 10^{-3}\cdot T^3$ |
| $- 1\overset{''}{.}6921\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}8\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $M_7=$ | $141\overset{\circ}{.}04971405 + 360\overset{\circ}{.}0\cdot T + 68\overset{\circ}{.}37767714\cdot T +(+ 3\overset{''}{.}234204\cdot T^2$ |
| $- 1\overset{''}{.}46753\cdot 10^{-3}\cdot T^3 - 6\overset{''}{.}6\cdot 10^{-6}\cdot T^4)/3600''$ | |
| $L_7=$ | $314\overset{\circ}{.}05500511 + 360\overset{\circ}{.}0\cdot T + 69\overset{\circ}{.}8640561\cdot T +(+ 1\overset{''}{.}0940272\cdot T^2$ |
| $+ 9\overset{''}{.}474\cdot 10^{-5}\cdot T^3 - 2\overset{''}{.}3521\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}8\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $n_{7.m}=$ | $0\overset{\circ}{.}01172834160540726 + \text{höhere Terme von } M_7$ |
| $n_{7.l}=$ | $0\overset{\circ}{.}01176903644353943 + \text{höhere Terme von } L_7$ |
| Neptun | |
| $a_8=$ | $30.1103868694\text{ AE} - 1.6635\cdot 10^{-7}\text{ AE}\cdot T + 6.86\cdot 10^{-10}\text{ AE}\cdot T^2$ |
| $\epsilon_8=$ | $0.009455747 + 6.03263\cdot 10^{-6}\cdot T + 0.0\cdot T^2 - 4.83\cdot 10^{-11}\cdot T^3$ |
| $i_8=$ | $1\overset{\circ}{.}76995259 + (- 33.509412\cdot T - 2\overset{''}{.}54991\cdot 10^{-2}\cdot T^2 + 9\overset{''}{.}845\cdot 10^{-5}\cdot T^3$ |
| $+ 1\overset{''}{.}01\cdot 10^{-7}\cdot T^4 - 5\overset{''}{.}0\cdot 10^{-10}\cdot T^5 - 1\overset{''}{.}0\cdot 10^{-11}\cdot T^6)/3600''$ | |
| $\Omega_8=$ | $131\overset{\circ}{.}78405702 + (+ 3967\overset{''}{.}934159\cdot T + 9\overset{''}{.}342773\cdot 10^{-1}\cdot T^2 - 2\overset{''}{.}29323\cdot 10^{-3}\cdot T^3$ |
| $- 3\overset{''}{.}3948\cdot 10^{-5}\cdot T^4 - 4\overset{''}{.}79\cdot 10^{-8}\cdot T^5 - 6\overset{''}{.}0\cdot 10^{-11}\cdot T^6)/3600''$ | |
| $\omega_8=$ | $276\overset{\circ}{.}33621852 + (+ 1167\overset{''}{.}730286\cdot T + 4\overset{''}{.}493376\cdot 10^{-1}\cdot T^2 + 2\overset{''}{.}36686\cdot 10^{-3}\cdot T^3$ |
| $+ 1\overset{''}{.}0434\cdot 10^{-5}\cdot T^4 + 2\overset{''}{.}99\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}6\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $\varpi_8=$ | $48\overset{\circ}{.}12027554 + 1\overset{\circ}{.}426295679\cdot T +(+ 1\overset{''}{.}3836149\cdot T^2 + 7\overset{''}{.}363\cdot 10^{-5}\cdot T^3$ |
| $- 0\overset{''}{.}16914\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}8\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $M_8=$ | $256\overset{\circ}{.}22838994 + 0\overset{\circ}{.}0\cdot T + 218\overset{\circ}{.}4570136\cdot T +(- 0\overset{''}{.}2718613\cdot T^2$ |
| $- 8\overset{''}{.}95\cdot 10^{-6}\cdot T^3)/3600''$ | |
| $L_8=$ | $304\overset{\circ}{.}34866548 + 0\overset{\circ}{.}0\cdot T + 219\overset{\circ}{.}8833092\cdot T +(+ 1\overset{''}{.}1117536\cdot T^2$ |
| $+ 6\overset{''}{.}468\cdot 10^{-5}\cdot T^3 - 2\overset{''}{.}3514\cdot 10^{-5}\cdot T^4 - 1\overset{''}{.}8\cdot 10^{-8}\cdot T^5 + 2\overset{''}{.}0\cdot 10^{-10}\cdot T^6)/3600''$ | |
| $n_{8.m}=$ | $0\overset{\circ}{.}005981027065419423 + \text{höhere Terme von } M_8$ |
| $n_{8.l}=$ | $0\overset{\circ}{.}00602007691289832 + \text{höhere Terme von } L_8$ |
| Pluto | |
| $a_9=$ | $39.544625\text{ AE} + 4.3852\cdot 10^{-3}\text{ AE}\cdot T - 6.889\cdot 10^{-5}\text{ AE}\cdot T^2$ |
| $- 1.1164\cdot 10^{-5}\text{ AE}\cdot T^3 - 1.204\cdot 10^{-7}\text{ AE}\cdot T^4 + 2.89\cdot 10^{-9}\text{ AE}\cdot T^5$ | |
| $\epsilon_9=$ | $0.249084 + 6.51\cdot 10^{-5}\cdot T - 1.11\cdot 10^{-6}\cdot T^2 - 1.15\cdot 10^{-7}\cdot T^3$ |
| $+ 1.1\cdot 10^{-9}\cdot T^4 + 1.0\cdot 10^{-10}\cdot T^5$ | |
| $i_9=$ | $17\overset{\circ}{.}139804 + (- 0\overset{''}{.}99216\cdot T + 9\overset{''}{.}936\cdot 10^{-2}\cdot T^2 + 2\overset{''}{.}11356\cdot 10^{-2}\cdot T^3$ |
| $+ 1\overset{''}{.}03032\cdot 10^{-3}\cdot T^4 + 1\overset{''}{.}548\cdot 10^{-5}\cdot T^5)/3600''$ | |
| $\Omega_9=$ | $110\overset{\circ}{.}308843 + (+ 28\overset{''}{.}67148\cdot T - 0\overset{''}{.}542448\cdot T^2 - 6\overset{''}{.}23916\cdot 10^{-2}\cdot T^3$ |
| $- 2\overset{''}{.}75256\cdot 10^{-3}\cdot T^4 - 3\overset{''}{.}9888\cdot 10^{-5}\cdot T^5)/3600''$ | |
| $\omega_9=$ | $113\overset{\circ}{.}794573 + (- 15\overset{''}{.}93972\cdot T - 0\overset{''}{.}764856\cdot T^2 + 1\overset{''}{.}47924\cdot 10^{-2}\cdot T^3$ |
| $+ 6\overset{''}{.}984\cdot 10^{-4}\cdot T^4 - 6\overset{''}{.}48\cdot 10^{-7}\cdot T^5)/3600''$ | |
| $\varpi_9=$ | $224\overset{\circ}{.}103416 + (+ 12\overset{''}{.}73176\cdot T - 1\overset{''}{.}307268\cdot T^2 - 4\overset{''}{.}75992\cdot 10^{-2}\cdot T^3$ |
| $- 2\overset{''}{.}05452\cdot 10^{-3}\cdot T^4 - 4\overset{''}{.}0536\cdot 10^{-5}\cdot T^5)/3600''$ | |
| $M_9=$ | $14\overset{\circ}{.}813282 + 145\overset{\circ}{.}2063327\cdot T + (- 41\overset{''}{.}366124\cdot T^2 - 0\overset{''}{.}3820176\cdot T^3$ |
| $- 2\overset{''}{.}230272\cdot 10^{-2}\cdot T^4 - 2\overset{''}{.}72232\cdot 10^{-4}\cdot T^5)/3600''$ | |
| $L_9=$ | $238\overset{\circ}{.}916698 + 145\overset{\circ}{.}2098693\cdot T + (- 42\overset{''}{.}673392\cdot T^2 - 0\overset{''}{.}4296168\cdot T^3$ |
| $- 2\overset{''}{.}435724\cdot 10^{-2}\cdot T^4 - 3\overset{''}{.}12768\cdot 10^{-4}\cdot T^5)/3600''$ | |
| $n_9=$ | $0\overset{\circ}{.}00396357 + (- 2\overset{''}{.}3724\cdot 10^{-3}\cdot T + 3\overset{''}{.}636\cdot 10^{-5}\cdot T^2 + 5\overset{''}{.}976\cdot 10^{-6}\cdot T^3$ |
| $+ 6\overset{''}{.}48\cdot 10^{-8}\cdot T^4 - 1\overset{''}{.}4\cdot 10^{-9}\cdot T^5)/3600''$ | |
| $P=$ | $238\overset{\circ}{.}88593433 + 145\overset{\circ}{.}182978\cdot T + (- 40\overset{''}{.}7518596\cdot T^2 + 0\overset{''}{.}122662792\cdot T^3)/3600''$ |
Die mittleren Bahnelemente II (JPL)
Die Bahnelemente des Jet Propulsion Laboratory beziehen sich auf eine mittlere dynamische Ekliptik und ein mittleres Äquinoktium des Datums. Ferner bezieht sich die Epoche auf den 01. Januar 2000 um $12^{h}$ dynamischer Weltzeit ($J2000.0 = JDE\; 2451545.0$). Sie sind im Zeitraum von 3000 v.Chr. bis 3000 n.Chr. gültig. Die mittleren Bahnelemente wurden E.M. Standish & J.G. Williams entnommen. In diesen Bahnelementen fehlen (außer bei der mittleren Länge $L_k$) im Gegensatz zu den vorangegangenen Bahnelementen die Terme höherer Ordnung. Es müssen noch die Störungsterme für die beiden größten Gasriesen Jupiter und Saturn addiert werden:
$$V_5 = 0\overset{\circ}{.}329\cdot\sin(G)\tag{2}$$ $$V_6 = -0\overset{\circ}{.}814\cdot\sin(G)\tag{3}$$ mit \[\begin{align} G =&\;172\overset{\circ}{.}74\\ &+0\overset{\circ}{.}00111588\cdot (JDE - 2451545.0) \end{align}\tag{4}\]
| Tabelle 2 | |
|---|---|
| Merkur | |
| $a_1 = 0.38709843\text{ AE}$ | $\epsilon_1 = 0.20563661 +2.123\cdot 10^{-5}\cdot T$ |
| $i_1 = 7\overset{\circ}{.}00559432 -5\overset{\circ}{.}90158\cdot 10^{-3}\cdot T$ | $\Omega_1 = 48\overset{\circ}{.}33961819 -0\overset{\circ}{.}12214182\cdot T$ |
| $\omega_1 = 29\overset{\circ}{.}11810076 +0\overset{\circ}{.}28154195\cdot T$ | $\varpi_1 = 77\overset{\circ}{.}45771895 +0\overset{\circ}{.}15940013\cdot T$ |
| $M_1 = 174\overset{\circ}{.}79394829 + n_m\cdot d$ | $L_1 = 252\overset{\circ}{.}25166724 + n_l\cdot d +1\overset{''}{.}084\cdot T^2/3600''$ |
| $n_{1.m} = 4\overset{\circ}{.}09233444123477$ | $n_{1.l} = 4\overset{\circ}{.}09233880537248$ |
| Venus | |
| $a_2 = 0.72332102\text{ AE} -2.6\cdot 10^{-7}\text{ AE}\cdot T$ | $\epsilon_2 = 0.00676399 -5.107\cdot 10^{-5}\cdot T$ |
| $i_2 = 3\overset{\circ}{.}39777545 +4\overset{\circ}{.}3494\cdot 10^{-4}\cdot T$ | $\Omega_2 = 76\overset{\circ}{.}67261496 -0\overset{\circ}{.}27274174\cdot T$ |
| $\omega_2 = 55\overset{\circ}{.}09494217 +0\overset{\circ}{.}32953822\cdot T$ | $\varpi_2 = 131\overset{\circ}{.}76755713 +0\overset{\circ}{.}05679648\cdot T$ |
| $M_2 = 50\overset{\circ}{.}21215137 + n_m\cdot d$ | $L_2 = 181\overset{\circ}{.}97970850 + n_l\cdot d +1\overset{''}{.}115\cdot T^2/3600''$ |
| $n_{2.m} = 1\overset{\circ}{.}60212892008542$ | $n_{2.l} = 1\overset{\circ}{.}6021304750883$ |
| Erde Mond Baryzentrum | |
| $a_3 = 1.00000018\text{ AE} -3.0\cdot 10^{-8}\text{ AE}\cdot T$ | $\epsilon_3 = 0.01673163 -3.661\cdot 10^{-5}\cdot T$ |
| $i_3 = -0\overset{\circ}{.}00054346 -0\overset{\circ}{.}01337178\cdot T$ | $\Omega_3 = 359\overset{\circ}{.}75876144 -5\overset{\circ}{.}11260389\cdot T$ |
| $\omega_3 = 103\overset{\circ}{.}17129741 +5\overset{\circ}{.}43055649\cdot T$ | $\varpi_3 = 102\overset{\circ}{.}93005885 +0\overset{\circ}{.}31795260\cdot T$ |
| $M_3 = 357\overset{\circ}{.}53685687 + n_m\cdot d$ | $L_3 = 100\overset{\circ}{.}46691572 + n_l\cdot d +1\overset{''}{.}089\cdot T^2/3600''$ |
| $n_{3.m} = 0\overset{\circ}{.}985600413708145$ | $n_{3.l} = 0\overset{\circ}{.}985609118775907$ |
| Mars | |
| $a_4 = 1.52371243\text{ AE} +9.7\cdot 10^{-7}\text{ AE}\cdot T$ | $\epsilon_4 = 0.09336511 +9.149\cdot 10^{-5}\cdot T$ |
| $i_4 = 1\overset{\circ}{.}85181869 -0\overset{\circ}{.}00724757\cdot T$ | $\Omega_4 = 49\overset{\circ}{.}71320984 -0\overset{\circ}{.}26852431\cdot T$ |
| $\omega_4 = 286\overset{\circ}{.}36934232 +0\overset{\circ}{.}72076056\cdot T$ | $\varpi_4 = 336\overset{\circ}{.}08255216 +0\overset{\circ}{.}45223625\cdot T$ |
| $M_4 = 19\overset{\circ}{.}3493162 + n_m\cdot d$ | $L_4 = 355\overset{\circ}{.}43186836 + n_l\cdot d +1\overset{''}{.}111\cdot T^2/3600''$ |
| $n_{4.m} = 0\overset{\circ}{.}52402045465243$ | $n_{4.l} = 0\overset{\circ}{.}52403283620616$ |
| Jupiter | |
| $a_5 = 5.20248019\text{ AE} -2.864\cdot 10^{-5}\text{ AE}\cdot T$ | $\epsilon_5 = 0.04853590 +1.8026\cdot 10^{-4}\cdot T$ |
| $i_5 = 1\overset{\circ}{.}29861416 -0\overset{\circ}{.}00322699\cdot T$ | $\Omega_5 = 100\overset{\circ}{.}29282654 +0\overset{\circ}{.}13024619\cdot T$ |
| $\omega_5 =273\overset{\circ}{.}9821259 +0\overset{\circ}{.}05174577\cdot T$ | $\varpi_5 = 14\overset{\circ}{.}27495244 +0\overset{\circ}{.}18199196\cdot T$ |
| $M_5 = 20\overset{\circ}{.}05983908 + n_m\cdot d + V_5$ | $L_5 = 34\overset{\circ}{.}33479152 + n_l\cdot d -0\overset{''}{.}448272\cdot T^2/3600''$ |
| $n_{5.m} = 0\overset{\circ}{.}083086152651882$ | $n_{5.l} = 0\overset{\circ}{.}083091135320192$ |
| Saturn | |
| $a_6 = 9.54149883\text{ AE} -3.065\cdot 10^{-5}\text{ AE}\cdot T$ | $\epsilon_6 = 0.05550825 -3.2044\cdot 10^{-4}\cdot T$ |
| $i_6 = 2\overset{\circ}{.}49424102 +0\overset{\circ}{.}00451969\cdot T$ | $\Omega_6 = 113\overset{\circ}{.}63998702 -0\overset{\circ}{.}25015002\cdot T$ |
| $\omega_6 = 339\overset{\circ}{.}22137361 +0\overset{\circ}{.}7919448\cdot T$ | $\varpi_6 = 92\overset{\circ}{.}86136063 +0\overset{\circ}{.}54179478\cdot T$ |
| $M_6 = 317\overset{\circ}{.}21435266 + n_m\cdot d + V_6$ | $L_6 = 50\overset{\circ}{.}07571329 + n_l\cdot d +0\overset{''}{.}932364\cdot T^2/3600''$ |
| $n_{6.m} = 0\overset{\circ}{.}033444850170021$ | $n_{6.l} = 0\overset{\circ}{.}033459683702669$ |
| Uranus | |
| $a_7 = 19.18797948\text{ AE} -2.0455\cdot 10^{-4}\text{ AE}\cdot T$ | $\epsilon_7 = 0.04685740 -1.55\cdot 10^{-5}\cdot T$ |
| $i_7 = 0\overset{\circ}{.}77298127 -0\overset{\circ}{.}00180155\cdot T$ | $\Omega_7 = 73\overset{\circ}{.}96250215 +0\overset{\circ}{.}05739699\cdot T$ |
| $\omega_7 = 98\overset{\circ}{.}47154226 +0\overset{\circ}{.}03527286\cdot T$ | $\varpi_7 = 172\overset{\circ}{.}43404441 +0\overset{\circ}{.}09266985\cdot T$ |
| $M_7 = 141\overset{\circ}{.}76872184 + n_m\cdot d$ | $L_7 = 314\overset{\circ}{.}20276625 + n_l\cdot d +2\overset{''}{.}099916\cdot T^2/3600''$ |
| $n_{7.m} = 0\overset{\circ}{.}011729020016427$ | $n_{7.l} = 0\overset{\circ}{.}011731557178645$ |
| Neptun | |
| $a_8 = 30.06952752\text{ AE} +6.447\cdot 10^{-5}\text{ AE}\cdot T$ | $\epsilon_8 = 0.00895439 +8.18\cdot 10^{-6}\cdot T$ |
| $i_8 = 1\overset{\circ}{.}77005520 +0\overset{\circ}{.}00022400\cdot T$ | $\Omega_8 = 131\overset{\circ}{.}78635853 -0\overset{\circ}{.}00606302\cdot T$ |
| $\omega_8 = 274\overset{\circ}{.}89522871 +0\overset{\circ}{.}0161624\cdot T$ | $\varpi_8 = 46\overset{\circ}{.}68158724 +0\overset{\circ}{.}01009938\cdot T$ |
| $M_8 = 257\overset{\circ}{.}54130563 + n_m\cdot d$ | $L_8 = 304\overset{\circ}{.}22289287 + n_l\cdot d -1\overset{''}{.}488528\cdot T^2/3600''$ |
| $n_{8.m} = 0\overset{\circ}{.}005980973408898$ | $n_{8.l} = 0\overset{\circ}{.}005981249914853$ |
| Pluto | |
| $a_9 = 39.48686035\text{ AE} + 6.447\cdot 10^{-5}\text{ AE}\cdot T$ | $\epsilon_9 = 0.24885238 + 6.016\cdot 10^{-5}\cdot T$ |
| $i_9 = 17\overset{\circ}{.}14104260 + 5\overset{\circ}{.}01\cdot 10^{-6}\cdot T$ | $\Omega_9 = 110\overset{\circ}{.}30167986 - 0\overset{\circ}{.}00809981\cdot T$ |
| $\omega_9 = 113\overset{\circ}{.}79534612 - 0\overset{\circ}{.}00158846\cdot T$ | $\varpi_9 = 224\overset{\circ}{.}09702598 - 0\overset{\circ}{.}00968827\cdot T$ |
| $M_9 = 14\overset{\circ}{.}86832413 + 145\overset{\circ}{.}1901173\cdot T$ | $L_9 = 238\overset{\circ}{.}96535011 + 145\overset{\circ}{.}18042903\cdot T$ |
| $n_m = 0\overset{\circ}{.}003975088769336$ | $n_l = 0\overset{\circ}{.}005980973408898$ |
Die mittleren Bahnelemente III
Diese mittleren Bahnelemente stammen vom Schweden P. Schlyter. Sie gehören keiner eigenen Theorie an. Er behauptet, sie aus dem Paper von T.C. Van Flandern & K.F. Pulkkinen (Low Precision Formulae for Planetary Position) zu haben. Der Gültigkeitszeitraum geht von 1700 bis 2300 bei einer Genauigkeit von ca. $1'$. Es gilt das Äquinoktium des Datums mit der Epoche 31. Dezember 1999 um $0^h\;UT$. $$d = JDE + 1\overset{d}{.}5 - 2451545.0\tag{5}$$
Vorangestellt ist eine Spalte mit der neuberechneten Epoche $J2000.0$, die zum Vergleich mit den anderen oben genannten Bahnelementen dient.
| Tabelle 3 | ||
|---|---|---|
| Sonne: | $J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_0 =$ | $ 1.0\textsf{ AE} $ | $ 1.0\textsf{ AE} $ |
| $\epsilon_0 =$ | $ 0.0167089982735 $ | $ 0.016709 - 1.151\cdot 10^{-9}\cdot d $ |
| $i_0 =$ | $ 0\overset{\circ}{.}0 $ | $ 0\overset{\circ}{.}0 $ |
| $\Omega_0 =$ | $ 0\overset{\circ}{.}0 $ | $ 0\overset{\circ}{.}0 $ |
| $\omega_0 =$ | $282\overset{\circ}{.}94047064025$ | $282\overset{\circ}{.}9404 + 4\overset{\circ}{.}70935\cdot 10^{-5}\cdot d $ |
| $\varpi_0 =$ | $282\overset{\circ}{.}94047064025$ | $282\overset{\circ}{.}9404 + 4\overset{\circ}{.}70935\cdot 10^{-5}\cdot d $ |
| $M_0 =$ | $357\overset{\circ}{.}52540038775$ | $356\overset{\circ}{.}047 + 0\overset{\circ}{.}9856002585\cdot d $ |
| $L_0 =$ | $280\overset{\circ}{.}465871028 $ | $278\overset{\circ}{.}9874 + 0\overset{\circ}{.}985647352\cdot d $ |
| Mond: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_m =$ | $ 60.2666\textsf{ Erdradien} $ | $ 60.2666\textsf{ Erdradien} $ |
| $\epsilon_m =$ | $ 0.0549 $ | $ 0.0549 $ |
| $i_m =$ | $ 5\overset{\circ}{.}1454 $ | $ 5.1454 $ |
| $\Omega_m =$ | $125\overset{\circ}{.}04336928755$ | $125\overset{\circ}{.}1228 - 0\overset{\circ}{.}0529538083\cdot d $ |
| $\omega_m =$ | $318\overset{\circ}{.}30993598345$ | $318\overset{\circ}{.}0634 + 0\overset{\circ}{.}1643573223\cdot d $ |
| $\varpi_m =$ | $ 83\overset{\circ}{.}353305271 $ | $ 83\overset{\circ}{.}1862 + 0\overset{\circ}{.}.111403514\cdot d $ |
| $M_m =$ | $134\overset{\circ}{.}96288942635$ | $115\overset{\circ}{.}3654 + 13\overset{\circ}{.}0649929509\cdot d $ |
| $L_m =$ | $218\overset{\circ}{.}31619469735$ | $198\overset{\circ}{.}5516 + 13\overset{\circ}{.}1763964649\cdot d $ |
| Merkur: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_1 =$ | $ 0.387098\textsf{ AE} $ | $ 0.387098\textsf{ AE} $ |
| $\epsilon_1 =$ | $ 0.2056350008385 $ | $ 0.205635 + 5.59\cdot 10^{-10}\cdot d $ |
| $i_1 =$ | $ 7\overset{\circ}{.}004700075 $ | $ 7\overset{\circ}{.}0047 + 5\overset{\circ}{.}0\cdot 10^{-8}\cdot d $ |
| $\Omega_1 =$ | $ 48\overset{\circ}{.}33134868805$ | $ 48\overset{\circ}{.}3313 + 3\overset{\circ}{.}24587\cdot 10^{-5}\cdot d $ |
| $\omega_1 =$ | $ 29\overset{\circ}{.}1241152166 $ | $ 29\overset{\circ}{.}1241 + 1\overset{\circ}{.}01444\cdot 10^{-5}\cdot d $ |
| $\varpi_1 =$ | $ 77\overset{\circ}{.}45546390465$ | $ 77\overset{\circ}{.}4554 + 4\overset{\circ}{.}26031\cdot 10^{-5}\cdot d $ |
| $M_1 =$ | $174\overset{\circ}{.}7947016552 $ | $168\overset{\circ}{.}6562 + 4\overset{\circ}{.}0923344368\cdot d $ |
| $L_1 =$ | $252\overset{\circ}{.}25016555985$ | $246\overset{\circ}{.}1116 + 4\overset{\circ}{.}0923770399\cdot d $ |
| Venus: | $ J2000.0 0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_2 =$ | $ 0.72333\textsf{ AE} $ | $ 0.72333\textsf{ AE} $ |
| $\epsilon_2 =$ | $ 0.006772998047 $ | $ 0.006773 - 1.302\cdot 10^{-9}\cdot d $ |
| $i_2 =$ | $ 3\overset{\circ}{.}39460004125$ | $ 3\overset{\circ}{.}3946 + 2\overset{\circ}{.}75\cdot 10^{-8}\cdot d $ |
| $\Omega_2 =$ | $ 76\overset{\circ}{.}6799369885 $ | $ 76\overset{\circ}{.}6799 + 2\overset{\circ}{.}4659\cdot 10^{-5}\cdot d $ |
| $\omega_2 =$ | $ 54\overset{\circ}{.}8910207561 $ | $ 54\overset{\circ}{.}891 + 1\overset{\circ}{.}38374\cdot 10^{-5}\cdot d $ |
| $\varpi_2 =$ | $131\overset{\circ}{.}5709577446 $ | $131\overset{\circ}{.}5709 + 3\overset{\circ}{.}84964\cdot 10^{-5}\cdot d $ |
| $M_2 =$ | $ 50\overset{\circ}{.}4083953366 $ | $ 48\overset{\circ}{.}0052 + 1\overset{\circ}{.}6021302244\cdot d $ |
| $L_2 =$ | $181\overset{\circ}{.}9793530812 $ | $179\overset{\circ}{.}5761 + 1\overset{\circ}{.}6021687208\cdot d $ |
| Erde: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_3 =$ | $ 1.0\textsf{ AE} $ | $ 1.0\textsf{ AE} $ |
| $\epsilon_3 =$ | $ 0.0167089982735 $ | $ 0.016709 - 1.151\cdot 10^{-9}\cdot d $ |
| $i_3 =$ | $ 0\overset{\circ}{.}0 $ | $ 0\overset{\circ}{.}0 $ |
| $\Omega_3 =$ | $ 0\overset{\circ}{.}0 $ | $ 0\overset{\circ}{.}0 $ |
| $\omega_3 =$ | $102\overset{\circ}{.}94047064025$ | $102\overset{\circ}{.}9404 + 4\overset{\circ}{.}70935\cdot 10^{-5}\cdot d $ |
| $\varpi_3 =$ | $102\overset{\circ}{.}94047064025$ | $102\overset{\circ}{.}9404 + 4\overset{\circ}{.}70935\cdot 10^{-5}\cdot d $ |
| $M_3 =$ | $357\overset{\circ}{.}52540038775$ | $356\overset{\circ}{.}047 + 0\overset{\circ}{.}9856002585\cdot d $ |
| $L_3 =$ | $100\overset{\circ}{.}465871028 $ | $ 98\overset{\circ}{.}9874 + 0\overset{\circ}{.}985647352\cdot d $ |
| Mars: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_4 =$ | $ 1.523688\textsf{ AE} $ | $ 1.523688\textsf{ AE} $ |
| $\epsilon_4 =$ | $ 0.093405003774 $ | $ 0.093405 + 2.516\cdot 10^{-9}\cdot d $ |
| $i_4 =$ | $ 1\overset{\circ}{.}8496999733 $ | $ 1\overset{\circ}{.}8497 - 1\overset{\circ}{.}78\cdot 10^{-8}\cdot d $ |
| $\Omega_4 =$ | $ 49\overset{\circ}{.}5577166215 $ | $ 49\overset{\circ}{.}5574 + 2\overset{\circ}{.}11081\cdot 10^{-5}\cdot d $ |
| $\omega_4 =$ | $286\overset{\circ}{.}5020394415 $ | $286\overset{\circ}{.}5016 + 2\overset{\circ}{.}92961\cdot 10^{-5}\cdot d $ |
| $\varpi_4 =$ | $336\overset{\circ}{.}059756063 $ | $336\overset{\circ}{.}059 + 5\overset{\circ}{.}04042\cdot 10^{-5}\cdot d $ |
| $M_4 =$ | $19\overset{\circ}{.}3881311649 $ | $ 18\overset{\circ}{.}6021 + 0\overset{\circ}{.}5240207766\cdot d $ |
| $L_4 =$ | $355\overset{\circ}{.}4472067712 $ | $354\overset{\circ}{.}6611 + 0\overset{\circ}{.}5240711808\cdot d $ |
| Jupiter: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_5 =$ | $ 5.20256\textsf{ AE} $ | $ 5.20256\textsf{ AE} $ |
| $\epsilon_5 =$ | $ 0.0484980067035 $ | $ 0.048498 + 4.469\cdot 10^{-9}\cdot d $ |
| $i_5 =$ | $ 1\overset{\circ}{.}30299976645$ | $ 1\overset{\circ}{.}303 - 1\overset{\circ}{.}557\cdot 10^{-7}\cdot d $ |
| $\Omega_5 =$ | $100\overset{\circ}{.}4542415281 $ | $100\overset{\circ}{.}4542 + 2\overset{\circ}{.}76854\cdot 10^{-5}\cdot d $ |
| $\omega_5 =$ | $273\overset{\circ}{.}87772467575$ | $273\overset{\circ}{.}8777 + 1\overset{\circ}{.}64505\cdot 10^{-5}\cdot d $ |
| $\varpi_5 =$ | $ 14\overset{\circ}{.}33196620385$ | $ 14\overset{\circ}{.}3319 + 4\overset{\circ}{.}41359\cdot 10^{-5}\cdot d $ |
| $M_5 =$ | $ 20\overset{\circ}{.}01962795015$ | $ 19\overset{\circ}{.}895 + 0\overset{\circ}{.}0830853001\cdot d $ |
| $L_5 =$ | $ 34\overset{\circ}{.}351594154 $ | $ 34\overset{\circ}{.}2269 + 0\overset{\circ}{.}083129436\cdot d $ |
| Saturn: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_6 =$ | $ 9.55475\textsf{ AE} $ | $ 9.55475\textsf{ AE} $ |
| $\epsilon_6 =$ | $ 0.0555459857515 $ | $ 0.055546 - 9.499\cdot 10^{-9}\cdot d $ |
| $i_6 =$ | $ 2\overset{\circ}{.}48859983785$ | $ 2\overset{\circ}{.}4886 - 1\overset{\circ}{.}081\cdot 10^{-7}\cdot d $ |
| $\Omega_6 =$ | $113\overset{\circ}{.}66375847 $ | $113\overset{\circ}{.}6634 + 2\overset{\circ}{.}3898\cdot 10^{-5}\cdot d $ |
| $\omega_6 =$ | $339\overset{\circ}{.}3943464915 $ | $339\overset{\circ}{.}3939 + 2\overset{\circ}{.}97661\cdot 10^{-5}\cdot d $ |
| $\varpi_6 =$ | $ 93\overset{\circ}{.}0580349615 $ | $ 93\overset{\circ}{.}05723 + 5\overset{\circ}{.}36641\cdot 10^{-5}\cdot d $ |
| $M_6 =$ | $317\overset{\circ}{.}0171663423 $ | $316\overset{\circ}{.}967 + 0\overset{\circ}{.}0334442282\cdot d $ |
| $L_6 =$ | $ 50\overset{\circ}{.}07454683845$ | $ 50\overset{\circ}{.}0243 + 0\overset{\circ}{.}0334978923\cdot d $ |
| Uranus: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_7 =$ | $ 19.18170997675\textsf{ AE} $ | $ 19.18171\textsf{ AE} - 1.55\textsf{ AE}\cdot 10^{-8}\cdot d $ |
| $\epsilon_7 =$ | $ 0.047318011175 $ | $ 0.047318 + 7.45\cdot 10^{-9}\cdot d $ |
| $i_7 =$ | $ 0\overset{\circ}{.}7733000285 $ | $ 0\overset{\circ}{.}7733 + 1\overset{\circ}{.}9\cdot 10^{-8}\cdot d $ |
| $\Omega_7 =$ | $ 74\overset{\circ}{.}000520967 $ | $ 74\overset{\circ}{.}0005 + 1\overset{\circ}{.}3978\cdot 10^{-5}\cdot d $ |
| $\omega_7 =$ | $ 96\overset{\circ}{.}6612458475 $ | $ 96\overset{\circ}{.}6612 + 3\overset{\circ}{.}0565\cdot 10^{-5}\cdot d $ |
| $\varpi_7 =$ | $170\overset{\circ}{.}6617668145 $ | $170\overset{\circ}{.}6617 + 4\overset{\circ}{.}4543\cdot 10^{-5}\cdot d $ |
| $M_7 =$ | $142\overset{\circ}{.}608088709 $ | $142\overset{\circ}{.}5905 + 0\overset{\circ}{.}011725806\cdot d $ |
| $L_7 =$ | $313\overset{\circ}{.}2698555235 $ | $313\overset{\circ}{.}2522 + 0\overset{\circ}{.}011770349\cdot d $ |
| Neptun: | $ J2000.0 $ | $ \textsf{Epoche 31.12.1999} $ |
| $a_8 =$ | $30.058260049695\textsf{ AE} $ | $ 30.05826\textsf{ AE} + 3.313\textsf{ AE}\cdot 10^{-8}\cdot d $ |
| $\epsilon_8 =$ | $ 0.008606003225 $ | $ 0.008606 + 2.15\cdot 10^{-9}\cdot d $ |
| $i_8 =$ | $ 1\overset{\circ}{.}7699996175 $ | $ 1\overset{\circ}{.}77 - 2\overset{\circ}{.}55\cdot 10^{-7}\cdot d $ |
| $\Omega_8 =$ | $131\overset{\circ}{.}781052595 $ | $131\overset{\circ}{.}7806 + 3\overset{\circ}{.}0173\cdot 10^{-5}\cdot d $ |
| $\omega_8 =$ | $272\overset{\circ}{.}846009595 $ | $272\overset{\circ}{.}8461 - 0\overset{\circ}{.}6027\cdot 10^{-5}\cdot d $ |
| $\varpi_8 =$ | $ 44\overset{\circ}{.}62706219 $ | $ 44\overset{\circ}{.}6267 + 2\overset{\circ}{.}4146\cdot 10^{-5}\cdot d $ |
| $M_8 =$ | $260\overset{\circ}{.}2560927205 $ | $260\overset{\circ}{.}2471 + 0\overset{\circ}{.}005995147\cdot d $ |
| $L_8 =$ | $304\overset{\circ}{.}8828289395 $ | $304\overset{\circ}{.}8738 + 0\overset{\circ}{.}006019293\cdot d $ |