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====== 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 =====

<imgcaption image2|>{{ :bahnelemente_neu.png |Skizze zu den Bahnelementen und weiteren Größen}}</imgcaption>

<WRAP center round box 100%>
$a$ = 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 {{:fruehlingspunkt.png?nolink&20|}} 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 = $\overline{SP}$\\
{{:fruehlingspunkt.png?nolink&20|}} = Richtung zum Frühlingspunkt\\
☊ = aufsteigender Knoten (Übergang Süd -> Nord)\\
☋ = absteigender Knoten (Übergang Nord -> Süd)\\
</WRAP>

<imgcaption image1|>{{ :punkte_ekliptik_bahnebene.png |Größen von der Sonne aus gesehen}}</imgcaption>

<WRAP center round box 100%>
{{:fruehlingspunkt.png?nolink&20|}} = 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 {{:fruehlingspunkt.png?nolink&20|}} -> ☊ \\
$\omega$ = Argument des Perihels, Winkel ☊ -> $P$ \\
$\varpi = \Omega + \omega$ = Länge des Perihels, Winkel {{:fruehlingspunkt.png?nolink&20|}} -> $P$ (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
</WRAP>

===== 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 [[:literaturhinweise#paper_simbret|J.L. Simon, P. Bretagnon & al.]] entnommen und stammen vom [[https://www.imcce.fr/|Institut de mécanique céleste et de calcul des éphémérides]].

{{tablelayout?rowsHeaderSource=1&colwidth="80px,850px"&float=center}}
^  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 [[https://ssd.jpl.nasa.gov/planets/eph_export.html|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 [[literaturhinweise#paper_stanwill|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
\[\begin{align}
G =&\;172\overset{\circ}{.}74\\
   &+0\overset{\circ}{.}00111588\cdot (JDE - 2451545.0)
\end{align}\tag{4}\]

{{tablelayout?rowsHeaderSource=1&colwidth="420px,500px"&float=center}}
^  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 [[http://www.stjarnhimlen.se|P. Schlyter]]. Sie gehören keiner eigenen Theorie an. Er behauptet, sie aus dem Paper von [[:literaturhinweise#paper_holland|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.

{{tablelayout?rowsHeaderSource=1&colwidth="100px,220px,320px"&float=center}}
^  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           $  |

<WRAP center round info 100%>
P. Schlyters Reihenentwicklungen über den Mond und Pluto stammen komplett von J. Meeus; diese sind [[mondposition_nach_meeus|hier]] und [[pluto#berechnungsmethode_i|da]] zu finden.
</WRAP>