Die Helligkeiten und die Rotationselemente werden von der IAU in dem dreijährig erscheinenden Report neu bestimmt. Die alten Angaben gelten dann als veraltet, allerdings nur fast. Um einen Vergleich mit alten Ephemeriden zu bekommen, ist eine Beibehaltung notwendig. Dieser Abschnitt zeigt die Zeiträume der Gültigkeit auf.
Laut O. Montenbruck, 2. Auflage:
| Tabelle 1: Rotationselemente | ||||||
|---|---|---|---|---|---|---|
| Planet | $\alpha_0$ | $\Delta\alpha$ | $\delta_0$ | $\Delta\delta$ | ||
| Sonne: | $+285\overset{\circ}{.}96$ | $+0\overset{\circ}{.}183 \cdot T$ | $+63\overset{\circ}{.}96$ | $+0\overset{\circ}{.}154 \cdot T$ | ||
| Merkur: | $+281\overset{\circ}{.}01$ | $-0\overset{\circ}{.}033 \cdot T$ | $+0\overset{\circ}{.}276 \cdot T$ | $+61\overset{\circ}{.}45$ | $-0\overset{\circ}{.}005 \ T$ | $+0\overset{\circ}{.}107 \cdot T$ |
| Venus: | $+272\overset{\circ}{.}76$ | $-0\overset{\circ}{.}043 \cdot T$ | $+67\overset{\circ}{.}21$ | $+0\overset{\circ}{.}027 \cdot T$ | ||
| Mars: | $+317\overset{\circ}{.}681$ | $-0\overset{\circ}{.}108 \cdot T$ | $+0\overset{\circ}{.}786 \cdot T$ | $+52\overset{\circ}{.}886$ | $-0\overset{\circ}{.}061 \cdot T$ | $+0\overset{\circ}{.}413 \cdot T$ |
| Jupiter: | $+268\overset{\circ}{.}05$ | $-0\overset{\circ}{.}009 \cdot T$ | $+0\overset{\circ}{.}116 \cdot T$ | $+64\overset{\circ}{.}49$ | $+0\overset{\circ}{.}003 \cdot T$ | $-0\overset{\circ}{.}018 \cdot T$ |
| Saturn: | $+40\overset{\circ}{.}66$ | $-0\overset{\circ}{.}036 \cdot T$ | $+4\overset{\circ}{.}731 \cdot T$ | $+83\overset{\circ}{.}52$ | $-0\overset{\circ}{.}004 \cdot T$ | $+0\overset{\circ}{.}407 \cdot T$ |
| Uranus: | $+257\overset{\circ}{.}43$ | $+1\overset{\circ}{.}429 \cdot T$ | $-15\overset{\circ}{.}10$ | $-0\overset{\circ}{.}114 \cdot T$ | ||
| Neptun: | $+295\overset{\circ}{.}33$ | $+0\overset{\circ}{.}849 \cdot T$ | $+40\overset{\circ}{.}65$ | $+0\overset{\circ}{.}242 \cdot T$ | ||
| Tabelle 2: Nullmeridiane | ||||
|---|---|---|---|---|
| Planet | System | $W_0$ | $\textrm{d}W$ | $\Delta W$ |
| Sonne: | $84\overset{\circ}{.}11$ | $+14\overset{\circ}{.}1844$ | $1\overset{\circ}{.}222$ | |
| Merkur: | $329\overset{\circ}{.}71$ | $+6\overset{\circ}{.}1385025$ | $1\overset{\circ}{.}145$ | |
| Venus: | $159\overset{\circ}{.}91$ | $-1\overset{\circ}{.}4814205$ | $1\overset{\circ}{.}436$ | |
| Mars: | $176\overset{\circ}{.}655$ | $+350\overset{\circ}{.}8919830$ | $0\overset{\circ}{.}620$ | |
| Jupiter: | System I: | $67\overset{\circ}{.}10$ | $+877\overset{\circ}{.}9$ | $1\overset{\circ}{.}291$ |
| System II: | $43\overset{\circ}{.}30$ | $+870\overset{\circ}{.}27$ | $1\overset{\circ}{.}291$ | |
| System III: | $284\overset{\circ}{.}95$ | $+870\overset{\circ}{.}536642$ | $1\overset{\circ}{.}291$ | |
| Saturn: | System I: | $227\overset{\circ}{.}2037$ | $+844\overset{\circ}{.}3$ | $3\overset{\circ}{.}470$ |
| System II: | $105\overset{\circ}{.}4857$ | $+812\overset{\circ}{.}0$ | $3\overset{\circ}{.}470$ | |
| System III: | $38\overset{\circ}{.}90$ | $+810\overset{\circ}{.}7939024$ | $3\overset{\circ}{.}470$ | |
| Uranus: | $261\overset{\circ}{.}62$ | $-554\overset{\circ}{.}913$ | $0\overset{\circ}{.}564$ | |
| Neptun: | $107\overset{\circ}{.}21$ | $+468\overset{\circ}{.}75$ | $0\overset{\circ}{.}662$ | |
Laut Explanatory Supplement, 2. Auflage:
| Tabelle 3: Rotationselemente | ||||||
|---|---|---|---|---|---|---|
| Planet | $\alpha_0$ | $\Delta\alpha$ | $\delta_0$ | $\Delta\delta$ | ||
| Sonne: | $+286\overset{\circ}{.}13$ | $+0\overset{\circ}{.}183\cdot T$ | $+63\overset{\circ}{.}87$ | $+0\overset{\circ}{.}154\cdot T$ | ||
| Merkur: | $+281\overset{\circ}{.}01$ | $-0\overset{\circ}{.}033\cdot T$ | $+0\overset{\circ}{.}276\cdot T$ | $+61\overset{\circ}{.}45$ | $-0\overset{\circ}{.}005\cdot T$ | $+0\overset{\circ}{.}107\cdot T$ |
| Venus: | $+272\overset{\circ}{.}76$ | $-0\overset{\circ}{.}043\cdot T$ | $+67\overset{\circ}{.}16$ | $+0\overset{\circ}{.}027\cdot T$ | ||
| Erde: | $+0\overset{\circ}{.}00$ | $-0\overset{\circ}{.}641\cdot T$ | $+0\overset{\circ}{.}640\cdot T$ | $+90\overset{\circ}{.}00$ | $-0\overset{\circ}{.}557\cdot T$ | $+0\overset{\circ}{.}557\cdot T$ |
| Mars: | $+317\overset{\circ}{.}68143$ | $-0\overset{\circ}{.}1061\cdot T$ | $+0\overset{\circ}{.}786\cdot T$ | $+52\overset{\circ}{.}8865$ | $0\overset{\circ}{.}0609\cdot T$ | $+0\overset{\circ}{.}413\cdot T$ |
| Jupiter: | $+268\overset{\circ}{.}05$ | $-0\overset{\circ}{.}009\cdot T$ | $+0\overset{\circ}{.}116\cdot T$ | $+64\overset{\circ}{.}49$ | $+0\overset{\circ}{.}003\cdot T$ | $-0\overset{\circ}{.}018\cdot T$ |
| Saturn: | $+40\overset{\circ}{.}589$ | $-0\overset{\circ}{.}036\cdot T$ | $+4\overset{\circ}{.}731\cdot T$ | $+83\overset{\circ}{.}537$ | $-0\overset{\circ}{.}004\cdot T$ | $+0\overset{\circ}{.}407\cdot T$ |
| Uranus: | $+257\overset{\circ}{.}311$ | $+1\overset{\circ}{.}429\cdot T$ | $-15\overset{\circ}{.}175$ | $-0\overset{\circ}{.}114\cdot T$ | ||
| Neptun: | $+299\overset{\circ}{.}36$ | $+0\overset{\circ}{.}70\cdot\sin(\color{#ff00ff}{N})$ | $+0\overset{\circ}{.}849\cdot T$ | $+43\overset{\circ}{.}46$ | $-0\overset{\circ}{.}51\cdot\cos(\color{#ff00ff}{N})$ | $+0\overset{\circ}{.}242\cdot T$ |
mit $\color{#ff00ff}N = 357\overset{\circ}{.}85 + 52\overset{\circ}{.}316\cdot T$.
Die Rotationselemente $\alpha_0$ und $\delta_0$ bezeichnen die erdäquatoriale Lage der Rotationsachsen der Planeten. Es folgt die Tabelle mit den Nullmeridianen $W$:
| Tabelle 4: Nullmeridiane | ||||
|---|---|---|---|---|
| Planet | System | $W_0$ | $\text{d}W$ | $\Delta W$ |
| Sonne: | $+84\overset{\circ}{.}182$ | $+14\overset{\circ}{.}1844$ | $+1\overset{\circ}{.}222$ | |
| Merkur: | $+329\overset{\circ}{.}68$ | $+6\overset{\circ}{.}1385025$ | $+1\overset{\circ}{.}145$ | |
| Venus: | $+160\overset{\circ}{.}20$ | $-1\overset{\circ}{.}4813688$ | $+1\overset{\circ}{.}436$ | |
| Erde: | $+190\overset{\circ}{.}147$ | $+360\overset{\circ}{.}9856235$ | $+0\overset{\circ}{.}624$ | |
| Mars: | $+176\overset{\circ}{.}630$ | $+350\overset{\circ}{.}89198226$ | $+0\overset{\circ}{.}620$ | |
| Jupiter: | System I: | $+67\overset{\circ}{.}10$ | $+877\overset{\circ}{.}9$ | $+1\overset{\circ}{.}291$ |
| System II: | $+43\overset{\circ}{.}30$ | $+870\overset{\circ}{.}27$ | $+1\overset{\circ}{.}291$ | |
| System III: | $+284\overset{\circ}{.}95$ | $+870\overset{\circ}{.}5366420$ | $+1\overset{\circ}{.}291$ | |
| Saturn: | System I: | $+227\overset{\circ}{.}2037$ | $+844\overset{\circ}{.}3$ | $+3\overset{\circ}{.}470$ |
| System II: | $+105\overset{\circ}{.}4857$ | $+812\overset{\circ}{.}0$ | $+3\overset{\circ}{.}470$ | |
| System III: | $+38\overset{\circ}{.}90$ | $+810\overset{\circ}{.}7939024$ | $+3\overset{\circ}{.}470$ | |
| Uranus: | $+203\overset{\circ}{.}81$ | $-501\overset{\circ}{.}1600928$ | $+0\overset{\circ}{.}564$ | |
| Neptun: | $+253\overset{\circ}{.}18$ | $+536\overset{\circ}{.}3128492$ | $+0\overset{\circ}{.}662$ | |
Laut Report der IAU ab 2009 bis heute:
| Tabelle 5: Rotationselemente und Nullmeridiane in der Übersicht | ||||
|---|---|---|---|---|
| Planet | Variable | Epoche | zeitliche Änderung | Störungsterm |
| Sonne: | $\alpha_0 =$ | $+286\overset{\circ}{.}13$ | ||
| $\delta_0 =$ | $+63\overset{\circ}{.}87$ | |||
| (2009) | $W =$ | $+84\overset{\circ}{.}182$ | $+14\overset{\circ}{.}1844 \cdot d$ | |
| (2015) | $W =$ | $+84\overset{\circ}{.}176$ | $+14\overset{\circ}{.}1844 \cdot d$ | |
| Merkur: | $\alpha_0 =$ | $+281\overset{\circ}{.}0097$ | $-0\overset{\circ}{.}0328 \cdot T$ | |
| (2009) | $\delta_0 =$ | $+61\overset{\circ}{.}443$ | $-0\overset{\circ}{.}0049 \cdot T$ | |
| $W =$ | $+329\overset{\circ}{.}5469$ | $+6\overset{\circ}{.}1385025 \cdot d$ | $+0\overset{\circ}{.}00993822 \cdot \sin(M_1)$ | |
| $-0\overset{\circ}{.}00104581 \cdot \sin(M_2)$ | ||||
| $-0\overset{\circ}{.}00010280 \cdot \sin(M_3)$ | ||||
| $-0\overset{\circ}{.}00002364 \cdot \sin(M_4)$ | ||||
| $-0\overset{\circ}{.}00000532 \cdot \sin(M_5)$ | ||||
| Merkur: | $\alpha_0 =$ | $+281\overset{\circ}{.}0103$ | $-0\overset{\circ}{.}0328 \cdot T$ | |
| (2015) | $\delta_0 =$ | $+61\overset{\circ}{.}4155$ | $-0\overset{\circ}{.}0049 \cdot T$ | |
| $W =$ | $+329\overset{\circ}{.}5988$ | $+6\overset{\circ}{.}1385108 \cdot d$ | $+0\overset{\circ}{.}01067257 \cdot \sin(M_1)$ | |
| $-0\overset{\circ}{.}00112309 \cdot \sin(M_2)$ | ||||
| $-0\overset{\circ}{.}00011040 \cdot \sin(M_3)$ | ||||
| $-0\overset{\circ}{.}00002539 \cdot \sin(M_4)$ | ||||
| $-0\overset{\circ}{.}00000571 \cdot \sin(M_5)$ | ||||
| Venus: | $\alpha_0 =$ | $+272\overset{\circ}{.}76$ | ||
| $\delta_0 =$ | $+67\overset{\circ}{.}16$ | |||
| $W =$ | $+160\overset{\circ}{.}20$ | $-1\overset{\circ}{.}4813688 \cdot d$ | ||
| Erde: | $\alpha_0 =$ | $+0\overset{\circ}{.}00$ | $-0\overset{\circ}{.}641 \cdot T$ | |
| $\delta_0$ = | $+90\overset{\circ}{.}00$ | $-0\overset{\circ}{.}557 \cdot T$ | ||
| $W =$ | $+190\overset{\circ}{.}147$ | $+360\overset{\circ}{.}9856235 \cdot d$ | ||
| Mars: | $\alpha_0 =$ | $+317\overset{\circ}{.}68143$ | $-0\overset{\circ}{.}1061 \cdot T$ | |
| (2009) | $\delta_0 =$ | $+52\overset{\circ}{.}8865$ | $-0\overset{\circ}{.}0609 \cdot T$ | |
| $W =$ | $+176\overset{\circ}{.}630$ | $+350\overset{\circ}{.}89198226 \cdot d$ | ||
| Mars: | $\alpha_0 =$ | $+317\overset{\circ}{.}269202$ | $-0\overset{\circ}{.}10927547 \cdot T$ | $+0\overset{\circ}{.}000068 \cdot \sin(P_1)$ |
| (2015) | $-0\overset{\circ}{.}000238 \cdot \sin(P_2)$ | |||
| $-0\overset{\circ}{.}000052 \cdot \sin(P_3)$ | ||||
| $-0\overset{\circ}{.}000009 \cdot \sin(P_4)$ | ||||
| $-0\overset{\circ}{.}419057 \cdot \sin(P_5)$ | ||||
| $\delta_0 =$ | $+54\overset{\circ}{.}432516$ | $-0\overset{\circ}{.}05827105 \cdot T$ | $+0\overset{\circ}{.}000051 \cdot \cos(P_6)$ | |
| $-0\overset{\circ}{.}000141 \cdot \cos(P_7)$ | ||||
| $-0\overset{\circ}{.}000031 \cdot \cos(P_8)$ | ||||
| $-0\overset{\circ}{.}000005 \cdot \cos(P_9)$ | ||||
| $-1\overset{\circ}{.}591274 \cdot \cos(P_{10})$ | ||||
| $W =$ | $+176\overset{\circ}{.}049863$ | $+350\overset{\circ}{.}891982443297 \cdot d$ | $+0\overset{\circ}{.}000145 \cdot \sin(P_{11})$ | |
| $-0\overset{\circ}{.}000157 \cdot \sin(P_{12})$ | ||||
| $-0\overset{\circ}{.}000040 \cdot \sin(P_{13})$ | ||||
| $-0\overset{\circ}{.}000001 \cdot \sin(P_{14})$ | ||||
| $-0\overset{\circ}{.}000001 \cdot \sin(P_{15})$ | ||||
| $-0\overset{\circ}{.}584542 \cdot \sin(P_{16})$ | ||||
| Jupiter: | $\alpha_0 =$ | $+268\overset{\circ}{.}056595$ | $-0\overset{\circ}{.}006499 \cdot T$ | $+0\overset{\circ}{.}000117 \cdot \sin(J_1)$ |
| $+0\overset{\circ}{.}000938 \cdot \sin(J_2)$ | ||||
| $+0\overset{\circ}{.}001432 \cdot \sin(J_3)$ | ||||
| $+0\overset{\circ}{.}000030 \cdot \sin(J_4)$ | ||||
| $+0\overset{\circ}{.}002150 \cdot \sin(J_5)$ | ||||
| $\delta_0 =$ | $+64\overset{\circ}{.}495303$ | $+0\overset{\circ}{.}002413 \cdot T$ | $+0\overset{\circ}{.}000050 \cdot \cos(J_1)$ | |
| $+0\overset{\circ}{.}000404 \cdot \cos(J_2)$ | ||||
| $+0\overset{\circ}{.}000617 \cdot \cos(J_3)$ | ||||
| $-0\overset{\circ}{.}000013 \cdot \cos(J_4)$ | ||||
| $+0\overset{\circ}{.}000926 \cdot \cos(J_5)$ | ||||
| $W =$ | $+284\overset{\circ}{.}95$ | $+870\overset{\circ}{.}5360000 \cdot d$ | ||
| Saturn: | $\alpha_0 =$ | $+40\overset{\circ}{.}589$ | $-0\overset{\circ}{.}036 \cdot T$ | |
| (2015) | $\delta_0 =$ | $+83\overset{\circ}{.}537$ | $-0\overset{\circ}{.}004 \cdot T$ | |
| $W =$ | $+38\overset{\circ}{.}90$ | $+810\overset{\circ}{.}7939024 \cdot d$ | ||
| Saturn: | $\alpha_0 =$ | $+40\overset{\circ}{.}594872$ | $-0\overset{\circ}{.}0554684758 \cdot T$ | $+0\overset{\circ}{.}031424 \cdot \sin(S_1)$ |
| (2022) | $+0\overset{\circ}{.}000371 \cdot \sin(S_3)$ | |||
| $+0\overset{\circ}{.}000093 \cdot \sin(S_4)$ | ||||
| $+0\overset{\circ}{.}000508 \cdot \sin(S_6)$ | ||||
| $+0\overset{\circ}{.}000064 \cdot \sin(S_8)$ | ||||
| $-0\overset{\circ}{.}000257 \cdot \sin(S_9)$ | ||||
| $-0\overset{\circ}{.}000018 \cdot \sin(S_{10})$ | ||||
| $\delta_0 =$ | $+83\overset{\circ}{.}534351$ | $-0\overset{\circ}{.}0063345961 \cdot T$ | $+0\overset{\circ}{.}003627 \cdot \cos(S_1)$ | |
| $-0\overset{\circ}{.}000019 \cdot \cos(S_2)$ | ||||
| $+0\overset{\circ}{.}000042 \cdot \cos(S_3)$ | ||||
| $-0\overset{\circ}{.}000009 \cdot \cos(S_5)$ | ||||
| $-0\overset{\circ}{.}000057 \cdot \cos(S_6)$ | ||||
| $+0\overset{\circ}{.}000005 \cdot \cos(S_7)$ | ||||
| $-0\overset{\circ}{.}000007 \cdot \cos(S_8)$ | ||||
| $-0\overset{\circ}{.}000029 \cdot \cos(S_9)$ | ||||
| $W =$ | $+38\overset{\circ}{.}814623$ | $+818\overset{\circ}{.}1387763691 \cdot d$ | $-0\overset{\circ}{.}031224 \cdot \sin(S_1)$ | |
| $-0\overset{\circ}{.}000368 \cdot \sin(S_3)$ | ||||
| $-0\overset{\circ}{.}000093 \cdot \sin(S_4)$ | ||||
| $-0\overset{\circ}{.}000505 \cdot \sin(S_6)$ | ||||
| $-0\overset{\circ}{.}000063 \cdot \sin(S_8)$ | ||||
| $+0\overset{\circ}{.}000255 \cdot \sin(S_9)$ | ||||
| $+0\overset{\circ}{.}000018 \cdot \sin(S_{10})$ | ||||
| Uranus: | $\alpha_0 =$ | $+257\overset{\circ}{.}311$ | ||
| (2009) | $\delta_0 =$ | $-15\overset{\circ}{.}175$ | ||
| $W =$ | $+203\overset{\circ}{.}81$ | $-501\overset{\circ}{.}1600928 \cdot d$ | ||
| Uranus: | $\alpha_0 =$ | $+257\overset{\circ}{.}309980$ | $+0\overset{\circ}{.}000173 \cdot T$ | $+0\overset{\circ}{.}000895 \cdot \sin(U_1)$ |
| (2014) | $+0\overset{\circ}{.}000180 \cdot \sin(U_2)$ | |||
| $+0\overset{\circ}{.}000098 \cdot \sin(U_3)$ | ||||
| $+0\overset{\circ}{.}000075 \cdot \sin(U_4)$ | ||||
| $-0\overset{\circ}{.}000818 \cdot \sin(K)$ | ||||
| $\delta_0 =$ | $-15\overset{\circ}{.}172395$ | $+0\overset{\circ}{.}000019 \cdot T$ | $+0\overset{\circ}{.}000851 \cdot \cos(U_1)$ | |
| $+0\overset{\circ}{.}000173 \cdot \cos(U_2)$ | ||||
| $+0\overset{\circ}{.}000094 \cdot \cos(U_3)$ | ||||
| $+0\overset{\circ}{.}000072 \cdot \cos(U_4)$ | ||||
| $+0\overset{\circ}{.}000818 \cdot \cos(K)$ | ||||
| Neptun: | $\alpha_0 =$ | $+299\overset{\circ}{.}364$ | $+0\overset{\circ}{.}70 \cdot \sin(N_1)$ | |
| (2015) | $\delta_0 =$ | $+43\overset{\circ}{.}449$ | $-0\overset{\circ}{.}51 \cdot \cos(N_1)$ | |
| $W =$ | $+253\overset{\circ}{.}18$ | $+536\overset{\circ}{.}3128492 \cdot d$ | $-0\overset{\circ}{.}48 \cdot \sin(N_1)$ | |
| Neptun: | $\alpha_0 =$ | $+299\overset{\circ}{.}295650$ | $+0\overset{\circ}{.}635398 \cdot \sin(N_2)$ | |
| (2009) | $-0\overset{\circ}{.}138711 \cdot \sin(2 \ N_2)$ | |||
| $\delta_0 =$ | $+43\overset{\circ}{.}443282$ | $-0\overset{\circ}{.}461627 \cdot \cos(N_2)$ | ||
| $+0\overset{\circ}{.}050388 \cdot \cos(2 \ N_2)$ | ||||
Die Nullmeridiane Jupiters und Saturns beziehen sich auf des System III. Die Hilfswerte für die Planeten Merkur ($M_i$), Mars ($P_i$), Jupiter ($J_i$), Saturn ($S_i$), Uranus ($U_i$) und Neptun ($N_i$) lauten:
| Tabelle 6: Störungsterme | |||
|---|---|---|---|
| Mars: | Mars: | ||
| $P_1=$ | $198\overset{\circ}{.}991226 + 19139\overset{\circ}{.}4819985 \cdot T$ | $P_9=$ | $79\overset{\circ}{.}476401 + 76560\overset{\circ}{.}6495004 \cdot T$ |
| $P_2=$ | $226\overset{\circ}{.}292679 + 38280\overset{\circ}{.}8511281 \cdot T$ | $P_{10}=$ | $166\overset{\circ}{.}325722 + 0\overset{\circ}{.}5042615 \cdot T$ |
| $P_3=$ | $249\overset{\circ}{.}663391 + 57420\overset{\circ}{.}7251593 \cdot T$ | $P_{11}=$ | $129\overset{\circ}{.}071773 + 19140\overset{\circ}{.}0328244 \cdot T$ |
| $P_4=$ | $266\overset{\circ}{.}183510 + 76560\overset{\circ}{.}6367950 \cdot T$ | $P_{12}=$ | $36\overset{\circ}{.}352167 + 38281\overset{\circ}{.}0473591 \cdot T$ |
| $P_5=$ | $79\overset{\circ}{.}398797 + 0\overset{\circ}{.}5042615 \cdot T$ | $P_{13}=$ | $56\overset{\circ}{.}668646 + 57420\overset{\circ}{.}9295360 \cdot T$ |
| $P_6=$ | $122\overset{\circ}{.}433576 + 19139\overset{\circ}{.}9407476 \cdot T$ | $P_{14}=$ | $67\overset{\circ}{.}364003 + 76560\overset{\circ}{.}2552215 \cdot T$ |
| $P_7=$ | $43\overset{\circ}{.}058401 + 38280\overset{\circ}{.}8753272 \ T$ | $P_{15}=$ | $104\overset{\circ}{.}792680 + 95700\overset{\circ}{.}4387578 \cdot T$ |
| $P_8=$ | $57\overset{\circ}{.}663379 + 57420\overset{\circ}{.}7517205 \ T$ | $P_{16}=$ | $95\overset{\circ}{.}391654 + 0\overset{\circ}{.}5042615 \cdot T$ |
| Merkur: | Jupiter: | ||
| $M_1=$ | $174\overset{\circ}{.}7910857 + 4\overset{\circ}{.}092335 \cdot d$ | $J_1=$ | $99\overset{\circ}{.}360714 + 4850\overset{\circ}{.}4046 \cdot T$ |
| $M_2=$ | $349\overset{\circ}{.}5821714 + 8\overset{\circ}{.}184670 \cdot d$ | $J_2=$ | $175\overset{\circ}{.}895369 + 1191\overset{\circ}{.}9605 \cdot T$ |
| $M_3=$ | $164\overset{\circ}{.}3732571 + 12\overset{\circ}{.}277005 \cdot d$ | $J_3=$ | $300\overset{\circ}{.}323162 + 262\overset{\circ}{.}5475 \cdot T$ |
| $M_4=$ | $339\overset{\circ}{.}1643429 + 16\overset{\circ}{.}369340 \cdot d$ | $J_4=$ | $114\overset{\circ}{.}012305 + 6070\overset{\circ}{.}2476 \cdot T$ |
| $M_5=$ | $153\overset{\circ}{.}9554286 + 20\overset{\circ}{.}461675 \cdot d$ | $J_5=$ | $49\overset{\circ}{.}511251 + 64\overset{\circ}{.}3000 \cdot T$ |
| Saturn: | Saturn: | ||
| $S_1=$ | $335\overset{\circ}{.}808773 + 51\overset{\circ}{.}9103094 \cdot T$ | $S_6=$ | $159\overset{\circ}{.}835383 + 2445\overset{\circ}{.}2110530 \cdot T$ |
| $S_2=$ | $348\overset{\circ}{.}007386 + 94\overset{\circ}{.}9443923 \cdot T$ | $S_7=$ | $249\overset{\circ}{.}835383 + 2445\overset{\circ}{.}2110530 \cdot T$ |
| $S_3=$ | $9\overset{\circ}{.}358404 + 1004\overset{\circ}{.}8552318 \cdot T$ | $S_8=$ | $117\overset{\circ}{.}222690 + 3667\overset{\circ}{.}7456916 \cdot T$ |
| $S_4=$ | $129\overset{\circ}{.}758867 + 1222\overset{\circ}{.}9801079 \cdot T$ | $S_9=$ | $280\overset{\circ}{.}168047 + 7226\overset{\circ}{.}4296474 \cdot T$ |
| $S_5=$ | $219\overset{\circ}{.}758867 + 1222\overset{\circ}{.}9801079 \cdot T$ | $S_{10}=$ | |
| Uranus: | Neptun: | ||
| $U_1=$ | $328\overset{\circ}{.}616724 + 26\overset{\circ}{.}9601 \cdot T$ | $K=$ | $137\overset{\circ}{.}359959$ |
| $U_2=$ | $259\overset{\circ}{.}275089 + 2024\overset{\circ}{.}7285 \cdot T$ | $N_1=$ | $357\overset{\circ}{.}85 + 52\overset{\circ}{.}316 \cdot T$ |
| $U_3=$ | $102\overset{\circ}{.}827444 + 182\overset{\circ}{.}8030 \cdot T$ | $N_2=$ | $357\overset{\circ}{.}598765 + 52\overset{\circ}{.}383622 \cdot T$ |
| $U_4=$ | $185\overset{\circ}{.}361668 + 276\overset{\circ}{.}4108 \cdot T$ | ||
\[\begin{align} \text{System I: } W &= 58\overset{\circ}{.}04 + 877\overset{\circ}{.}8169088 d \\ \text{System II: } W &= 34\overset{\circ}{.}82 + 870\overset{\circ}{.}1869088 d \\ \text{System III: } W &= 275\overset{\circ}{.}91 + 870\overset{\circ}{.}4529088 d \end{align}\]
\[\begin{align} \text{System I: } W &= 315\overset{\circ}{.}82 + 844\overset{\circ}{.}2999153 d \\ \text{System II: } W &= 192\overset{\circ}{.}82 + 811\overset{\circ}{.}9999153 d \\ \text{System III: } W &= 126\overset{\circ}{.}82 + 810\overset{\circ}{.}7938177 d \end{align}\]
Nochmal die Helligkeitsausdrücke zum Vergleich:
| Tabelle 7: Helligkeiten ab 2012 | |
|---|---|
| Planet | Absolute Helligekeit |
| Merkur: | $\begin{align} m_0 = &-0\overset{m}{.}42 \\ &+3\overset{m}{.}8 \left(\frac{\varphi}{100^{\circ}} \right) \\ &-2\overset{m}{.}73 \left(\frac{\varphi}{100^{\circ}} \right)^2 \\ &+2\overset{m}{.}00 \left(\frac{\varphi}{100^{\circ}} \right)^3\end{align}$ |
| Venus: | $\begin{align} m_0 = &-4\overset{m}{.}40 \\ &+0\overset{m}{.}009 \left(\frac{\varphi}{100^{\circ}} \right) \\ &+2\overset{m}{.}39 \left(\frac{\varphi}{100^{\circ}} \right)^2 \\ &-0\overset{m}{.}65 \left(\frac{\varphi}{100^{\circ}} \right)^3\end{align}$ |
| Erde: | $\begin{align} m_0 = &-3\overset{m}{.}87 \\ &+1\overset{m}{.}30 \left(\frac{\eta}{100^{\circ}} \right) \\ &+0\overset{m}{.}19 \left(\frac{\eta}{100^{\circ}} \right)^2 \\ &+0\overset{m}{.}48 \left(\frac{\eta}{100^{\circ}} \right)^3\end{align}$ |
| Mars: | $\begin{align} m_0 = &-1\overset{m}{.}52 \\ &+1\overset{m}{.}60 \left(\frac{\varphi}{100^{\circ}} \right)\end{align}$ |
| Jupiter: | $\begin{align} m_0 = &-9\overset{m}{.}40 \\ &+0\overset{m}{.}50 \left(\frac{\varphi}{100^{\circ}} \right)\end{align}$ |
| Saturn: | $\begin{align} m_0 = &-8\overset{m}{.}88 \\ &+4\overset{m}{.}40 \left(\frac{\varphi}{100^{\circ}} \right) \\ &-2\overset{m}{.}60 \cdot\sin (|\color{#ff00ff}{D_E}|) \\ &+1\overset{m}{.}25 \cdot\sin (|\color{#ff00ff}{D_E}|)^2\end{align}$ |
| Uranus: | $\begin{align} m_0 = &-7\overset{m}{.}19 \\ &+0\overset{m}{.}28 \left(\frac{\varphi}{100^{\circ}} \right)\end{align}$ |
| Neptun: | $\begin{align} m_0 = &-7\overset{m}{.}05 \\ &+0\overset{m}{.}54 \left(\frac{\varphi}{100^{\circ}} \right)\end{align}$ |
| Tabelle 8 | |
|---|---|
| Ältere Gleichungen (von 1992 bis 2012) | |
| Merkur: | $m_0 = -0\overset{m}{.}42 +3\overset{m}{.}80 \left(\frac{\varphi}{100^{\circ}}\right) -2\overset{m}{.}73 \left(\frac{\varphi}{100^{\circ}}\right)^2 +2\overset{m}{.}00 \left(\frac{\varphi}{100^{\circ}}\right)^3$ |
| Venus: | $m_0 = -4\overset{m}{.}40 +0\overset{m}{.}09 \left(\frac{\varphi}{100^{\circ}}\right) +2\overset{m}{.}39 \left(\frac{\varphi}{100^{\circ}}\right)^2 -0\overset{m}{.}65 \left(\frac{\varphi}{100^{\circ}}\right)^3$ |
| Jüngere Gleichungen (nach 2012) | |
| Merkur: | $m_0 = -0\overset{m}{.}60 -4\overset{m}{.}98 \left(\frac{\varphi}{100^{\circ}}\right) -4\overset{m}{.}88 \left(\frac{\varphi}{100^{\circ}}\right)^2 +3\overset{m}{.}02 \left(\frac{\varphi}{100^{\circ}}\right)^3$ |
| Venus 1: | $m_0 = -4\overset{m}{.}47 -1\overset{m}{.}03 \left(\frac{\varphi}{100^{\circ}}\right) +0\overset{m}{.}57 \left(\frac{\varphi}{100^{\circ}}\right)^2 +0\overset{m}{.}13 \left(\frac{\varphi}{100^{\circ}}\right)^3$ |
| Venus 2: | $m_0 = +0\overset{m}{.}18 -1\overset{m}{.}02 \left(\frac{\varphi}{100^{\circ}}\right)$ |
| Gleichungen (1992 bis 2012) | |
| Erde: | $m_0 = -3\overset{m}{.}87 +1\overset{m}{.}30 \left(\frac{\eta}{100^{\circ}}\right) +0\overset{m}{.}19 \left(\frac{\eta}{100^{\circ}}\right)^2 +0\overset{m}{.}48 \left(\frac{\eta}{100^{\circ}}\right)^3$ |
| Mars: | $m_0 = -1\overset{m}{.}52 +1\overset{m}{.}60 \left(\frac{\varphi}{100^{\circ}}\right)$ |
| Jupiter: | $m_0 = -9\overset{m}{.}40 +0\overset{m}{.}50 \left(\frac{\varphi}{100^{\circ}}\right)$ |
| Saturn: | $m_0 = -8\overset{m}{.}88 +4\overset{m}{.}40 \left(\frac{|\Delta A|}{100^{\circ}}\right) -2\overset{m}{.}60 \sin\left(|\color{#ff00ff}{D_E}|\right) +1\overset{m}{.}25 \sin^2\left(|\color{#ff00ff}{D_E}|\right)$ |
| Uranus: | $m_0 = -7\overset{m}{.}19 +0\overset{m}{.}28 \left(\frac{\varphi}{100^{\circ}}\right)$ |
| Neptun: | $m_0 = -6\overset{m}{.}87 +0\overset{m}{.}54 \left(\frac{\varphi}{100^{\circ}}\right)$ |
Venus 1: $2\overset{\circ}{.}2 \leq \varphi \leq 163\overset{\circ}{.}6$
Venus 2: $163\overset{\circ}{.}6 \leq \varphi \leq 170\overset{\circ}{.}2$
Aufgrund der schwierigen Photometrie bei kleinen Phasenwinkeln gibt es unterschiedliche Darstellungen für $m_0$. Die obigen Gleichungen stammen aus O. Montenbruck & T. Pfleger und dem Explanatory Supplement (2. und 3. Auflage). J. Meeus, O. Montenbruck (2. Auflage) und das alte Explanatory Supplement (1. Auflage) geben folgende alternative, aber unvollständige Gleichungen aufgrund fehlender Phasenterme mit der Gültigkeitsdauer von 1966 bis 1992 an:
| Tabelle 9: Gleichungen von 1966 bis 1992 | |
|---|---|
| Merkur: | $m_0 = -0\overset{m}{.}00325 +1\overset{m}{.}815 \left(\frac{\varphi}{100^{\circ}}\right) -1\overset{m}{.}023 \left(\frac{\varphi}{100^{\circ}}\right)^2$ |
| Venus: | $m_0 = -4\overset{m}{.}00 +1\overset{m}{.}322 \left(\frac{\varphi}{100^{\circ}}\right) +0\overset{m}{.}4247 \left(\frac{\varphi}{100^{\circ}}\right)^3$ |
| Mars: | $m_0 = -1\overset{m}{.}30 +1\overset{m}{.}486 \left(\frac{\varphi}{100^{\circ}}\right)$ |
| Jupiter: | $m_0 = -8\overset{m}{.}93$ |
| Saturn: | $m_0 = -8\overset{m}{.}68 + 4\overset{m}{.}40 \left(\frac{|\Delta A|}{100^{\circ}}\right) -2\overset{m}{.}60 \sin\left(|\color{#ff00ff}{D_E}|\right) +1\overset{m}{.}25 \sin^2\left(|\color{#ff00ff}{D_E}|\right)$ |
| Uranus: | $m_0 = -6\overset{m}{.}85$ |
| Neptun: | $m_0 = -7\overset{m}{.}05$ |
Aus dem Abschnitt über die Rotation kommt noch ein Hinweis:
$\color{#ff00ff}{D_E}$ ist die planetozentrische Breite der Erde über der Äquatorebene des Saturn (Ringneigung) und $\Delta A = A_{\oplus} - A_{\odot}$ ist die Differenz zwischen der planetozentrischen Länge der Erde bzw. der Sonne.