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0β€ž(0.05) = 0.000 β€” 4.000 x 0.050 = β€”0.200;
0β€ž(0.05) = 1.630 + 0.000 x 0.050=β€”=_ 1.630.
Now our main calculations begin:
z(0.1) = 0.500β€”0.20x0.1 =_ 0.480
(0.1) = 0.0 + 1.63 x 0.1 =_ 0.163
r= V0.4802+0.1632 =_ 0.507
1/rα»Ά = 7.677
ΓΈβ€ž(0.1) = β€”0.480 x 7.677 = β€”3.685
aβ€ž(0.1) = β€”0.163 x 7.677 = β€”1.250
0β€ž(0.15) = β€”0.200 β€” 3.685 x 0.1 = β€”0.568
0u(0.15) = 1.680 β€” 1.250 x0.1 = 1.505
+(0.2) = 0.480 β€” 0.568 x01 =_ 0.4238
(0.2) = 0.163 + 1.505x0.1 = 0.313
In this way we obtain the values given in Table 9-2, and in 20 steps or so we
have chased the planet halfway around the sunl In Eig. 9-6 are plotted the z-
and -coordinates given in Table 9-2. "The dots represent the positions at the
succession of times a tenth of a unit apart; we see that at the start the planet
moves rapidly and at the end it moves slowly, and so the shape of the curve 1s
determined. Thus we see that we real do know how to calculate the motion of
planetsl
Table 9-2
Solution of duβ€ž/d‑ = β€”ΓΈΕ“/rαΊΊ, duy/dt β€”= β€”α»₯/rαΊΊ, r = +2 + 92.
Interval: = 0.100
Ởrbiα»―t uy = 1.63 β€ž=0 z=05 =0 at =0
‑ un Uz Δ‘β€ž α»₯ Uy Δ‘α»₯ r 1/rΕ 
0.0 0.500 β€”4.000 0.000 0.000 |[ 0.500 | 8.000
β€”0.200 1.630
--- Trang 191 ---
Table 9-2
t un Uz Δ‘β€ž α»₯ Uy Δ‘α»₯ r 1/rαΊΊ
0.1 0.480 β€”3.685 0.163 β€”1.251 || 0.507 | 7.677
β€”0.568 1.505
0.2 0.423 β€”2.897 0.313 β€”2.146 || 0.527 | 6.847
β€”0.858 1.290
0.3 0.337 β€”1.958 0.443 β€”2.569 || 0.556 | 5.805
β€”1.054 1.033
0.4 0.232 β€”1.112 0.546 β€”2.617 || 0.593 | 4.794
β€”1.165 0.772
0.5 0.115 β€”0.454 0.623 β€”2.449 || 0.634 | 3.931
β€”1.211 0.527
0.6 | β€”0.006 -+0.018 0.676 β€”2.190 || 0.676 | 3.241
β€”1.209 0.308
0.7 | β€”0.127 +0.342 0.706 β€”1.911 || 0.718 | 2.705
β€”1.175 0.117
0.8 | β€”0.244 -+0.559 0.718 β€”1.646 || 0.758 | 2.292
β€”1.119 β€”0.048
0.9 | β€”0.356 +0.702 0.713 β€”1.408 || 0.797 | 1.974
β€”1.048 β€”0.189
1.0 | β€”0.461 -+0.796 0.694 β€”1.200 || 0.833 | 1.728
β€”0.969 β€”0.309
1.1 | β€”0.558 -+0.856 0.664 β€”1.019 || 0.867 | 1.536
β€”0.883 β€”0.411
1.2 | β€”0.646 -+0.895 0.623 β€”0.862 || 0.897 | 1.385
β€”0.794 β€”0.497
1.3 | β€”0.725 -+0.919 0.573 β€”0.726 || 0.924 | 1.267
β€”0.702 β€”0.569
1.4 | β€”0.795 -+0.933 0.516 β€”0.605 || 0.948 | 1.174
β€”0.608 β€”0.630
1.5 | β€”0.856 +0.942 0.453 β€”0.498 || 0.969 | 1.100
β€”0.514 β€”0.680
1.6 | β€”0.908 -+0.947 0.385 β€”0.402 || 0.986 | 1.043
β€”0.420 β€”0.720
1.7 | β€”0.950 -+0.950 0.313 β€”0.313 || 1.000 | 1.000
β€”0.325 β€”0.7Γ°1
1.8 | β€”0.982 +0.952 0.238 β€”0.230 || 1.010 | 0.969
β€”0.229 β€”0.774
1.9 | β€”1.005 -+0.953 0.160 β€”0.152 || 1.018 | 0.949
--- Trang 192 ---
Table 9-2
t un Uz Δ‘β€ž α»₯ Uy Δ‘α»₯ r 1/rαΊΊ
β€”0.134 β€”0.790
2.0 | β€”1.018 +0.955 0.081 β€”0.076 || 1.022 | 0.938
β€”0.038 β€”0.797
2.1 | β€”1.022 +0.957 0.002 β€”0.002 || 1.022 | 0.936
+0.057 β€”0.797
2.2 | β€”1.017 +0.959 || β€”0.078 +0.074 || 1.020 | 0.944
β€”0.790
2.3
Crossed zΓΈ-axis at 2.101 sec, .'. period = 4.20 sec.
β€ž = 0 at 2.086 sec.
Cross ΓΈ at β€”1.022, .'. semimajor axis = ... = 0.761.
0y = 0.T9T.
Predicted time z(0.761)3⁄/2 = x(0.663) = 2.082.
=1.0 α»Έ
t= β€” _t=05
t=15β€”N * 05 7
t= 20^" =0
β€”1.0 β€”0.5 SUN 0.5 x
Fig. 9-6. The calculated motion of a planet around the sun.
Now let us see how we can calculate the motion of Neptune, Jupiter, UỦranus,
or any other planet. lÝ we have a great many planets, and let the sun move
too, can we do the same thing? Of course we can. We calculate the force on
a particular planet, let us say planet number Β‘, which has a position #ΒΏ, ΒΏ, ZΒΏ
(2= 1 may represent the sun, ΒΏ = 2 Mercury, ΒΏ = 3 Venus, and so on). We must
know the positions of all the planets. The force acting on one is due to all the
other bodies which are located, let us say, at positions #;,;,z;. Therefore the
--- Trang 193 ---
equations are
mị TU — N¬_ GmimjVi S17)
Δ‘t = Tα»­