25
th
i
th
a
i
=
N
X
j6=i
GM
j
|r
ij
|
3
r
ij
j
i
=
N
X
i6=j
GM
j
|r
ij
|
3
Ã
v
ij
−
3 (r
ij
.v
ij
) r
ij
|r
ij
|
2
!
|r
ij
|
3
= |r
ji
|
3
n
n(n−1)
2
O(n
2
)
| r
ij
|
3
¡
x
2
+ y
2
+ z
2
¢
3
2
n
O(m
2
),
2 × 10
30
10
22
− 10
27
×10
22
10
15
− 10
19
1.8 × 10
15
10
13
10
3
10
17
Kg 10
−5
< 0.07ms
−2
O(n
2
)
O(n
2
)
O(n
2
)
O(n
2
)
×2.5
O(n
2
)
¯
¯
¯
f
i
¯
¯
¯
=
P
N
j6=i
GM
j
|~r
ij
|
2
+e
e
t
t + δt
E
i
=
1
2
m
i
v
2
i
+
N
X
i6=j
Gm
i
m
j
|r
ij
|
h
i
= m
i
| r
ij
× v
ij
|
N = 2
v
1
= v
0
+ a
0
δt
s
1
= s
0
+
δt
2
(v
1
+ v
0
)
v s
s
1
= s
0
+ v
1
2
δt
v
3
2
= v
1
2
+ a
1
δt
s
1
= s
0
+ v
0
δt +
(δt)
2
2
a
0
v
1
= v
0
+
δt
2
(a
1
+ a
0
)
4E
simple euler
= 0.3
4E
leapf rog
= 1.3 × 10
−10
4AM
simple euler
= 0.19
4AM
leapf rog
= 4.1 × 10
−15
10
7
s
pred
= s
0
+ v
0
δt + a
0
(δt)
2
2
+ j
0
(δt)
3
6
v
pred
= a
0
δt + j
0
(δt)
2
2
a j s
pred
v
1
= v
0
+
δt
2
(a
0
+ a
pred
) +
(δt)
2
12
(j
0
− j
pred
)
s
1
= s
0
+
δt
2
(v
0
+ v
1
) +
(δt)
2
12
(a
0
− a
pred
)
P EC
P (EC)
2
δt
e = 0.99
∆t
c
=
v
u
u
t
η |a|.
¯
¯
a
(2)
¯
¯
+
¯
¯
a
(1)
¯
¯
2
¯
¯
a
(1)
¯
¯
.
¯
¯
a
(3)
¯
¯
+
¯
¯
a
(2)
¯
¯
2
η a a
(1)
j
a j
a
(2)
1
=
−6(a
0
− a
1
) − δt(4j
0
+ 2j
1
)
(δt)
2
a
(3)
1
=
12(a
0
− a
1
) + 6δt(j
0
+ j
1
)
(δt)
3
T
T
T δt
δt T
T
T δt δt δt ¿ T
r =
l
1 + e cos θ
θ
e =
p
1 + 2Eh
2
l =
h
2
GM
|r ×v| 0.5v
2
+
GM
r
θ
θ = cos
−1
µ
1
e
µ
l
|r |
− 1
¶¶
π
r v
2π
θ
•
•
•
•
|
~
f
approx
|
−
|
~
f
true
|
|
~
f
true
|
8.7 × 10
−14
8.5 × 10
−7
5.6 × 10
−10
< 10
−15
2.7 × 10
−15
8.5 × 10
−7
3.9 × 10
−15
< 10
−15
1.6 × 10
−14
< 10
−15
8.0 × 10
−15
< 10
−15
3.4 × 10
−13
1.2 × 10
−6
2.3 × 10
−7
5.8 × 10
−6
4.9 × 10
−6
2.8 × 10
−4
8.9 × 10
−8
1.6 × 10
−5
1.0 × 10
−9
1.2 × 10
−5
8.1 × 10
−7
4.2 × 10
−5
1.6 × 10
−6
1.3 × 10
−4
1.1 × 10
−5
5.2 × 10
−4
1.2 × 10
−7
1.2 × 10
−5
7.7 × 10
−9
8.5 × 10
−7
2.6 × 10
−8
1.2 × 10
−6
3.6 × 10
−8
8.5 × 10
−7
5.7 × 10
−8
3.6 × 10
−6
2.9 × 10
−7
1.4 × 10
−5
1.7 × 10
−6
1.2 × 10
−4
4.7 × 10
−6
2.1 × 10
−4
5.8 × 10
−5
9.4 × 10
−4
3 × 10
−3
8.9 × 10
−2
1.8 × 10
−8
< 10
−15
3.2 × 10
−10
< 10
−15
1.0 × 10
−8
< 10
−15
1.3 × 10
−8
1.7 × 10
−6
1.2 × 10
−7
3.3 × 10
−6
2.3 × 10
−7
1.2 × 10
−5
8.0 × 10
−9
8.5 × 10
−7
2.5 × 10
−9
< 10
−15
8.2 × 10
−9
8.5 × 10
−7
8.1 × 10
−9
< 10
−15
δt = 1 Day
P (EC)
2
4.4 × 10
−4
×10
−5
4.1 × 10
−4
5 × 10
−5
P EC 1.1 × 10
−3
9.0 × 10
−5
8.9 × 10
−4
1.0 × 10
−4
2.5 × 10
−4
5.9 × 10
−6
2.3 × 10
−4
8.0 × 10
−6
6.7 × 10
−3
4.7 × 10
−4
6.9 × 10
−3
4.9 × 10
−4
δt = 0.02 Day
P (EC)
2
2 × 10
−12
1.9 × 10
−6
4.3 × 10
−10
1.9 × 10
−6
P EC 2 × 10
−9
×10
−6
2.7 × 10
−9
×10
−6
2.9 × 10
−9
1 × 10
−6
3.4 × 10
−9
1.1 × 10
−6
3.1 × 10
−4
4.7 × 10
−6
3.2 × 10
−4
4.7 × 10
−6
P (EC)
2
10
−15
δt = 0.02
3.4 ×
10
−9
δt = 0.02 Day
P (EC)
2
P (EC)
2
8.1 × 10
−10
2.7 × 10
−7
P EC 5.6 × 10
−7
×10
−7
1.26 × 10
−9
3.2 × 10
−6
9.0 × 10
−3
7.0 × 10
−4
P (EC)
2
P EC
P (EC)
2
•
•
•
•
•
•
•
•
•
•
•
•
•
p
x
2
+ y
2
+ z
2
¡
x
2
+ y
2
+ z
2
¢
∞
e ≥
1
e > 1
e ≥ 1
0.5v
2
+
GM
r
|r ×v|
G ∗ Mass
√
1 + 2Eh
2
h
2
GM
i
th
δt
t+interv al
δt