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16.346 Astrodynamics
Fall 2008

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Lecture 2

The Two Body Problem Continued

The Eccentricity Vector

or

The Laplace Vector

µe = v × h −

µ
r
r

Explicit Form of the Velocity Vector

#3.1

Using the expansion of the triple vector product a × (b × c) = (a · c)b − (a · b)c we have
h × µe = h × (v × h) −

µ
h × r = h2 v − (h · v)h − µh × ir = h2 v − µh ih × ir
r

since h and v are perpendicular. Therefore:
h × µe

=⇒

v=

µ
i × (e + ir )
h h

or
hv
= ih × (e ie + ir ) = e ih × ie + ih × ir = e ip + iθ
µ
Then since
ip = sin f ir + cos f iθ
we have
hv
= e sin f ir + (1 + e cos f ) iθ
µ
which is the basic relation for representing the velocity vector in the Hodograph Plane.
See Page 1 of Lecture 4
Conservation of Energy
p
p

hv hv
·
= v · v = 2(1 + e cos f ) + e2 − 1 = 2 × − (1 − e2 ) = p
µ
µ
µ
r

2 1
−
r
a

which can be written in either of two separate forms each having its own name:

1
µ
1 2 µ
= c3
v − =−
2
2
r
2a

Energy Integral


Vis-Viva Integral

v2 = µ

2 1
−
r
a

The constant c3 is used by Forest Ray Moulton, a Professor at the University of
Chicago in his 1902 book “An Introduction to Celestial Mechanics” — the first book on
the subject written by an American.

16.346 Astrodynamics

Lecture 2


Conic Sections
Ellipse or Hyperbola in rectangular coordinates ( e = 1 )
y 2 = r2 − x2 = (p − ex)2 − x2 = (1 − e2 )[a2 − (x + ea)2 ]
y2
(x + ea)2
+
=1
a2 (1 − e2 )
a2
Semiminor Axis:


b2 = a2 (1 − e2 ) = |a| p

Ellipse

Fig. 3.1 from An Introduction to the Mathematics and
Methods of Astrodynamics. Courtesy of AIAA. Used
with permission.

Hyperbola
Fig. 3.2 from An Introduction to
the Mathematics and Methods
of Astrodynamics. Courtesy of
AIAA. Used with permission.

Parabola in rectangular coordinates ( e = 1)
y 2 = r2 − x2 = (p − x)2 − x2

=⇒

Parabola
16.346 Astrodynamics

Lecture 2


y 2 = 2p( 12 p − x)

Fig. 3.3 from An Introduction to the
Mathematics and Methods of
Astrodynamics. Courtesy of AIAA.
Used with permission.

Alternate Forms of the Equation of Orbit
#4.1
Origin at focus
r + ex = p
Origin at center
r + ex = a
With x now measured from the center which is at a distance ae from the focus, then
r + ex = p
r + e(x − ae) = p = a(1 − e2 )
r + ex = a
Origin at pericenter r + ex = q
With x now measured from pericenter which is at a distance of a from the center and a
distance of q = a(1 − e) from the focus, then
r + ex = p
r + e(x + q) = p = q(1 + e)
r + ex = q
These are useful to derive other properties of conic sections:
• Focus-Directrix Property:
r = p − ex:
Page 144

p

PF = r = e
− x = e × PN
e
or
PF

=e
PN
• Focal-Radii Property:

r = a − ex:

Page 145

P F 2 = (x − ea)2 + y 2
P F ∗2 = (x + ea)2 + y 2
P F ∗2 = P F 2 + 4aex


so that

= r2 + 4aex

= (a − ex)2 + 4aex = (a + ex)2

a + ex
−(a + ex)

PF∗ =

ellipse
hyperbola

a>0
a < 0, x < 0

Thus,
P F ∗ + P F = 2a
P F ∗ − P F = −2a
• Euler’s Universal Form:

ellipse
hyperbola

From r = q − ex:

Page 143

y 2 = r2 − (q + x)2 = (q − ex)2 − (q + x)2
Then

16.346 Astrodynamics

y 2 = −(1 + e)[2qx + (1 − e)x2 ]

Lecture 2


Basic Two-Body Relations
d2 r
µ
+ 3r = 0
2
dt
r

Vector Equations of Motion

dv
=0
dt

r×

Angular Momentum Vector

dv
×h
dt

Eccentricity Vector

µe · r

Equation of Orbit

=⇒
=⇒

h × µe

Velocity Vector

=⇒

p≡

h2
µ

dv
µ
= − 3r
dt
r

r × v = constant ≡ h

1
v × h − ir = constant ≡ e
µ
r=

=⇒

p
h2 /µ
=
1 + e cos f
1 + e cos f

v=

1
h × (e + ir )
p

p

Orbital Parameter
Dynamics Definition:

or

p = a(1 − e2 )

Geometric Definition:

Total Energy or Semimajor Axis or Mean Distance

a

1 2 µ
µ
v − = constant ≡ −
2
r
2a
2
2
(x + ea)
y
Geometric Definition:
=1
+ 2
2

a
a (1 − e2 )
Dynamics Definition:

dθ
d2 r
−
r
dt2
dt

Eqs. of Motion in Polar Coord.

2

+

µ
=0
r2

d 2 dθ
r
dt
dt

Kepler’s Laws
dA
1 dθ
h

= r2
= constant =
dt
2 dt
2

Second Law
First Law

r=

Third Law

16.346 Astrodynamics

p
1 + e cos f

h
πab
=
P
2

or
=⇒

r = p − ex
a3
µ

= constant =
2
P
4π 2

Lecture 2

=0