Developed by Jim Beasley – Mathematics & Science Center – http://mathscience.k12.va.us
1
Space Flight
Basis for modern space flight had it’s
origin in ancient times
 Until about 50 years ago, space flight
was just the stuff of fiction

– Jules Verne’s From the Earth to the Moon
– Buck Rogers and Flash Gordon movies
and cartoons

Russians launched the first artificial
Earth satellite in 1957
2
Celestial Mechanics

Origin in early astronomical observations
Ptolemy (AD 140), Egyptian astronomer,
mathematician

Thought Earth the center of Universe
• Possibly skewed his data to support theory
 Copernicus (AD 1473 - AD 1543), Polish
physician, mathematician and astronomer
• Proposed a heliocentric model of solar system
with planets in circular orbits
•
Tycho Brahe (AD 1546 - AD 1601), Danish
astronomer

Developed theory that Sun orbits the Earth
while other planets revolve about the Sun
•
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Celestial Mechanics
– Telescope invented in 1608; Galileo
improved it and used it to observe 3 moons of
Jupiter in 1610.
– Johannes Kepler (1571 - 1630), German
astronomer, used Brahe’s data to formulate
his basic laws of planetary motion
– Sir Isaac Newton (1642 - 1727), English
physicist, astronomer and mathematician,
built upon the work of his predecessor Kepler
to derive his laws of motion and universal
gravitation
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Orbital Mechanics
 Based Upon Knowledge of Celestial Mechanics
 Not a Trivial Problem
– Time consuming and compute intensive
– Lesson makes assumptions and uses
simplifications
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Orbital Mechanics
Newton’s Concept of Orbital Flight
The Cannonball Analogy
Cannonball
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Orbital Mechanics
Newton’s Concept of Orbital Flight
The Cannonball Analogy
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Orbital Mechanics
Newton’s Concept of Orbital Flight
The Cannonball Analogy
Apoapsis
Periapsis
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Orbital Mechanics
Newton’s Concept of Orbital Flight
The Cannonball Analogy
Apoapsis
Cannonball
Periapsis
Basic Concept of Space Flight:
- Increase in Speed at Apoapsis – Raises the Periapsis Altitude
- Decrease in Speed at Apoapsis – Lowers the Periapsis Altitude
- Increase in Speed at Periapsis – Raises the Apoapsis Altitude
- Decrease in Speed at Periapsis – Lowers the Apoapsis Altitude
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Earth Orbiting Satellites

Question: What keeps a satellite in
orbit?
– Velocity
– Gravity (Centripetal Force)
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Earth Orbiting Satellites
“I” is Inclination
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Space Travel

Question: From what you know now,
how could you move from point A to
point B in space?
Through a series of “change of
velocity” manuevers
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Rendezvous In Space
Objective: Perform Calculations to Simulate Space Shuttle Orbit
Transfer
Rendezvous In Space
Space Station Orbit
Original Shuttle Orbit
Space Shuttle Transfer Orbit
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Rendezvous In Space

Simplifying Assumptions
1) Perfectly round Earth
2) Perfectly circular orbits for Shuttle and Space
Station
3) Both orbits are in the same plane
4) Neglect gravitational force from moon and
planets
5) Increased velocity (Delta V Burn) applied to
Shuttle at periapsis and rendezvous at apoapsis
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Rendezvous In Space

Kepler’s Laws of Planetary Motion
1) All planets move in elliptical orbits about
the sun, with the sun at one focus.
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Basic Properties of an Ellipse Related to
Orbiting Bodies
Ellipse Construction
P
Pencil Point
String
a
b
A
Pin at
Focus
B
c
Pin at
Focus
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Basic Properties of an Ellipse Related to
Orbiting Bodies
Ellipse Construction
P
Pencil Point
String
a
b
A
Pin at
Focus
B
c
Pin at
Focus
a = Semi-Major Axis
b = Semi-Minor Axis
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Basic Properties of an Ellipse Related to
Orbiting Bodies
Ellipse Construction
P
Pencil Point
String
a
b
A
Pin at
Focus
B
c
Pin at
Focus
a = Semi-Major Axis
Eccentricity (e) = Dist (A to B)/ (Dist A to P to B)
b = Semi-Minor Axis
(An Ellipse Whose Eccentricity = 0 is a Circle)
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Basic Properties of an Ellipse
Ellipse Construction
P
Pencil Point
String
a
b
A
Pin at
Focus
B
c
Pin at
Focus
a = Semi-Major Axis
Eccentricity (e) = Dist (A to B)/ (Dist A to P to B)
b = Semi-Minor Axis
(An Ellipse Whose Eccentricity = 0 is a Circle)
Length of String = 2 x Semi-Major Axis (a) is Width of Ellipse
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Basic Properties of an Ellipse
P
Pencil Point
String
a
b
A
Pin at
Focus
B
c
Pin at
Focus
Triangle With Sides a-b-c is a Right Triangle; Therefore,
a2 = b2 + c2 (Knowing Any Two Sides, We Can Calculate the Third)
Developed by Jim Beasley – Mathematics & Science Center – http://mathscience.k12.va.us
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Basic Properties of an Ellipse
P
Pencil Point
String
a
b
A
Pin at
Focus
B
c
Pin at
Focus
Triangle With Sides a-b-c is a Right Triangle; Therefore,
a2 = b2 + c2 (Knowing Any Two Sides, We Can Calculate the Third)
From Previous Chart - e = 2c / 2a = c / a or c = e x a
Developed by Jim Beasley – Mathematics & Science Center – http://mathscience.k12.va.us
21
Basic Properties of an Ellipse
P
Pencil Point
String
a
b
A
Pin at
Focus
B
c
Pin at
Focus
Triangle With Sides a-b-c is a Right Triangle; Therefore,
a2 = b2 + c2 (Knowing Any Two Sides, We Can Calculate the Third)
From Previous Chart - e = 2c / 2a = c / a or c = e x a
Substituting and Solving, b (Semi-Minor Axis) = a x (1-e2)1/2
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Rendezvous In Space
Orbit Transfer:
Space
Station
Orbit
Transfer
Orbit
Space
Shuttle
Orbit
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Rendezvous In Space
Orbit Transfer:
Space
Station
Orbit
Apoapsis
Periapsis
Transfer
Orbit
Space
Shuttle
Orbit
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Rendezvous In Space
Orbit Transfer:
Space
Station
Orbit
Apoapsis
Periapsis
Transfer
Orbit
Space
Shuttle
Orbit
Semi-major Axis (a) = (rperiapsis + rapoapsis) / 2
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Rendezvous In Space
Orbit Transfer:
Space
Station
Orbit
Apoapsis
Periapsis
Transfer
Orbit
Space
Shuttle
Orbit
Semi-major Axis (a) = (rperiapsis + rapoapsis) / 2
Eccentricity (e) = c / a = 1 - (rperiapsis / a)
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Rendezvous In Space
 Newton’s Laws of Motion and Universal Gravitation
Were Based Upon Kepler’s Work
– Newton’s First Law
An object at rest will remain at rest unless acted
upon by some outside force. A body in motion will
remain in motion in a straight line without being
acted upon by a outside force.
Once in motion satellites
remain in motion.
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Rendezvous In Space
- Newton’s Second Law
If a force is applied to a body, there will be a
change in acceleration proportional to the
magnitude of the force and in the direction in
which it is applied.
Force = Mass x Acceleration
υ
m
r
Explains why satellites move in
circular orbits. The acceleration
is towards the center of the
circle - called centripetal
acceleration.
Centripetal
Force
M
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Rendezvous In Space
- Second Law
If a force is applied to a body, there will be a
change in acceleration proportional to the
magnitude of the force and in the direction in
which it is applied.
F = ma
Explains why planets (or satellites) move in circular (or
elliptical) orbits. The acceleration is towards the center of the
circle - called centripetal acceleration and is provided by
mutual gravitational attraction between the Sun and planet.
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Rendezvous In Space

Newton’s Laws of Motion
- Third Law
If Body 1 exerts a force on Body 2, then Body 2
will exert a force of equal strength but opposite
direction, on Body 1.
For every action there is an equal and opposite
reaction.
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Rendezvous In Space

Newton’s Laws of Motion
- Third Law
If Body 1 exerts a force on Body 2, then Body 2
will exert a force of equal strength but opposite
direction, on Body 1.
For every action there is an equal and opposite
reaction.
Rocket - Exhaust gases in one direction; Rocket is propelled in
opposite direction.
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Rendezvous In Space

Universal Law of Gravitation
r
M
M
m
Force = G x (M * m / r2)
Where G is the universal constant of gravitation, M and
m are two masses and r is the separation distance
between them.
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Rendezvous In Space

Universal Law of Gravitation
F = G x (M * m / r2)
From Newton’s second law F=ma; we can solve for
acceleration (which is centripetal acceleration, g).
g = GM / r2
At the surface of the earth, this acceleration = 32.2 ft/sec2 or 9.81
m/sec2
GM = g x r2, where r is the average radius of the Earth (6375kM)
GM = 3.986 x1014 m3/s2
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Rendezvous In Space

Kepler’s Laws of Planetary Motion
2) A line joining any planet to the sun sweeps
out equal areas in equalωtime.
A
B
r
0
ωΔt
D
C
Area of Shaded Segment From A to B = Area From C to D
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Rendezvous In Space

Kepler’s Laws of Planetary Motion
2) A line joining any planet to the sun sweeps
out equal areas in equalωtime.
A
B
r
0
ωΔt
D
C
As the planet moves close to the sun in it’s orbit, it
speeds up.
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Rendezvous In Space

Kepler’s Laws of Planetary Motion
2) A line joining any planet to the sun sweeps
out equal areas in equal time.
vp
rperihelion
raphelion
va
And, at perihelion and aphelion the relative (or perpendicular)
velocities are inversely proportional to the respective
distances from the sun, by equation:
rperihelion x vperihelion = raphelion x v aphelion
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Rendezvous In Space
rperihelion = a ( 1 - e ) and raphelion = a (1 + e)

Therefore, the velocity in orbit at these two
points can be most easily related:
Using the geometry of ellipses, one can show the two velocities
as:
Vperiapsis = vcircular X [(1+e) / (1-e)] and
Vapoapsis = vcircular X [(1-e) / (1+e)]
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Rendezvous In Space

Kepler’s Laws of Planetary Motion
3) The square of the period of any planet
about the sun is proportional to the planet’s
mean distance from the sun.
P2 = a3
a
Period P is the time required
to make one revolution
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Rendezvous In Space
r
υ
The Period of an Object in a Circular Orbit is:
P = 2пr/ υ
Where, r is the Radius of Circle and υ is Circular
Velocity.
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Rendezvous In Space

Lesson Objectives:
1) Derive Equation for Shuttle’s Circular Velocity
υ
m
r
Centripetal
Force
M
Knowing that the force acting on Shuttle is centrepetal force (an
acceleration directed towards center of the Earth), we can describe it by
the following equation:
a = υ2 / r
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Rendezvous In Space

Lesson Objectives (Continued):
– 1) Derive Equation for Shuttle’s Circular Velocity (Continued)
Also knowing that centripetal acceleration (a) is simply Earth’s
gravity (g), we can express the equation as:
g = υ2 / r
In addition, we know that Newton expressed gravity in his
Universal Law of Gravity as:
g = GM / r2
Solving for Circular Velocity “υ”
υ = (GM / r)
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Rendezvous In Space

Lesson Objectives (Continued):
2) Execute TI-92 Program “rendevu” to Perform Orbit Transfer
Calculations
Write Down Answers on Work Sheet
3) Develop Parametric Equations for Space Station Circular Orbit,
Original Space Shuttle Circular Orbit and Transfer Elliptical
Orbit in Terms of Semi-Major Axis, Space Station Orbital
Radius, Shuttle Orbital Radius and Eccentricity.
4) Graph the Data by selecting Green Diamond and “E” Key.
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Rendezvous In Space

Equations Used in the TI-92, Cont’d
Parametric Equations for Circle:
x
r
α
y
cos α = x / r ; therefore, x = r * cos α
and
sin α = y / r; therefore,
y = r * sin α
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Rendezvous In Space

Equations Used in the TI-92, Cont’d
Parametric Equations for an Ellipse:
a
α
b
x = a * cos α
and, y = b * sin α
Where a is the semimajor axis and b is the semiminor axis.
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Rendezvous In Space

Equations Used in the TI-92, Cont’d
Parametric Equations for Circular Orbits:
Space Stationxt1 = ssorad * cos (t)
yt1 = ssorad * sin (t)
Space Shuttle xt2 = shtlorad * cos (t)
yt2 = shtlorad * sin (t)
Parametric Equations for an Ellipse:
x = a * cos α
and, y = b * sin α
Where a is the semimajor axis and b is the semiminor axis.
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