1 Developed by Jim Beasley – Mathematics & Science Center – .

1Developed by Jim Beasley – Mathematics & Science Center – http://mathscience.k12.va.us

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Space FlightSpace FlightBasis for modern space flight had it’s origin

in ancient timesUntil 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

cartoonsRussians launched the first artificial Earth

satellite in 1957

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Celestial MechanicsCelestial MechanicsOrigin 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 MechanicsCelestial 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 MechanicsOrbital 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 MechanicsOrbital Mechanics

Cannonball

Newton’s Concept of Orbital Flight The Cannonball Analogy

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Orbital MechanicsOrbital MechanicsNewton’s Concept of Orbital Flight The Cannonball Analogy

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Apoapsis

Periapsis


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Cannonball

Apoapsis

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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Question: What keeps a satellite in orbit?– Velocity– Gravity (Centripetal Force)

Earth Orbiting Earth Orbiting SatellitesSatellites

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Earth Orbiting Earth Orbiting SatellitesSatellites

“I” is Inclination

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Space TravelSpace 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 SpaceRendezvous In Space

Rendezvous In Space

Space Station Orbit

Original Shuttle Orbit

Space Shuttle Transfer Orbit

Objective: Perform Calculations to Simulate Space Shuttle Orbit Transfer

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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 SpaceRendezvous In SpaceKepler’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

A B

ab

c

String

Pin at Focus

Pin at Focus

PPencil Point

Ellipse Construction

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Orbiting Bodies

A B

ab

c

String

Pin at Focus

Pin at Focus

PPencil Point


a = Semi-Major Axis

b = Semi-Minor Axis

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Orbiting Bodies

A B

ab

c

String

Pin at Focus

Pin at Focus

PPencil Point


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

A B

ab

c

String

Pin at Focus

Pin at Focus

PPencil Point


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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A B

ab

c

String

Pin at Focus

Pin at Focus

PPencil Point

Developed by Jim Beasley – Mathematics & Science Center – http://mathscience.k12.va.us

Triangle With Sides a-b-c is a Right Triangle; Therefore,

a2 = b2 + c2 (Knowing Any Two Sides, We Can Calculate the Third)

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A B

ab

c

String

Pin at Focus

Pin at Focus

PPencil Point

Developed by Jim Beasley – Mathematics & Science Center – http://mathscience.k12.va.us



From Previous Chart - e = 2c / 2a = c / a or c = e x a

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A B

ab

c

String

Pin at Focus

Pin at Focus

PPencil Point



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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Orbit Transfer:

Space Shuttle Orbit

Space Station Orbit

Transfer Orbit

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Orbit Transfer:

Space Shuttle Orbit

Space Station Orbit

Transfer Orbit

Periapsis

Apoapsis

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Semi-major Axis (a) = (rperiapsis + rapoapsis) / 2

Orbit Transfer:

Space Shuttle Orbit

Space Station Orbit

Transfer Orbit

Periapsis

Apoapsis

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Semi-major Axis (a) = (rperiapsis + rapoapsis) / 2

Eccentricity (e) = c / a = 1 - (rperiapsis / a)

Orbit Transfer:

Space Shuttle Orbit

Space Station Orbit

Transfer Orbit

Periapsis

Apoapsis

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Rendezvous In SpaceRendezvous 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 SpaceRendezvous 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.

Explains why satellites move in circular orbits. The acceleration is towards the center of the circle - called centripetal acceleration.

υm

MCentripetal

Force

r

Force = Mass x Acceleration

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Rendezvous In SpaceRendezvous 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 = maF = 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 SpaceRendezvous 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 SpaceRendezvous 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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Universal Law of Gravitation

Force = G x (M * m / rForce = G x (M * m / r22))

Where G is the universal constant of gravitation, M and m are two masses and r is the separation distance between them.


Mr

Mm

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Rendezvous In SpaceRendezvous In Space Universal Law of Gravitation

F = G x (M * m / rF = G x (M * m / r22))

From Newton’s second law F=ma; we can solve for acceleration (which is centripetal acceleration, g).

g = GM / rg = GM / r22

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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2) A line joining any planet to the sun sweeps out equal areas in equal time.

Kepler’s Laws of Planetary Motion

ω

ωΔt

r

Area of Shaded Segment From A to B = Area From C to D

CD

0

AB

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ω

ωΔt

r

As the planet moves close to the sun in it’s orbit, it speeds up.

CD

0

AB

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raphelion

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

rperihelion

vp

va

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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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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

Rendezvous In SpaceRendezvous In Space Kepler’s Laws of Planetary Motion

a

Period P is the time required to make one revolution

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The Period of an Object in a Circular Orbit is:

P = 2пr/

Where, r is the Radius of Circle and is Circular Velocity.


r

υ

υ

υ

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Rendezvous In SpaceRendezvous In Space Lesson Objectives:

1) Derive Equation for Shuttle’s Circular Velocity

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

υm

MCentripetal

Force

r

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Rendezvous In SpaceRendezvous 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 SpaceRendezvous In Space Lesson Objectives (Continued):

2) Execute TI-92 Program “rendevu” to Perform Orbit Transfer CalculationsWrite 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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Equations Used in the TI-92, Cont’dParametric Equations for Circle:


αy

cos α = x / r ; therefore, x = r * cos α

and

sin α = y / r; therefore, y = r * sin α

r

x

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Equations Used in the TI-92, Cont’dParametric 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.


aα

b

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Rendezvous In SpaceRendezvous In SpaceEquations Used in the TI-92, Cont’d

Parametric Equations for Circular Orbits:

Space Station-

xt1 = 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.

1 Developed by Jim Beasley – Mathematics & Science Center – .

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