1 Developed by Jim Beasley – Mathematics & Science Center –

2 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

3 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 AD 1543), Polish physician, mathematician and astronomer Proposed a heliocentric model of solar system with planets in circular orbits Tycho Brahe (AD AD 1601), Danish astronomer Developed theory that Sun orbits the Earth while other planets revolve about the Sun

4 Celestial Mechanics – Telescope invented in 1608; Galileo improved it and used it to observe 3 moons of Jupiter in – Johannes Kepler ( ), German astronomer, used Brahe’s data to formulate his basic laws of planetary motion – Sir Isaac Newton ( ), English physicist, astronomer and mathematician, built upon the work of his predecessor Kepler to derive his laws of motion and universal gravitation

5 Orbital Mechanics Based Upon Knowledge of Celestial Mechanics Not a Trivial Problem –Time consuming and compute intensive –Lesson makes assumptions and uses simplifications

6 Orbital Mechanics Cannonball Newton’s Concept of Orbital Flight The Cannonball Analogy

7 Orbital Mechanics Newton’s Concept of Orbital Flight The Cannonball Analogy

8 Orbital Mechanics Apoapsis Periapsis Newton’s Concept of Orbital Flight The Cannonball Analogy

9 Orbital Mechanics Cannonball Apoapsis Periapsis Newton’s Concept of Orbital Flight The Cannonball Analogy 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

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

11 Earth Orbiting Satellites “I” is Inclination

12 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

13 Rendezvous In Space Objective: Perform Calculations to Simulate Space Shuttle Orbit Transfer

14 Simplifying Assumptions Rendezvous In Space 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

15 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.

16 Basic Properties of an Ellipse Related to Orbiting Bodies A B a b c String Pin at Focus P Pencil Point Ellipse Construction

17 Basic Properties of an Ellipse Related to Orbiting Bodies A B a b c String Pin at Focus P Pencil Point Ellipse Construction a = Semi-Major Axis b = Semi-Minor Axis

18 Basic Properties of an Ellipse Related to Orbiting Bodies A B a b c String Pin at Focus P Pencil Point Ellipse Construction 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)

19 Basic Properties of an Ellipse A B a b c String Pin at Focus P Pencil Point Ellipse Construction 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

20 Basic Properties of an Ellipse A B a b c String Pin at Focus P Pencil Point Developed by Jim Beasley – Mathematics & Science Center – Triangle With Sides a-b-c is a Right Triangle; Therefore, a 2 = b 2 + c 2 (Knowing Any Two Sides, We Can Calculate the Third)

21 Basic Properties of an Ellipse A B a b c String Pin at Focus P Pencil Point Developed by Jim Beasley – Mathematics & Science Center – Triangle With Sides a-b-c is a Right Triangle; Therefore, a 2 = b 2 + c 2 (Knowing Any Two Sides, We Can Calculate the Third) From Previous Chart - e = 2c / 2a = c / a or c = e x a

22 Basic Properties of an Ellipse A B a b c String Pin at Focus P Pencil Point Triangle With Sides a-b-c is a Right Triangle; Therefore, a 2 = b 2 + c 2 (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-e 2 ) 1/2

23 Rendezvous In Space Orbit Transfer: Space Shuttle Orbit Space Station Orbit Transfer Orbit

24 Rendezvous In Space Orbit Transfer: Space Shuttle Orbit Space Station Orbit Transfer Orbit Periapsis Apoapsis

25 Rendezvous In Space Semi-major Axis (a) = (r periapsis + r apoapsis ) / 2 Orbit Transfer: Space Shuttle Orbit Space Station Orbit Transfer Orbit Periapsis Apoapsis

26 Rendezvous In Space Semi-major Axis (a) = (r periapsis + r apoapsis ) / 2 Eccentricity (e) = c / a = 1 - (r periapsis / a) Orbit Transfer: Space Shuttle Orbit Space Station Orbit Transfer Orbit Periapsis Apoapsis

27 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.

28 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. Explains why satellites move in circular orbits. The acceleration is towards the center of the circle - called centripetal acceleration. υ m M Centripetal Force r Force = Mass x Acceleration

29 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.

30 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.

31 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.

32 Universal Law of Gravitation Force = G x (M * m / r 2 ) Force = G x (M * m / r 2 ) Where G is the universal constant of gravitation, M and m are two masses and r is the separation distance between them. Rendezvous In Space M r M m

33 Rendezvous In Space Universal Law of Gravitation F = G x (M * m / r 2 ) From Newton’s second law F=ma; we can solve for acceleration (which is centripetal acceleration, g). g = GM / r 2 At the surface of the earth, this acceleration = 32.2 ft/sec 2 or 9.81 m/sec 2 GM = g x r 2, where r is the average radius of the Earth (6375kM) GM = x10 14 m 3 /s 2

34 Rendezvous In Space 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 C D 0 A B

35 Rendezvous In Space 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 As the planet moves close to the sun in it’s orbit, it speeds up. C D 0 A B

36 Rendezvous In Space 2) A line joining any planet to the sun sweeps out equal areas in equal time. Kepler’s Laws of Planetary Motion r aphelion And, at perihelion and aphelion the relative (or perpendicular) velocities are inversely proportional to the respective distances from the sun, by equation: r perihelion x v perihelion = r aphelion x v aphelion r perihelion vpvp vava

37 Rendezvous In Space r perihelion = a ( 1 - e ) and r aphelion = 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: V periapsis = v circular X  [(1+e) / (1-e)] and V apoapsis = v circular X  [(1-e) / (1+e)]

38 3) The square of the period of any planet about the sun is proportional to the planet’s mean distance from the sun. P 2 = a 3 Rendezvous In Space Kepler’s Laws of Planetary Motion a Period P is the time required to make one revolution

39 The Period of an Object in a Circular Orbit is: P = 2пr/ Where, r is the Radius of Circle and is Circular Velocity. Rendezvous In Space r υ υ υ

40 Rendezvous 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 = υ 2 / r υ m M Centripetal Force r

41 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 = υ 2 / r In addition, we know that Newton expressed gravity in his Universal Law of Gravity as: g = GM / r 2 Solving for Circular Velocity “υ” υ =  (GM / r)

42 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.

43 Equations Used in the TI-92, Cont’d Parametric Equations for Circle: Rendezvous In Space α y cos α = x / r ; therefore, x = r * cos α and sin α = y / r; therefore, y = r * sin α r x

44 Equations Used in the TI-92, Cont’d 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. Rendezvous In Space a α b

45 Rendezvous In Space Equations 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.