An1ma3er Propulsion. Why an1ma3er? An1ma3er rockets would have extremely high exhaust veloci1es (over 10 5 km/s)

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Alberta Sabina Thornton
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1 Advanced Propulsion

2 Why an1ma3er? An1ma3er has an extremely high energy density. An1ma3er Propulsion An#ma&er rocket uses the reac1on of ma3er and an1ma3er to create electricity, to generate thrust by expelling the products of the reac1on; or to heat a gas which well be expelled for thrust. An1ma3er rockets would have extremely high exhaust veloci1es (over 10 5 km/s) Also capable of producing high thrusts Storing requires magne1c field traps Fast missions to Mars or outer planets Poten1al for unmanned or manned interstellar missions 2

3 Many technical challenges to be overcome: Trapping an1ma3er is difficult The world produces between 1 and 10 nanograms of an1ma3er per year Most expensive substance on Earth: $62.5 trillion/gram An1ma3er Propulsion Energy conversion requires some technical miracles to be overcome PENN State is studying an1ma3er trapping and produc1on They also design an1ma3er rockets 3

4 Planetary Mo1on and Orbital Mechanics 4

5 Kepler s Laws 1. Planets move in elliptical orbits with the Sun at one focus of the ellipse. a Eccentricity = distance between foci/length of major axis 5

6 Kepler s Laws 2. The orbital period of a planet varies such that a line joining the Sun and the planet will sweep equal areas in equal time intervals. 6

7 Kepler s Laws 2. The orbital period of a planet varies such that a line joining the Sun and the planet will sweep equal areas in equal time intervals. 7

8 Kepler s Laws 3. The amount of time a planet takes to orbit the Sun is related to its orbit s size such that the period P, squared, is proportional to the semi-major axis, a, cubed Planets around the sun P 2 = a 3 where P is in years and a is in astronomical units (AU). In general P 2 a 3 P 2 = k a 3 P and a are in arbitrary units k was a measured quantity for Kepler 8

9 Kepler s 3 rd Law and the Planets Planet Period (years) Distance (AU) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Eccentricity

Kepler s 3 rd Law and the Planets Planet Period (years) Distance (AU) Mercury 0.24 0.38 Venus 0.62 0.72 Earth 1.00 1.0 Mars 1.88 1.52 Jupiter 11.

10 Kepler s 3 rd Law and the Planets Planet Period (years) Distance (AU) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Eccentricity

11 What s really governing planetary mo1on? Newton s 1 st Law of Motion: From Newton s Principia published in 1687: Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare. An object at rest will remain at rest unless acted upon by an external and unbalanced force. An object in motion will remain in motion unless acted upon by an external and unbalanced force. 11

12 What s really governing planetary mo1on? Newton s 2 nd Law of Motion: From Newton s Principia published in 1687: Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur. The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed. 12

13 What s really governing planetary mo1on? Newton s 3 rd Law of Motion: From Newton s Principia published in 1687: Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi. For a force there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions. 13

14 What s really governing planetary mo1on? Isaac Newton ( ): Discoveries are the core for most of our understanding of gravity and motion Law of Universal Gravity: Massive objects attract F is gravitational force of attraction (Newton) M = mass (kg) of one object m is mass (kg) of second object r = distance (m) between the two objects G = 6.7 x m 3 kg -1 s -2 (gravitational constant) 14

15 What s really governing planetary mo1on? Newton discovered that the planets are moving and that they are attracted to the Sun. This allows for the elliptical orbits and can prove Kepler s third law, which in general is P = orbital period (seconds) a = semimajor axis (m) M = mass of system (kg) G = 6.7 x m 3 kg -1 s -2 15

16 Geosynchronous Orbit For some applica1ons, we want to keep a satellite over a single point above the Earth. P = 24 hours = 86,400 s G = 6.67 x m 3 s - 2 kg - 1 What is the semimajor axis? a P 2π 2 = GMP = = 3 a GM 2 2 4π 1/3 Mass of the Earth: M = 5.97 x kg ( )( x10 m s kg 5.97x10 kg )( 86,400s) 4π 2 2 1/3 = 4.16x10 7 m = 41, 640km = 6.6 Re 16

17 What do we need for a mission to Mars? 17

18 Mo1ons to consider 1) Orbital mo8on of the Earth 2) Orbital mo1on of Mars 3) Launch of spacecrae off Earth 4) Escape of spacecrae from Earth 5) Orbital mo1on of the spacecrae 6) Capture of spacecrae by Mars 18

19 Orbital Velocity of Earth For orbits that are approximately circular, the orbital speed is given by: 2πa v orbit = P For Earth: a = 1.5x10 8 km P = 3.1x10 7 s (1 year) 2 ( = π v orbit 7 8 km) s v Earth = 29.8 km/s 19

20 Mo1ons to consider 1) Orbital mo1on of the Earth 2) Orbital mo8on of Mars 3) Launch of spacecrae off Earth 4) Escape of spacecrae from Earth 5) Orbital mo1on of the spacecrae 6) Capture of spacecrae by Mars 20

21 Orbital Velocity of Mars For orbits that are approximately circular, the orbital speed is given by: 2πa v orbit = P For Mars: a = 2.3x10 8 km P = 5.8x10 7 s Mars 2 ( = π v orbit 7 8 km) s v Mars = 24.2 km/s 21

22 Mo1ons to consider 1) Orbital mo1on of the Earth 2) Orbital mo1on of Mars 3) Launch of spacecrac off Earth 4) Escape of spacecrae from Earth 5) Orbital mo1on of the spacecrae 6) Capture of spacecrae by Mars 22

Escape Velocity Need kine1c energy to be greater than poten1al energy to escape Earth s gravity What speed do we need? v esape = 2GM R planet planet 1. Escape velocity is independent of rocket mass.

23 Escape Velocity Need kine1c energy to be greater than poten1al energy to escape Earth s gravity What speed do we need? v esape = 2GM R planet planet 1. Escape velocity is independent of rocket mass. 2. Only depends on planet mass and radius. 3. This does not include energy lost to the atmosphere. 4. This assumes the rocket is not fired con1nuously. 5. Less ini1al speed is needed to get to orbit. 23

24 Escape Veloci1es v esape = 2GM R planet planet Planet Mass (kg) Radius (m) v escape (km/s) Mercury 3.3x x Venus 4.9x x Earth 6.0x x Moon 7.36x x Mars 6.4x x Jupiter 1.9x x Saturn 5.7x x Uranus 8.7x x Neptune 1.0x x

25 Mo1ons to consider 1) Orbital mo1on of the Earth 2) Orbital mo1on of Mars 3) Launch of spacecrae off Earth 4) Escape of spacecrac from Earth 5) Orbital mo1on of the spacecrae 6) Capture of spacecrae by Mars 25

26 We also need enough energy to get into transfer orbit Two velocity requirements for gehng into transfer orbit: 1. Spacecrae must change it s velocity to get into Low Earth Orbit (LEO). Note that this change in velocity is less than the escape velocity of the Earth. 2. Spacecrae also needs an addi#onal change in velocity to get into the transfer orbit The spacecrae accelerates (remember that accelera#on is the change in velocity over #me) to these veloci1es using propulsion But what is this addi#onal change in velocity to get into the transfer orbit? 26

27 Mo1ons to consider 1) Orbital mo1on of the Earth 2) Orbital mo1on of Mars 3) Launch of spacecrae off Earth 4) Escape of spacecrae from Earth 5) Orbital mo8on of the spacecrac 6) Capture of spacecrae by Mars 27

28 Hohmann Transfer Kepler s and Newton s laws provide a way to calculate the path between to bodies in the solar system. Hohmann Transfer: transfer orbit that requires the minimum energy (usually) 1.5 AU 1.0 AU What is the semimajor axis of this orbit? 2a = 1.5 AU + 1 AU = 2.5 AU a = 1.25 AU Earth s orbit spacecrae s orbi Mars orbit 28

29 Hohmann Transfer Kepler s and Newton s laws provide a way to calculate the path between to bodies in the solar system. Hohmann Transfer: transfer orbit that requires the minimum energy (usually) 1.5 AU 1.0 AU What is the semimajor axis of this orbit? a = 1.25 AU What is the 1me required? Kepler s 3 rd Law: P 2 = a 3 P = (a 3 ) 1/2 P = ( ) 1/2 = 1.4 yrs Earth s orbit spacecrae s orbi Mars orbit Travel 1me = 0.7 years = 8.4 months 29

30 Earth Mars (Hohmann) Transfer Orbit: How much change in velocity is needed? For a circular orbit 2πa v orbit = P Transfer orbit is actually elliptical so velocity depends on location in orbit (this results from conservation of energy and Kepler s 2 nd law regarding equal areas in equal times) V2 1.5 AU 1.0 AU Earth s orbit V1 spacecrae s orbi Mars orbit 30

31 Earth Mars Transfer Orbit: How much change in velocity is needed? We can calculate this. V1 = 30.6 km/sec V2 = 21.8 km/sec Recall that the Earth and Mars are moving at 29.8 km/sec and 24.2 km/sec. Our satellite must leave going 0.8 km/sec faster than Earth and arrive at Mars going 2.4 km/sec slower than Mars. V2 1.5 AU 1.0 AU Earth s orbit V1 spacecrae s orbi Mars orbit 31

Lecture 4: Newton s Laws

Lecture 4: Newton s Laws Laws of motion Reference frames Law of Gravity Momentum and its conservation Sidney Harris This week: continue reading Chapter 3 of text 2/6/14 1 Newton s Laws & Galilean Relativity