Advanced Propulsion Anmaer Propulsion Anmaer rocket uses the reacon of maer and anmaer to create electricity, to generate thrust by expelling the products of the reacon; or to heat a gas which well be expelled for thrust. Why anmaer? Anmaer has an extremely high energy density.

Anmaer rockets would have extremely high exhaust velocies (over 105 km/s)

Also capable of producing high thrusts Fast missions to Mars or outer planets

Storing requires magnec Potenal for unmanned or manned field traps interstellar missions 2 Anmaer Propulsion Many technical challenges to be overcome:

Trapping anmaer is difficult

The world produces between 1 and 10 nanograms of anmaer per year

Most expensive substance on Earth: $62.5 trillion/gram

Energy conversion requires some technical miracles to be overcome PENN State is studying anmaer trapping and producon

They also design anmaer rockets 3 Planetary Moon and Orbital Mechanics

4 Kepler’s Laws

1. Planets move in elliptical orbits with the Sun at one a focus of the ellipse.

Eccentricity = distance between foci/length of major axis

5 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 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 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 In general

P2 = a3 P2 ∝ a3

where P is in years and a is in P2 = k a3 astronomical units (AU).

P and a are in arbitrary units k was a measured quantity for Kepler

8 Kepler’s 3rd Law and the Planets Planet Period (years) Distance (AU) Eccentricity Mercury 0.24 0.38 0.206 Venus 0.62 0.72 0.007 Earth 1.00 1.0 0.017 Mars 1.88 1.52 0.093 Jupiter 11.85 5.2 0.049 Saturn 29.46 9.54 0.056 Uranus 84.07 19.18 0.044 Neptune 164.82 30.06 0.011 Pluto 248.6 39.44 0.249

9 Kepler’s 3rd Law and the Planets Planet Period (years) Distance (AU) Eccentricity Mercury 0.24 0.38 0.206 Venus 0.62 0.72 0.007 Earth 1.00 1.0 0.017 Mars 1.88 1.52 0.093 Jupiter 11.85 5.2 0.049 Saturn 29.46 9.54 0.056 Uranus 84.07 19.18 0.044 Neptune 164.82 30.06 0.011 Pluto 248.6 39.44 0.249

10 What’s really governing planetary moon?

Newton’s 1st 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 What’s really governing planetary moon?

Newton’s 2nd 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 What’s really governing planetary moon?

Newton’s 3rd 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 What’s really governing planetary moon? Isaac Newton (1642-1727): 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

-11 3 -1 -2 G = 6.7 x 10 m kg s (gravitational constant) 14 What’s really governing planetary moon? 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 10-11 m3 kg-1 s-2

15 Geosynchronous Orbit For some applicaons, we want to keep a satellite over a single point above the Earth.

P = 24 hours = 86,400 s -11 3 -2 -1 G = 6.67 x 10 m s kg What is the semimajor axis? Mass of the Earth: 24 2 M = 5.97 x 10 kg ⎛ P ⎞ a3 ⎜ ⎟ = ⎝ 2π ⎠ GM 1/3 ⎛ GMP2 ⎞ a ⎜ ⎟ = ⎜ 2 ⎟ ⎝ 4π ⎠ 1/3 ⎛ 6.67x10 −11m3s−2kg −1 5.97x10 24 kg (86,400s)2 ⎞ = ⎜ ( )( ) ⎟ ⎜ 2 ⎟ ⎝ 4π ⎠

4.16x107 m 41, 640km 6.6Re = = = 16 What do we need for a mission to Mars?

17 Moons to consider

1) Orbital moon of the Earth 2) Orbital moon of Mars 3) Launch of spacecra off Earth 4) Escape of spacecra from Earth 5) Orbital moon of the spacecra 6) Capture of spacecra by Mars

18 Orbital Velocity of Earth For orbits that are approximately circular, the orbital speed is given by: 2πa v = orbit P For Earth: a = 1.5x108 km P = 3.1x107 s (1 year)

2π (1.5×108 km) vorbit = 7 v = 29.8 km/s 3.1×10 s Earth

19 Moons to consider

1) Orbital moon of the Earth 2) Orbital moon of Mars 3) Launch of spacecra off Earth 4) Escape of spacecra from Earth 5) Orbital moon of the spacecra 6) Capture of spacecra by Mars

20 Orbital Velocity of Mars For orbits that are approximately circular, the orbital speed is given by: 2πa v = orbit P

For Mars: Mars a = 2.3x108 km P = 5.8x107 s

2π (2.3×108 km) v = orbit 5.8×107 s vMars = 24.2 km/s

21 Moons to consider

1) Orbital moon of the Earth 2) Orbital moon of Mars 3) Launch of spacecra off Earth 4) Escape of spacecra from Earth 5) Orbital moon of the spacecra 6) Capture of spacecra by Mars

22 Escape Velocity Need kinec energy to be greater than potenal energy to escape Earth’s gravity What speed do we need? 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 2GM planet vesape = not fired connuously. Rplanet 5. Less inial speed is needed to get to orbit. 23 2GM v planet Escape Velocies esape = Rplanet

Planet Mass (kg) Radius v escape (m) (km/s) Mercury 3.3x1023 2.4x106 4.4 Venus 4.9x1024 6.0x106 10.4 Earth 6.0x1024 6.4x106 11.2 Moon 7.36x1022 1.7x106 2.6 Mars 6.4x1023 3.4x106 5.0 Jupiter 1.9x1027 7.1x107 59.7 Saturn 5.7x1026 6.0x107 35.5 Uranus 8.7x1025 2.6x107 21.3 Neptune 1.0x1026 2.5x107 23.5 24 Moons to consider

1) Orbital moon of the Earth 2) Orbital moon of Mars 3) Launch of spacecra off Earth 4) Escape of spacecra from Earth 5) Orbital moon of the spacecra 6) Capture of spacecra by Mars

25 We also need enough energy to get into transfer orbit

Two velocity requirements for geng into transfer orbit: 1. Spacecra 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. Spacecra also needs an addional change in velocity to get into the transfer orbit The spacecra accelerates (remember that acceleraon is the change in velocity over me) to these velocies using propulsion But what is this addional change in velocity to get into the transfer orbit?

26

Moons to consider

1) Orbital moon of the Earth 2) Orbital moon of Mars 3) Launch of spacecra off Earth 4) Escape of spacecra from Earth 5) Orbital moon of the spacecra 6) Capture of spacecra by Mars

27 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 1.5 AU 1.0 AU requires the minimum energy (usually) What is the semimajor axis of this orbit? 2a = 1.5 AU + 1 AU = 2.5 AU Earth’s orbit

a = 1.25 AU spacecra’s orbit

Mars’ orbit

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 1.5 AU 1.0 AU requires the minimum energy (usually) What is the semimajor axis of this orbit? a = 1.25 AU Earth’s orbit What is the me required? Kepler’s 3rd Law: spacecra’s orbit P2 = a3 P = (a3)1/2 P = (1.253)1/2 Mars’ orbit = 1.4 yrs

Travel me = 0.7 years = 8.4 months 29 Earth–Mars (Hohmann) Transfer Orbit: How much change in velocity is needed? For a circular orbit 2πa vorbit = 1.5 AU 1.0 AU P Transfer orbit is actually elliptical so velocity Earth’s orbit depends on location in V2 V1 orbit (this results from conservation of energy spacecra’s orbit and Kepler’s 2nd law regarding equal areas in equal times) Mars’ orbit

30 Earth–Mars Transfer Orbit: How much change in velocity is needed?

• We can calculate this.

V1 = 30.6 km/sec 1.5 AU 1.0 AU V2 = 21.8 km/sec

• Recall that the Earth and Mars are moving at Earth’s orbit V2 29.8 km/sec and 24.2 V1 km/sec. spacecra’s orbit • Our satellite must leave going 0.8 km/sec faster than Earth and arrive at Mars Mars’ orbit going 2.4 km/sec slower than Mars. 31