Download Lecture 27: Planetary Motion

Advanced Propulsion 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. Why an1ma3er? An1ma3er has an extremely high energy density. An1ma3er rockets would have extremely high exhaust veloci1es (over 105 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 An1ma3er Propulsion 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 Energy conversion requires some technical miracles to be overcome PENN State is studying an1ma3er trapping and produc1on They also design an1ma3er rockets 3 Planetary Mo1on and Orbital Mechanics 4 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 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
astronomical units (AU).
P2 = k a3
P and a are in arbitrary units
k was a measured quantity for Kepler 8 rd
Kepler’s 3 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 rd
Kepler’s 3 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 mo1on? 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 mo1on? 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 mo1on? 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 mo1on? 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
G = 6.7 x 10-11 m3 kg-1 s-2 (gravitational constant)
14 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 10-11 m3 kg-1 s-2
15 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 10-­‐11 m3 s-­‐2 kg-­‐1 What is the semimajor axis? Mass of the Earth: M = 5.97 x 1024 kg 2
3
a
⎛ P ⎞
⎜
⎟ =
GM
⎝ 2π ⎠
⎛ GMP
a = ⎜⎜
2
4
π
⎝
(
2
1/ 3
⎞
⎟⎟
⎠
⎛ 6.67 x10
= ⎜⎜
⎝
−11
3 −2
m s kg
−1
)(5.97 x10
4π 2
24
)
2
1/ 3
kg (86,400s ) ⎞
⎟
⎟
⎠
7
= 4.16x10 m = 41, 640km = 6.6 Re
16 What do we need for a mission to Mars? 17 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 Orbital Velocity of Earth For orbits that are approximately circular, the orbital speed is given by: vorbit
2πa
=
P
For Earth: a = 1.5x108 km P = 3.1x107 s (1 year) vorbit
2π (1.5 ×108 km)
=
7
3.1×10 s
vEarth = 29.8 km/s 19 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 Orbital Velocity of Mars For orbits that are approximately circular, the orbital speed is given by: vorbit
2πa
=
P
For Mars: a = 2.3x108 km P = 5.8x107 s vorbit
2π (2.3 ×108 km)
=
7
5.8 ×10 s
Mars vMars = 24.2 km/s 21 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? vesape =
2GM planet
R 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 23 to get to orbit. Escape Veloci1es vesape =
2GM planet
Planet Mass (kg) Radius (m) v escape (km/s) Mercury Venus Earth Moon Mars Jupiter Saturn Uranus Neptune 3.3x1023 4.9x1024 6.0x1024 7.36x1022 6.4x1023 1.9x1027 5.7x1026 8.7x1025 1.0x1026 2.4x106 6.0x106 6.4x106 1.7x106 3.4x106 7.1x107 6.0x107 2.6x107 2.5x107 4.4 10.4 11.2 2.6 5.0 59.7 35.5 21.3 23.5 R planet
24 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 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 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 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 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 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 3rd Law: P2 = a3 P = (a3)1/2 P = (1.253)1/2 = 1.4 yrs Travel 1me = 0.7 years = 8.4 months Earth’s orbit spacecrae’s orbit Mars’ orbit 29 Earth–Mars (Hohmann) Transfer Orbit: How much change in velocity is needed? For a circular orbit
vorbit
2πa
=
P
Transfer orbit is actually
elliptical so velocity
depends on location in
orbit (this results from
conservation of energy
and Kepler’s 2nd law
regarding equal areas in
equal times)
1.5 AU 1.0 AU V2 Earth’s orbit V1 spacecrae’s orbit 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
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 Earth’s orbit V1 spacecrae’s orbit Mars’ orbit 31