What do we need for a mission to Mars? 1 Mo3ons to consider 1)  Orbital mo3on of the Earth 2) Orbital mo3on of Mars 3) Launch of spacecra? oﬀ Earth 4) Escape of spacecra? from Earth 5) Orbital mo-on of the spacecra5 6) Capture of spacecra? by Mars 2 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 spacecra?’s orbit Mars’ orbit 3 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 3me required? Kepler’s 3rd Law: P2 = a3 P = (a3)1/2 P = (1.253)1/2 = 1.4 yrs Travel 3me = 0.7 years = 8.4 months Earth’s orbit spacecra?’s orbit Mars’ orbit 4 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 spacecra?’s orbit Mars’ orbit 5 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 spacecra?’s orbit Mars’ orbit 6 Mo3ons to consider 1)  Orbital mo3on of the Earth 2) Orbital mo3on of Mars 3) Orbital mo3on of the spacecra? 4) Launch of spacecra? oﬀ Earth 5) Escape of spacecra? from Earth 6) Capture of spacecra5 by Mars 7 Will Mars capture the spacecra?? •  Spacecraft traveling 2.4 km/s slower than Mars’ orbital velocity •  This is less than the escape velocity for Mars (5 km/s) •  Hence, spacecraft is captured by Mars’ gravity when it arrives near the planet 8 Mission Plan: How long would a round trip Earth-‐Mars mission take? •  Period of transfer orbit is 1.4 years (from P2 = a 3) •  So Earth to Mars takes 0.7 years or 8.4 months •  Planets are orbiting the sun, so we have to launch at just the right time for the spacecraft to rendezvous with Mars 9 This is a roundtrip 3cket right? •  Need to leave Mars when Earth is 8.4 months behind the location where the transfer orbit will take the spacecraft •  But when we arrive at Mars, Earth is only 3.6 months behind •  We need to wait 15.4 months after arriving for Earth and Mars to line up right 10 Mission to Mars! •  Total mission time is then 8.4 + 15.4 +8.4 =32.2 months or 2.6 years Courtesy of Touchstone Pictures What could be some of the hazards/problems associated with a 2.6 year journey? 11 There is an express ﬂight, but it will cost you! •  There is more than 1 transfer orbit •  After reaching escape velocity, accelerate spacecraft to 7 km/s instead of 0.8 km/s •  Earth to Mars in 3 months •  But higher velocity means higher fuel costs…because you have to slow down at Mars! 12 Return travel 3me = 4 months What about hazards for this journey? 13 How much more energy needed for fast mission? •  Kinetic energy is Ek = ½ mv2 •  If payload is mass is m=2000 kg, then slow mission: Ek = ½ (2000 kg) (0.8 km/s)2=6.4 x 108 Joules fast mission: Ek = ½ (2000 kg) (7 km/s)2=4.9 x1010 Joules Fast mission to Mars requires 76 times more energy 14 Inclined Orbits We have talked about orbits and implied that they were in the equatorial plane. These orbits are called equatorial orbits. You can have orbits with arbitrary inclina3ons. If the angle of inclina3on is 90°, the we call it a polar orbit. Why use a polar orbit? Orbital period is ~ 100 minutes Al3tude ~ 1000 km 15 Crea3ve Solu3on: Molniya Orbits Good orbit design can make other problems simpler. Russians wanted military communica3ons satellite network in the 1960s. Geosta3onary satellites are too far south from Russia – need an extremely powerful radio. Solu3on: Molniya orbit 1.  Highly ellip3cal 2.  Inclined orbit 63.4° 3.  Need only 3 satellites 4.  Each one spends about 8 hours over Russia 16 Hyperbolic & Parabolic Orbits All of the orbits so far have been ellipses (including circular orbits). When v = vescape at closest approach, we call this a parabolic orbit. Types of Parabolic Orbits: 1.  Escape orbit – object has the escape velocity and is moving away from the planet 2.  Capture orbit – object has the escape velocity and is moving towards the planet Hyperbolic orbits have a speed greater than the escape velocity at closest approach. 17 Lagrange Points & Halo Orbits Lagrange Points: a small body under the inﬂuence of gravity will remain stable Examples: L1: SOHO L2: WMAP L4 & L5: Trojan asteroids at Jupiter L1 – Gravity of two bodies is in balance L2 – Gravity of two bodies balances the centrifugal force L3 – Slightly inside orbit; aﬀected by both bodies L4 – 60° in front of orbi3ng body (distances to both masses are equal) L5 – 60° in front of orbi3ng body 18 Lagrange Points & Halo Orbits Halo Orbit: 3D orbit near L1, L2, or L3 Complicated structure due to 3-‐Body Problem. Orbits are unstable so sta3on keeping is required. Examples: SOHO, Genesis 19 Orbital Maneuvers We have already discussed one type of orbital maneuver, the Hohmann transfer, when we mapped out a poten3al mission to Mars. Hohmann transfer: 1.  Transfer between two coplanar circular orbits 2.  Requires two engine burns 3.  Lowest energy transfer if R’/R < 12 4.  Assume impulsive thrust 20 Orbital Maneuvers Other orbital maneuvers are required for diﬀerent applica3ons. R’ R Bi-‐ellip@c transfer: 1.  Transfer between two coplanar circular orbits 2.  Requires three engine burns 3.  Lower energy than Hohmann transfer if R’/R > 12 4.  Assume impulsive thrust 5.  Longer 3me than Hohmann transfer 21 Gravita3onal Assist Some3mes using a Hohmann transfer or bi-‐ellip3c transfer is too energy intensive. Our rocket cannot carry that much fuel. Gravita3onal Assist: 1.  Used to increase speed, decrease speed, and change direc3on 2.  Relies on using the rela3ve mo3on of the planet and spacecra? 3.  Saves fuel, 3me, and money 22 Gravita3onal Assist Some3mes using a Hohmann transfer or bi-‐ellip3c transfer is too energy intensive. Our rocket cannot carry that much fuel. How it works (simpliﬁed): 1.  Spacecra? moves towards planet with speed v (rela3ve to Sun) 2.  Planet moving towards spacecra? at speed u (rela3ve to Sun) 3.  Spacecra? moves at speed u + v with respect to planet’s surface (incoming) 4.  Spacecra? moves at speed u + v with respect to planet’s surface (outgoing) 5.  Spacecra? moves away from the planet with speed 2u+v (rela3ve to Sun) Oberth Eﬀect: gravita3onal assist with thrusters Is this all science ﬁc3on? 23 Gravita3onal Assist Notable uses: 1.  Mariner 10 2.  Voyager I & II 3.  Galileo 4.  Ulysses 5.  Cassini 6.  MESSANGER 24 Gravita3onal Assist 25 Gravita3onal Assist 26 Gravita3onal Assist 27 Aerobraking 28 Aerobraking 29 But is it really that simple… •  Lunar Mascons (aka mass concentra3ons) make stable low lunar obits diﬃcult (impossible?) to ﬁnd. Konopliv et al, Icarus 150, 1–18 (2001). 30 But is it really that simple… •  So why not just ﬂy high al3tude orbits? Because the Earth gravita3onally disturbs high al3tude, circular orbits. •  But there might be highly ellip3cal, high inclina3on orbits that could be stable for about 100 years. Todd Ely and Erica Lieb, Stable Constella3ons of Frozen Ellip3cal Inclined Lunar Orbits, Journal of the Astronau3cal Sciences, vol. 53, No. 3, July-‐Sept 2005, pp. 301-‐316 31 Other Orbital Maneuvers •  •  •  •  Orbital Inclina3on Change Phasing Rendezvous Docking 32 Orbital Sta3on-‐Keeping Any real orbit will change with 3me due to perturba3ons from other bodies in the solar system. Example: satellite is orbit around the Earth is perturbed by the Sun, Moon, Jupiter… Orbital Sta@on-‐Keeping: ﬁring thruster to keep a spacecra? in a par3cular orbit Typically a small set of thrusters are used. These are called the aDtude control system (ACS). Now this process is automated by an onboard computer that collects telemetry and makes correc3ons. Sta3on-‐keeping is cri3cal for satellites that must be oriented in a certain direc3on to communicate with Earth (communica3on satellites) 33