Orbital Mechanics Design Project
- 1. MAE 146: Astronautics Design Project Winter 2014 Interplanetary Orbiter Mission Neptune Atmospheric and Interior Science Orbiter (NAISO) PREPARED BY: Luke Guirguis Jose Sepulveda Sean Godinez
- 2. CONSTANTS and KNOWN VALUES Sun (S): 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑖𝑜𝑛𝑎𝑙 𝑃𝑎𝑟𝑎𝑚𝑡𝑒𝑟 ∶ 𝜇 !"# = 132,712,000,000 𝑘𝑚! 𝑠! Earth (E): 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑖𝑜𝑛𝑎𝑙 𝑃𝑎𝑟𝑎𝑚𝑡𝑒𝑟: 𝜇 ! = 398,600 𝑘𝑚! 𝑠! 𝐸𝑎𝑟𝑡ℎ 𝑂𝑟𝑏𝑖𝑡 𝑅𝑎𝑑𝑖𝑢𝑠 𝑎𝑟𝑜𝑢𝑛𝑑 𝑆𝑢𝑛: 𝑅! = 149.6 ∗ 10! 𝑘𝑚 𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝐸𝑎𝑟𝑡ℎ: 𝑟! = 6,378 𝑘𝑚 𝑃𝑎𝑟𝑘𝑖𝑛𝑔 𝑂𝑟𝑏𝑖𝑡 𝑅𝑎𝑑𝑖𝑢𝑠 𝑎𝑡 𝐸𝑎𝑟𝑡ℎ: 𝑟!"#$%&'#,! = 6,628 𝑘𝑚 𝑇𝑟𝑢𝑒 𝐴𝑛𝑎𝑜𝑚𝑙𝑦 𝑜𝑛 𝑀𝑎𝑟𝑐ℎ 13𝑡ℎ, 2014: 𝜃!,! = 70.90868° 𝑀𝑒𝑎𝑛 𝑀𝑜𝑡𝑖𝑜𝑛 𝑜𝑓 𝐸𝑎𝑟𝑡ℎ: 𝑛! = 1.1407 ∗ 10!! 𝑑𝑒𝑔 𝑠 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝐸𝑎𝑟𝑡ℎ 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡𝑜 𝑆𝑢𝑛: 𝑣! = 29.7844 𝑘𝑚 𝑠 Neptune (N): 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑖𝑜𝑛𝑎𝑙 𝑃𝑎𝑟𝑎𝑚𝑡𝑒𝑟: 𝜇 ! = 6,835,100 𝑘𝑚! 𝑠! 𝑁𝑒𝑝𝑡𝑢𝑛𝑒 𝑂𝑟𝑏𝑖𝑡 𝑅𝑎𝑑𝑖𝑢𝑠 𝑎𝑟𝑜𝑢𝑛𝑑 𝑆𝑢𝑛: 𝑅! = 4.495 ∗ 10! 𝑘𝑚 𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑁𝑒𝑝𝑡𝑢𝑛𝑒: 𝑟! = 24,760 𝑘𝑚 𝑇𝑎𝑟𝑔𝑒𝑡 𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑃𝑎𝑟𝑘𝑖𝑛𝑔 𝑂𝑟𝑏𝑖𝑡 𝑅𝑎𝑑𝑖𝑢𝑠 𝐴𝑟𝑜𝑢𝑛𝑑 𝑁𝑒𝑡𝑝𝑢𝑛𝑒 ∶ 𝑟!"#$%! = 25,060 𝑘𝑚 𝑇𝑟𝑢𝑒 𝐴𝑛𝑎𝑜𝑚𝑙𝑦 𝑜𝑛 𝑀𝑎𝑟𝑐ℎ 13𝑡ℎ, 2014: 𝜃!,! = 274.44241° 𝑀𝑒𝑎𝑛 𝑀𝑜𝑡𝑖𝑜𝑛 𝑜𝑓 𝑁𝑒𝑡𝑝𝑢𝑛𝑒: 𝑛! = 6.9260 ∗ 10!! 𝑑𝑒𝑔 𝑠 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑁𝑒𝑝𝑡𝑢𝑛𝑒 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡𝑜 𝑆𝑢𝑛: 𝑣! = 5.4336 𝑘𝑚 𝑠 Centaur Upper Stage: 𝐼!" = 451 𝑠 𝐶𝑒𝑛𝑡𝑎𝑢𝑟 𝑀𝑎𝑠𝑠 = 2,243 𝑘𝑔 NASIO: 𝐼!" = 318 𝑠 𝑁𝐴𝑆𝐼𝑂 𝑀𝑎𝑠𝑠 = 1,800 𝑘𝑔
- 3. VARIABLES Transfer Phase 𝑎!: 𝑆𝑒𝑚𝑖 − 𝑀𝑎𝑗𝑜𝑟 𝐴𝑥𝑖𝑠 𝑜𝑓𝐸𝑙𝑙𝑖𝑝𝑡𝑖𝑐𝑎𝑙 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑂𝑟𝑏𝑖𝑡 𝑣!" ! !: 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 ℎ𝑒𝑙𝑖𝑜𝑐𝑒𝑛𝑡𝑟𝑖𝑐 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑡 𝐸𝑎𝑟𝑡ℎ 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝑓𝑜𝑟 𝑎 𝐻𝑜ℎ𝑚𝑎𝑛𝑛 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑣!,!: 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑓𝑜𝑟 𝑡ℎ 𝑒 𝐸𝑎𝑟𝑡ℎ 𝑐𝑒𝑛𝑡𝑟𝑖𝑐 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 ℎ 𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑒𝑠𝑐𝑎𝑝𝑒 𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦 𝑣!"#$%&'#,!: 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝐶𝑖𝑟𝑢𝑙𝑎𝑟 𝑃𝑎𝑟𝑘𝑖𝑛𝑔 𝑂𝑟𝑏𝑖𝑡 𝑎𝑟𝑜𝑢𝑛𝑑 𝐸𝑎𝑟𝑡ℎ 𝑣!!",!: 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑤𝑖𝑡ℎ𝑖𝑛 𝑠𝑝ℎ𝑒𝑟𝑒 𝑜𝑓 𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑐𝑒 𝑜𝑓 𝐸𝑎𝑟𝑡ℎ Δ𝑣!: 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 ∆𝑣 𝑡𝑜 𝑖𝑛𝑠𝑒𝑟𝑡 𝑡ℎ𝑒 𝑠𝑝𝑎𝑐𝑒𝑐𝑟𝑎𝑓𝑡 𝑜𝑛𝑡𝑜 𝑎 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝐸𝑎𝑟𝑡ℎ 𝑒𝑠𝑐𝑎𝑝𝑒 𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦 𝑇!: 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑇𝑖𝑚𝑒 𝑜𝑓 𝑆𝑝𝑎𝑐𝑒𝑐𝑟𝑎𝑓𝑡 𝑇!: 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑁𝑒𝑡𝑝𝑢𝑛𝑒 𝑂𝑟𝑏𝑖𝑡 𝜙: 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑃𝑙𝑎𝑛𝑒𝑡𝑠 𝑡𝑜 𝑅𝑒𝑛𝑑𝑒𝑧𝑣𝑜𝑢𝑠 𝑎𝑡 𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝛽!: 𝐿𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑔𝑒𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑎 ∆!: 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑎𝑖𝑚𝑖𝑛𝑔 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑒𝑠𝑐𝑎𝑝𝑒 𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦 Arrival Phase 𝑣!" ! !: 𝐻𝑒𝑙𝑖𝑜𝑐𝑒𝑛𝑡𝑟𝑖𝑐 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑡 𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝑣!,!: 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒𝑡𝑜𝑐𝑒𝑛𝑡𝑟𝑖𝑐 𝑎𝑟𝑟𝑖𝑣𝑎𝑙 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦 𝛽!: 𝐿𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑔𝑒𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐴𝑟𝑟𝑖𝑣𝑎𝑙 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑎 ∆!: 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑎𝑖𝑚𝑖𝑛𝑔 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡ℎ𝑒 𝑠𝑝𝑎𝑐𝑒𝑐𝑟𝑎𝑓𝑡 𝑠ℎ𝑜𝑢𝑙𝑑 𝑒𝑛𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑒 𝑜𝑓 𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑐𝑒 ℎ!!",!: 𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦 𝑒!!",! ∶ 𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦 𝑣!: 𝑃𝑒𝑟𝑖𝑎𝑝𝑠𝑖𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑡𝑟𝑎𝑗𝑒𝑐𝑡𝑜𝑟𝑦 Δ𝑣!: 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 ∆𝑣 𝑡𝑜 𝑖𝑛𝑠𝑒𝑟𝑡 𝑖𝑛𝑡𝑜 𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑂𝑟𝑏𝑖𝑡 𝑎𝑟𝑜𝑢𝑛𝑑 𝑁𝑒𝑝𝑡𝑢𝑛𝑒 𝑣!"#$%&'#,!: 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝐶𝑖𝑟𝑢𝑙𝑎𝑟 𝑃𝑎𝑟𝑘𝑖𝑛𝑔 𝑇𝑎𝑟𝑔𝑒𝑡 𝑂𝑟𝑏𝑖𝑡 𝑎𝑟𝑜𝑢𝑛𝑑 𝑁𝑒𝑝𝑡𝑢𝑛𝑒
- 4. CONTENTS Section page 1.0 Mission Objectives………………………………………………… 1 2.0 Launch to Parking Orbit Phase………………………............ 2 3.0 Interplanetary Transfer Phase……………………………….. 4 4.0 Planetary Arrival Phase…………………………………………. 7 5.0 Propellant Mass Requirements………………………………. 9 6.0 Mission Summary…………………………………………………... 10 7.0 Team Contributions…………………………………................... 12 8.0 References……………………………………………………………. 13
- 5. 1 1.0 MISSION OBJECTIVE The Neptune Atmospheric and Interior Science Orbiter (NAISO) will be orbiting Neptune to perform an in depth study of the planet. The only other spacecraft that has flown by Neptune is Voyager 2, which revealed Neptune had four rings, five moons, and a "Great Dark Spot". The “Great Dark Spot” had vanished by the time the Hubble Space Telescope studied Neptune five years later. NAISO will be launched and inserted into a circular parking orbit around Earth, sent to Neptune using a Hohmann transfer, and inserted into a circular orbit around Neptune. NAISO’s science objectives are: • Study Neptune’s atmosphere and measure composition, temperature, and other atmospheric properties • Asses what the “Great Dark Spot” was and possibly search for other occurrences • Map Neptune’s gravity and magnetic fields to offer insight on its internal structure and core NAISO is equipped with various instruments similar to the JUNO spacecraft sent to Jupiter operated by JPL, in order to carry out these scientific objectives.
- 6. 2 2.0 LAUNCH TO PARKING ORBIT PHASE The launch of mission NAISO is scheduled to take place on June 13th, 2014 (calculations shown in Section 3: Interplanetary Transfer Phase) under the assumptions that the orbits of the planets around the Sun are coplanar and circular. On this date, the required angular separation of Earth from Neptune is approximately 113.1571°. While under realistic conditions a launch would not be scheduled until June 16th, 2014. The launch site is located at Cape Canaveral, Florida, also known as the Kennedy Space Center, at 28.° 30' N Latitude and 80° 33' W Longitude. The vehicle will be placed in parking orbit at a height altitude of 250km with a 28.5° inclination, which means we will launch with an Azimuth of 90° because of the equation: 𝐼𝑛𝑐𝑙𝑖𝑛𝑡𝑎𝑖𝑜𝑛: 𝑖 = 28.5° 𝐿𝑎 = 28.5° 𝐴𝑧 = sin!! cos 𝑖 cos 𝐿𝑎 = 90° This launch Azimuth will allow us to have a launch window of around 2 hours according to the graph below:
- 7. 3 According to NASA's Horizon web interface, on the date of departure the Sun relative to the geocentric frame will give us a hyperbolic asymptote with a right ascension of: 𝑅𝑖𝑔ℎ𝑡 𝐴𝑠𝑐𝑒𝑛𝑖𝑜𝑛: 𝑅𝐴 = 05: 24: 0.09 HMS and a declination of 𝐷 𝐸𝐶 = 23: 10: 46.7 DMS The location of the hyperbolic asymptote is : 𝛽! = 72.1256° , 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑆𝑒𝑐𝑡𝑖𝑜𝑛 03 and ∆!= 9101.366 𝑘𝑚, 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑆𝑒𝑐𝑡𝑖𝑜𝑛 03
- 8. 4 3.0 INTERPLANETARY TRANSFER PHASE Once in a circular parking orbit around Earth ( 𝑟!"#$%&'#,! = 6,628 𝑘𝑚), the NAISO satellite will need to attain a ∆v1 with the upper stage Centaur V to begin its Hohmann transfer to Neptune. In order to calculate this, we need to find the required heliocentric velocity of the spacecraft: 𝑆𝑒𝑚𝑖 − 𝑀𝑎𝑗𝑜𝑟 𝐴𝑥𝑖𝑠 𝑜𝑓 𝐸𝑙𝑙𝑖𝑝𝑡𝑖𝑐𝑎𝑙 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑂𝑟𝑏𝑖𝑡: 𝑎! = 1 2 𝑅! + 𝑅! = 2.3223e09 𝑘𝑚 𝐻𝑒𝑙𝑖𝑜𝑐𝑒𝑛𝑡𝑟𝑖𝑐 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑡 𝐸𝑎𝑟𝑡ℎ 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒: 𝑣!" ! ! = 2𝜇 !"# 𝑅! − 𝜇 !!" 𝑎! = 41.4376 𝑘𝑚 𝑠 Thus, the velocity for the hyperbolic departure from Earth is: 𝑣!,! = 𝑣!" ! ! − 𝑣! = 11.6532 𝑘𝑚 𝑠 The location of the perigee of the departure hyperbola is found using the equations below: 𝛽! = cos!! 1 1 + 𝑟!"#$%&'#,! 𝑣!,! ! 𝜇 ! = 72.1256° ∆!= 𝑟!"#$%&'#,! 1 + 2𝜇 ! 𝑟!"#$%&'#,! ∗ 𝑣!,! ! = 9101.366 𝑘𝑚 The initial hyperbolic velocity ( 𝑣!!",!) at Earth departure is found using conservation of energy, where 𝑟!"#$%&'#,! is the radius of the parking orbit : 𝑣!!",! ! 2 − 𝜇 ! 𝑟!"#$%&'#,! = 𝑣!,! ! 2 − 𝜇 !"# 𝑟! Solving for 𝑣!!",! , the equation reduces to:
- 9. 5 𝑣!!",! = 𝑣!,! ! + 2𝜇 ! 𝑟!"#$%&'#,! = 16.0023 𝑘𝑚 𝑠 And the required ∆v to attain 𝑣!,! is: Δ𝑣! = 𝑣!!",! − 𝑣!"#$%&'#,! , where 𝑣!"#$%&'#,! = ! ! !!"#$%&'#,! = Δ 𝑣! = 𝑣!!",! − ! ! !!"#$%&'#,! = 8.2474 !" ! The required angular separation ( 𝜙) between the Earth and Neptune at the time of Earth departure in order to arrive at Neptune’s orbit and rendezvous is found using the time of the Hohmann transfer, which is half of the transfer period of the elliptical orbit: 𝑇! = 1 2 𝑇 = 1 2 2𝜋 𝜇 !"# 𝑎! !/! = 965099874 𝑠 One period of Neptune’s orbit is: 𝑇! = !! ! !"# 𝑅! !/! = 5.1978𝑒09 s Thus the required angular separation is found using the relation: 180° − 𝜙 180° = 𝑇! ( 𝑇!/2) solving for 𝜙 yields: 𝜙 = 180° − !!∗!"#° !! ! = 113.1571°
- 10. 6 With the angular separation, we can now calculate the launch date. Using the true anomalies of Earth and Neptune recorded on March 13th, 2014 (𝜃!,! = 70.90868° 𝑎𝑛𝑑 𝜃!,! = 274.44241°), and also the mean motion of each planet ( 𝑛! = ∗ 10!! !"# ! 𝑎𝑛𝑑 𝑛! = ∗ 10!! !"# ! ), we can define the positions of Earth and Neptune relative to these true anomalies with the equations: 𝜃! = 𝜃!,! + 𝑛! 𝑡 𝑎𝑛𝑑 𝜃! = 𝜃!,! + 𝑛! 𝑡 The separation angle is the difference of these two equations: 𝜙 = 𝜃! − 𝜃! = 𝜃!,! − 𝜃!,! + 𝑛! − 𝑛! 𝑡 Solving for time t, this will give us the time from the true anomaly date in seconds to launch: 𝑡 = 𝜙 + 𝜃!,! − 𝜃!,! 𝑛! − 𝑛! = 7971148 𝑠 Using the MATLAB function “addtodate( )”, this time t was added to the initial date of March 13th, 2014 to yield a launching date of June 13th, 2014. The arrival date is simply calculated by adding the transfer time ( 𝑇!) to the launch date. Using MATLAB again, the arrival date is calculated to be January 11th, 2045.
- 11. 7 4.0 PLANETARY ARRIVAL PHASE Upon arrival to Neptune, the heliocentric velocity of the spacecraft is: 𝑣!" ! ! = 2𝜇 !"# 𝑅! − 𝜇 !"# 𝑎! = 1.3791 𝑘𝑚 𝑠 Thus, the velocity for the hyperbolic arrival to Neptune is: 𝑣!,! = 𝑣!" ! ! − 𝑣! = −4.0545 𝑘𝑚 𝑠 The minus sign indicates that Neptune is travelling faster than the orbiter and catching up to it, and thus 𝑣!,! is pointed opposite of Neptune’s velocity. The target orbit radius around Neptune ( 𝑟!"#$%! = 25,060 𝑘𝑚) is 300 km above the surface. The location of the perigee of the departure hyperbola is found using the equation below: 𝛽! = cos!! 1 1 + 𝑟!"#$%&'#,! 𝑣!,! ! 𝜇 ! = 19.4118 ° The required aiming distance (∆) with which the spacecraft should encounter the sphere of influence of Neptune in order achieve the target orbit radius is found using the equation: ∆!= 𝑟!"#$%! 1 + 2𝜇 ! 𝑟!"#$%! ∗ 𝑣!,! ! = 1.4652𝑒05 𝑘𝑚 This provides a hyperbolic arrival trajectory with an eccentricity of : 𝑒!!",! = 𝑟!"#$%! ∗ 𝑣!,! ! 𝜇 ! + 1 = 1.0603 and a specific angular momentum of: ℎ!!",! = 𝑟!"#$%! 𝑣!,! ! + 2𝜇 ! 𝑟!"#$%! = 5.9405𝑒05 𝑘𝑚! 𝑠
- 12. 8 Thus, the periapsis velocity of the hyperbolic arrival is: 𝑣! = ℎ!!",! 𝑟!"#$%! = 23.7052 𝑘𝑚 𝑠 The velocity required for the circular target orbit around Neptune is: 𝑣!"#$%&'#,! = 𝜇 ! 𝑟!"#$%! = 16.5151 𝑘𝑚 𝑠 Which means the required ∆v to circularize the orbiter’s path is: Δ𝑣! = 𝑣!"#$%&'#,! − 𝑣! = −7.1901 𝑘𝑚 𝑠
- 13. 9 5.0 PROPELLANT MASS REQUIREMENTS The NAISO satellite is estimated to have a mass that is half of the JUNO spacecraft. In addition, the NAISO is using 𝑁𝐴𝐼𝑆𝑂 𝐼!" = 318 𝑠 𝑁𝐴𝑆𝐼𝑂 𝑀𝑎𝑠𝑠: 𝑚!" = 1800 𝑘𝑔 Thus, the required mass of propellant ( 𝑚!,!) needed to achieve Δ 𝑣! = 7.1901 !" ! is found with the equation: 𝑚!,! 𝑚!" + 𝑚!,! = 1 − 𝑒 !( !!! !!"∗!! ) Solving for 𝑚!,! yields: 𝑚!,! = 16,240.48 𝑘𝑔 which gives a total mass of the spacecraft with propellant to be: 𝑚!"!! = 𝑚!" + 𝑚!,! = 18040.48 𝑘𝑔 Now the required amount of fuel for Δ 𝑣! = 8.2474 !" ! can be calculated: 𝐶𝑒𝑛𝑡𝑎𝑢𝑟 𝐼!" = 451 𝑠 𝐶𝑒𝑛𝑡𝑎𝑢𝑟 𝑀𝑎𝑠𝑠: 𝑚!"#$%&' = 1800 𝑘𝑔 𝑚!,! 𝑚!"#$%&' + 𝑚!,! + 𝑚!"!! = 1 − 𝑒 !( !!! !!"∗!! ) Again, solving for 𝑚!,! yields: 𝑚!,! = 110,549.34 𝑘𝑔 This yields a total mass of the upper stage prior to the execution of the ∆v1 of: 𝑚!"!#$ = 𝑚!"#$%&' + 𝑚!,! + 𝑚!"!! = 130,832.82 𝑘𝑔
- 14. 10 6.0 MISSION SUMMARY Objectives: • Insert Satellite into circular orbit 300 km above Neptune’s surface with a Hohmann transfer • Study Neptune’s atmosphere and measure composition, temperature, and other atmospheric properties • Asses what the “Great Dark Spot” was and possibly search for other occurrences • Map Neptune’s gravity and magnetic fields to offer insight on its internal structure and core Important Launching Parameters: Launch Date: June 13th, 2014 Arrival Date: January 11th, 2045 𝐼𝑛𝑐𝑙𝑖𝑛𝑡𝑎𝑖𝑜𝑛: 𝑖 = 28.5° 𝐿𝑎𝑢𝑛ℎ 𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒: 𝐿𝑎 = 28.5° 𝐴𝑧𝑖𝑚𝑢𝑡ℎ ∶ 𝐴𝑧 = 90° Important Transfer Parameters: 𝑣!,! = 11.6532 𝑘𝑚 𝑠 𝛽! = 72.1256° ∆!= 9101.366 𝑘𝑚 Δ𝑣! = 8.2474 𝑘𝑚 𝑠 𝜙 = 113.1571° Important Arrival Parameters: 𝑣!,! = −4.0545 𝑘𝑚 𝑠 𝑟!"#$%! = 25,060 𝑘𝑚
- 15. 11 𝛽! = 19.4118 ° ∆!= 1.4652𝑒05 𝑘𝑚 𝑣!"#$%&'#,! = 16.5151 𝑘𝑚 𝑠 Δ𝑣! = −7.1901 𝑘𝑚 𝑠 Mass Budget: 𝑁𝐴𝐼𝑆𝑂 𝐼!" = 318 𝑠 𝑁𝐴𝑆𝐼𝑂 𝑀𝑎𝑠𝑠: 𝑚!" = 1800 𝑘𝑔 𝑚!,! = 16,240.48 𝑘𝑔 𝑚!"!! = 18040.48 𝑘𝑔 𝐶𝑒𝑛𝑡𝑎𝑢𝑟 𝐼!" = 451 𝑠 𝐶𝑒𝑛𝑡𝑎𝑢𝑟 𝑀𝑎𝑠𝑠: 𝑚!"#$%&' = 1800 𝑘𝑔 𝑚!,! = 110,549.34 𝑘𝑔 𝑚!"!#$ = 130,832.82 𝑘𝑔
- 16. 12 7.0 TEAM CONTRIBUTIONS Everyone contributed to the calculations required for every section. The following members wrote the Sections: • Luke Guirguis: Sections 1, 3 and 4 • Jose Sepulveda: Sections 1,5, and 6 • Sean Godinez: Sections 1, 2, and 8
- 17. 13 8.0 REFERENCES “Atlas V 551.” Spaceflight 101. n.d. Web. 13 Mar. 2014. http://www.spaceflight101.com/atlas-‐v-‐551.html Brown, Charles D. Elements of Spacecraft Design. Web. 13 Mar. 2014. Brown, Charles D. Spacecraft Mission Design. 2nd ed. Web. 13 Mar. 2014. Curtis, Howard D. , Orbital Mechanics for Engineering Students Third Edition Goodman H.S. Pre-‐Flight Interplanetary Mission Analysis. 21 Jan. 1969. Web. 13 Mar. 2014 “HORIZONS System.” Solar System Dynamics. Jet Propulsion Lab, n.d. Web. 13 Mar. 2014. http://ssd.jpl.nasa.gov/?horizons “Juno Spacecraft Information.” Spaceflight 101. n.d. Web. 13 Mar. 2014. http://www.spaceflight101.com/juno-‐spacecraft-‐information.html