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SP 110 Orbits Orbits Basic requirements Orbital motion in the traditional sense requires two components 1. Attractive force 2. Relative motion away from the common axis Energy is easiest to use for the description of orbits 1. Attractive force – potential energy 2. Relative motion – kinetic energy Orbits Gravity is the attractive force for objects in orbit in space, although other attractive forces allow orbits such as electrically charged particles with relative motion that can orbit. These include subatomic particles, atoms and molecules, and larger particles Gravitational force – use gravitational potential energy Ep = -GMm/r G = gravitational constant, M and m are the two masses, and r is the separation Relative motion – use kinetic energy Ek = 1/2mV2 m = one of the masses, V is the relative motion Orbits The sum of the kinetic and potential energies is also important Etotal = Ep + Ek = -GMm/r + ½ mV2 If Etotal is negative, the negative component (gravitational potential an attractive force) is greater than the kinetic energy (relative motion), and the two objects can be in orbit If Etotal is positive, the kinetic energy (relative velocity) component is larger than the potential energy component (attractive force), and the two objects are not bound and there is no repeated orbit Orbits Two restrictions on these orbits needed to keep the discussion simple 1. Two-body orbits only 2. The off-axis relative motion must be sufficient to keep the two bodies from colliding Orbits - Kepler Johannes Kepler (1570-1630) first described planetary orbits with three laws 1. The planets orbit the Sun in elliptical orbits with the Sun at one focus 2. The planets sweep out equal areas in these ellipses in equal times (conservation of angular motion) 3. The period of orbit squared equals the semimajor axis (average separation) cubed P2 = a3 P = period in Earth-years, a = semimajor axis in Astronomical Units Orbits - Kepler Implications of the three Keplerain laws 1. The planets orbit the Sun in elliptical orbits with the Sun at one focus Each body orbits about the center of mass, but each body follows an elliptical path around the other, with the other at one focus 2. The planets sweep out equal areas in equal times Angular momentum is conserved Total orbital energy is constant In the elliptical orbit, decreasing separation corresponds to decreasing potential energy (less negative), and therefore faster orbital speeds (increasing kinetic energy to maintain the constant), and vice-versa for greater separation 3. The period of orbit squared equals the semimajor axis cubed Orbital velocity decreases with increasing separation Orbits - Kepler Keplerain orbits P2 = a3 Although the assumption here has been that there are only two bodies involved in orbits, orbital motion described as “Keplerian” can involve many bodies in orbit, including the planets solar system Kepler’s P2=a3 law is limited to objects in orbit around the Sun (planets, asteroids, comets, etc) Orbits - Newton Issac Newton (1643-1727) developed the basic laws of motion and the theory of gravity, and the relationship between mass and force Newton’s equation relating period to semimajor axis is generalized for any two-body orbit (satellites around Earth or the Moon, stars around the galaxy centers, etc.) P2 = a3 [4π2/G(M+m)] Orbits - Newton Newton’s period-semimajor axis equation can be used for any gravitational orbit that approximates a two-body orbit Example: Find the semimajor axis of a geosynchronous orbit P2 = a3 [4π2/G(M+m)] a = [p2G(M+m)/4π2]1/3 p = 24 hours (86,400 sec), MEarth = 5.97x1024 kg, G=6.67x10-11Nm2/kg2, m is mass of satellite and is insignificant (compared to Earth) a = 3.54x107m = 35,400 km (22,500 mi) Orbit Definitions Orbits - Definitions Basic ellipse Longest axis = major axis a = semi major axis = 1/2 major axis Smallest axis = minor axis (perpendicular to major axis) b = semi minor axis = 1/2 minor axis Orbits - Definitions Eccentricity – a measure of the flatness of the orbit e = [1- (b/a)2]1/2 e varies from zero to 1 for a bound orbit and greater than 1 for an unbound orbit (e = 1 for a parabolic orbit) e = 0 for a circular orbit e = 0.999999999 = flat (highly eccentric) orbit Orbits - Definitions Inclination angle – the angle between orbit plane and reference plane, or between the orbit plane and the equator i = 0 to 360o (0 to 2π) Orbits - Definitions Inclination - the angular difference between the orbital plane and a reference plane Nodes - the two intersecting points between an orbit and a reference plane Descending node - point of intersection of object in orbit that is descending through the reference plane Ascending node - point of intersection of object in orbit that is ascending through the reference plane Line of nodes - the line intersecting the two nodes, (also the intersection of the orbit plane and the reference plane) Orbits - Definitions Periapsis (or periastron) is the closest approach distance for a 2-body orbit Perigee - closest point in the orbit between the Earth and the object in orbit Perihelion - closest point in orbit of planets to the Sun Orbits - Definitions Apoapsis (or apoastron) is the farthest point in orbit between two bodies in a 2-body orbit Apogee - farthest point in orbit between an object and the Earth Aphelion - farthest point in orbit between an object and the Sun Orbits - Definitions Eccentricity – a measure of the flatness of the orbit e = [1- (b/a)2]1/2 If a = b, the orbit is circular, thus (b/a)2 = 1, and therefore e = [1- (1)2]1/2 = 0 Orbit and Position Reference Orbit Reference The most common type of orbit position reference is the geocentric equatorial reference coordinate system that employs the primary reference plane as the Earth’s equator This is a Cartesian reference with the X and Y axis on the equatorial plane, and the polar axis along the Z axis The X-axis is the primary axis pointing to the vernal equinox – the position of the Sun at the definition of spring (apparent passage through the plane of the ecliptic) Orbit Reference Orbit Reference To identify the position of a satellite in orbit, the geocentric equatorial coordinates are used as the positional reference system The orbit and orbit plane is used to identify the satellite position using a set of lengths and angles known as the orbital elements Can also be done in Cartesian coordinates A second coordinate system is used to identify an observer’s position with respect to the geocentric equatorial reference Provides relative angles to satellite from the observer’s position Orbit Reference The seven orbital elements are: a = semimajor axis i = inclination angle e = eccentricity Ω = Right Ascension (or longitude) of the ascending node angle between X-axis (vernal equinox) and the ascending node ω = Argument (or longitude) of the perigee - angle between the ascending node and the perigee ν = True anomaly - angle between perigee and position of orbiting object T = Time since perigee passage Orbit Reference Orbital Energy Orbital Energy Changing an orbit requires energy An orbit has a fixed semimajor axis based on the sum of the kinetic and potential energies Changing the semimajor axis changes the average potential energy and the constant Therefore, it takes energy (thrust) to boost to a higher or lower orbit Higher orbit requires added energy (forward thrust = increased speed) Lower orbit requires lower energy (retrograde thrust = decreased speed) Orbital Energy Based on the total energy, there are three types of orbits - bound, unbound , and escape (neither bound nor unbound) •Bound Orbits - Elliptical orbits - potential greater than kinetic energy •Unbound Orbits - Hyperbolic orbits - potential less than kinetic •Escape orbits - Parabolic orbits - potential equal to kinetic Orbit Circular Elliptical Escape /Parabolic Hyperbolic Condition Bound Bound Neither Unbound Total Energy Negative (Ek + Ep) (Ek < Ep) Negative (Ek < Ep) Zero (Ek = Ep) Positive (Ek > Ep) Eccentricity 0≤ e <1 e=1 e>1 e=0 Orbit Types Orbit Types The various types of orbits can describe the orbital energy and the orbital shape (eccentricity mostly), or by reference orbit orientation, orbital period or planetary surface coverage for the orbiting satellites Communication, remote sensing, and surveillance all require specific orientation throughout the satellite operation. A number of the types and uses for these orbits are given below. 1. Bound (elliptical) orbit A bound orbit, which is also an elliptical orbit, has relative kinetic energy less than the combined gravitational potential energy 2. Unbound (hyperbolic) orbit A hyperbolic orbit is unbound, meaning that the relative kinetic energy is greater than the combined gravitational energy 3. Escape (parabolic) orbit A parabolic orbit is neither bound nor unbound since the relative kinetic energy is exactly equal to the combined gravitational potential energy. The parabolic orbit conditions are the same as escape velocity. Orbit Types 4. Prograde orbit A prograde orbit has an inclination angle less than 900 which follows the same direction as the Earth's or the orbited planet’s rotation 5. Retrograde orbit A retrograde orbit has an inclination greater than 900 which travels in reverse direction to Earth's rotation 6. Polar orbit A polar orbit has an inclination of 900 which allows world-wide coverage over a period of hours to days depending on altitude. This is an orbit commonly used for meteorological and surveillance satellites Orbit Types 7. Geosynchronous orbit A geosynchronous orbit has an orbital period equal to the Earth's rotation period of 24 hours (23h56m4.09s). The semimajor axis of this orbit is 42,164 km. The inclination for this orbit is often but not required to be 0o 8. Geostationary orbit A geostationary orbit is also geosynchronous, but has an equatorial orbit (i = 00), with an eccentricity of zero. This provides a fixed communications platform with respect to the Earth, and is used for communications, remote sensing (whether satellites, for example.), and surveillance. Orbit Types 9. Sun-synchronous orbit A sun-synchronous satellite orbit maintains a constant orientation between the Sun and Earth which is useful for several applications. The most common are the remote sensing applications satellites (Landsat, for example) and astronomical observation satellites (IRAS, for example) These applications require either full back-illumination, or complete shadowing from the Sun. To accomplish this, a nearly polar orbit is used. If the orbit were polar, the orientation of the orbit would be fixed with respect to the Sun, but not the Earth This would show a 0.98o per day change in orientation as the Earth does an make the Sun-satellite orientation seasonal, as the Earth is. Hence, the spacecraft needs a -0.98o per day retrograde motion in the orbital plane to counter the Earth's orbital motion around the Sun. To do this, an orbital retrograde rotation can be made by using the oblateness of the Earth to place a torque on the orbit and produce a precession of the nodes (rotation of the orbital plane) A range of altitudes and corresponding inclinations are available for this type of orbit. Landsat satellites use a = 709 km and I = 98.2o Orbit Types 10. Molniya orbit Russia has traditionally used communication satellite orbits that are highly-inclined with high eccentricity for their high-latitude ground stations with the extended orbit (apogee) over the higher latitudes The Molniya's 12 hour orbit period also allows for communications between the Asian and North American continents, since it is one-half of the sidereal day (43,082 sec) The inclination for this orbit is 63.4o, with a semimajor axis of 26,562 km, and with varying apogee and perigee values that satisfy the desired period and semimajor axis 11. Tundra orbit A tundra orbit is an eccentric, high-inclination (63o) orbit similar to the Molniya orbit but with a period twice as long (one sidereal day) Like the Molniya orbit, this tundra orbit is used for communications at latitudes far from the equator Orbit Types 12. Parking orbit A parking orbit is a temporary orbit commonly used for spacecraft checkout operations before departure from Earth 13. Graveyard orbit A graveyard orbit is a permanent, higher-thannormal orbit used to remove defective or aging spacecraft from the busy geostationary region Orbit Types 14. Walking orbit A walking orbit gets its name from the rotation or precessional motion of the orbit due to the asymmetrical shape of the planet. An example of this is the precession of the sun-synchronous orbit due to the Earth's oblate shape (larger equatorial diameter than polar diameter because of its rapid rotation). 15. Halo orbit A halo orbit is found not around a celestial object but around either the Lagrange L1 or L2 stability regions (objects within or close to the L1 and L2 regions are not in a stable orbit) Orbit Transfers Transfer Orbits To get from one planet to another, a spacecraft must follow an elliptical, heliocentric orbit for most of its flight path As the spacecraft departs the planet of origin, the target planet must be in correct position so that spacecraft arrival is coincident with the target planet This simple requirement allows planet-to-planet flights only at specific times, and similarly, any return flights only at times of correct alignment of the two planets Transfer Orbits Basic orbit transfers 1. Hohmann Transfer - the most energy efficient coplanar orbit transfer has an arrival point 180o from the departure point 2. Spiral (low thrust) - this is a long-period spiral orbit from launch to arrival that is best suited for high efficiency, low thrust propulsion systems (ion engines are a good example) 3. Direct (high thrust) - the direct, short-period transfer requires greater thrust to accelerate the spacecraft quickly to desired orbit at the expense of greater transfer energy and a larger propulsion system 4. Gravity assist augmented propulsion - the gravitational attraction of a spacecraft in a close encounter with a large planet can be used to boost the kinetic energy of the spacecraft with reference to the Sun This technique can also be used to slow spacecraft to get to a closer heliocentric orbit Transfer Orbits Traditional orbit transfer definitions Type I (direct) transfer - transfer angle is less than the Hohmann transfer ellipse (<180o) This is also known as a direct transfer orbit Hohmann transfer - transfer angle of 180o Type II transfer - greater than 180o and less than 360o (spiral) Type III transfer - greater than 360o and less than 540o (spiral) Type IV transfer - greater than 540o and less than 720o (spiral) Transfer Orbits A sketch of the orbital paths of the three common types of transfer orbits Transfer Orbits Hohmann transfer for interplanetary missions The simple expression of a Hohmann transfer can be calculated in Earth years since the transfer orbit (trajectory cruise) occurs within the Sun's gravitational influence The ellipse representing the transfer will have the perihelion at the inner planet orbit and the aphelion at the larger orbit planet This technique also is used for Earth-orbit satellite transfers from a low-Earth orbit to higher orbits, an example being a boost from low-Earth parking orbit to a more distant geostationary orbit Note that the transfer ellipse touches both the larger and smaller circular orbits. That means that the semimajor axis of the transfer orbit is one-half of the two orbits added together Transfer Orbits Calculations for the Hohmann transfer to the planets atransfer = ½ (aA + aB) A period calculation uses the same third law of Kepler, but only one-half since the second half represents a return path to the departure planet ptransfer = ½ (atransfer3/2) Start with the semimajor axis values for the planets aEarth = 1.00 aJupiter = 5.20AU aNeptune = 30.11 AU aVenus = 0.72 AU aSaturn = 9.56 AU aPluto = 39.55 AU aMars = 1.52 AU aUranus = 19.22 AU Transfer Orbits Earth-Venus pHohmann = ½ [(aHohmann )3/2 ] = ½ [(aA + aB)/2]3/2 = ½ [(1 AU + 0.72 AU)/2]3/2 = 0.40 yr Earth-Mars PHohmann = ½ [(1 AU + 1.52 AU)/2]3/2 = 0.72 yr Earth-Jupiter PHohmann = ½ [(1 AU + 5.20 AU)/2]3/2 = 2.73 yr Earth-Saturn PHohmann = ½ [(1 AU + 9.56 AU)/2]3/2 = 6.07 yr Earth-Uranus PHohmann = ½ [(1 AU + 19.22 AU)/2]3/2 = 16.1 yr Earth-Neptune PHohmann = ½ [(1 AU + 30.11 AU)/2]3/2 = 30.7 yr Earth-Pluto PHohmann = ½ [(1 AU + 39.55 AU)/2]3/2 = 45.7 yr Gravity Assist Gravity Assist Gravity assist Augmenting spacecraft propulsion with a gravity-assisted boost is a technique that exchanges momentum between an orbiting planet and a spacecraft during a close flyby of a moving planet The gravity assist propulsion boost can be used to either increase or decrease the spacecraft velocity relative to the Sun which can provide significant ΔV changes that may not be possible with conventional boosters Missions to Mercury, for example, used or are using gravity assist to augment propulsion to overcome the extremely high gravitational pull from the Sun to reach the planet Gravity Assist Missions beyond Jupiter are also not possible without gravity assist since boosters are not available that are capable of taking interplanetary spacecraft beyond Jupiter, depending on the spacecraft mass Gravity assist was first used on the Mariner 10 mission to Mercury, and has since been used for all missions to the Giant Planets, with the exception of the Pioneer 10 which targeted only Jupiter Although the mission targeted Jupiter in a close approach trajectory, the resulting boost was simply a part of the observation objectives and not an intentional boost to reach a more distant planet Gravity Assist Even the Galileo spacecraft that was launched to Jupiter required a gravity assist because of its huge mass A close flyby of the Earth and Venus were used in a sequence labeled VEEGA (Venus-Earth-Earth Gravity Assist) since the launcher and upperstage booster were insufficient to get the spacecraft to Jupiter The Voyager II spacecraft was boosted to Jupiter, then used gravity assists to get to Saturn, Uranus, and Neptune, then outside of the solar system Gravity Assist In graphic form, with three of the four Jovian planets' gravity added, the solar system gravitational potential looks like this, with colored arrows showing the energy to/from orbits Gravity Assist A plot of the escape velocity throughout the solar system can be made by taking the square root of the absolute value of the potential energy, as shown below Gravity Assist Voyager II spacecraft was boosted to Jupiter, with gravity assists to get to Saturn, Uranus, and Neptune, then outside the solar system which is depicted below Synodic Period and Interplanetary Launch Opportunities Launch Opportunities The period that the position of a planet is repeated in the sky is called the synodic period The calculation of the synodic period is simple but has two forms. One is for orbits inside the Earth's orbit, the other for outside. To calculate the synodic period of Venus, for example, and the launch opportunity cycle between Earth and Venus, use: 1/Psynodic = 1/PEarth - 1/PVenus or Psynodic = 1/(1/PEarth - 1/PVenus ) = 1/(1 - 1/0.615) yr = -1.60 yr = 584 days Launch Opportunities Similar calculations produce the synodic periods for the other planets with respect to Earth. The same calculations can be made for relative position periods of any two planets The practical application of the synodic period is that it provides the frequency of launch opportunities from one planet to another Mercury Venus Mars Jupiter Saturn Uranus Neptune Pluto Moon 116 days 584 days (19.2 mo) 780 days (25.6 mo) 399 days (13.1 mo) 378 days 370 days 367 days 367 days 30 days Launch Opportunities Launch opportunities from Earth to the planets are dictated primarily by the synodic period and the phase position between the two planets Since the Earth is in a different orbital plane than all other planets, the alignment of the intersection of the planes (line of nodes) and the phase between the planets is an important consideration in launch timing In addition, none of the planets including Earth are in circular orbits, making the optimum departure and arrival points a compromise between three primary variables. Those are: 1. Orbit phase between launch and arrival planets 2. Line of nodes of the two planetary orbit planes 3. Position in orbit of the two planets Launch Opportunities Finding the minimum time or a minimum in transfer energy of an actual transfer is a compromise from the ideal orbits - circular and coplanar – using actual orbits The actual planetary orbit variables generate a range of solutions that are plotted for ease in planning and interpretation The plots are called pork chop plots because of their shape and have two general best solutions corresponding to the Type I and Type II transfers From missions to Mars, the Type I would be optimal for manned missions because of their shortest transfer period The Type II solution would be optimal for cargo or instrumentation missions since the energy is lower than the Type I solutions Launch Opportunities On the right is a graph of the Type I and Type II flight limitations for the 2005 opportunity to Mars The minimum energy is shown in the closed concentric shapes, with the lowest for Type II missions centered near 12 km2/s2 that corresponds to a flight duration of approximately 370 days Launch Opportunities A plot of the future 2.1 year flight opportunities for Mars is shown on the right for Type II (cargo) missions. Flight energy (C3 is total energy in km2/s2) is listed in the top series and time of flight shown on the bottom Mission to Mars Mission to Mars A quick and dirty estimate of the flight time to Mars and back using 2-dimensional orbits begins with a simple Hohmann transfer calculation Transfer orbit semimajor axis is simply the average of the Earth and Mars orbits, or (1+ 1.52)/2 AU = 1.26 AU The corresponding transfer period is 1.263/2 years = 1.41 years for the complete orbit The actual transfer period is one-half of this, or 0.71 years (8.5 months) each way Mission to Mars A Hohmann transfer requires that the target planet be at a position 180o from the point of spacecraft launch from Earth, therefore the launch can only take place during the correct alignment of Mars and Earth for the spacecraft to reach the target planet Mars which repeats with the 25.6 month (2.13 yr) synodic period Mission to Mars Phasing and synodic period To calculate the approximate time of flight to Mars and back and the time needed for the two planets to realign correctly for the return trip, begin by finding the mean motion between the two planets in their orbit around the Sun in degrees/day, or similar units The Earth's mean motion is simply 360o/year x 1 year/365.24days, or 0.986 deg/day For Mars, this is 360 degrees/1.88 years x 1 year/365.24 days or 0.524 deg/day Next, calculate the transfer period flight time to Mars using the circular orbit approximation. This was calculated earlier with a Hohmann transfer period of 0.71 years, or 259.3 days Mission to Mars Phasing and synodic period The next step is to find how far Mars will travel in its orbit during the spacecraft transfer period from Earth to Mars. This is just Mars' mean motion times the transfer period, or 259.3 days x 0.524 deg/day = 135.87o Mars must arrive at the 180o position during the transfer orbit; therefore, the spacecraft, which is launched from Earth at the 0o point, would be behind Mars by 180o - 135.87o, or 44.13o (alternatively, Mars would be 44.13o ahead of Earth) This alignment can be approximated from the Earth's and Mars' orbital elements, being careful to not confuse the 2-dimensional exercise with the 3-dimensional orbital elements Mission to Mars Phase angles for the Earth-MarsEarth interplanetary mission Mission to Mars Phasing and synodic period To calculate the time needed for the phase difference for the two planets to move into a correct alignment for the return trip, use the difference in mean motion between the two planets This relative motion is just the difference in the mean motion of the two planets, or 0.986 deg/day - 0.524 deg/day = 0.462 deg/day During the spacecraft transfer that took it to Mars, the Earth has traveled 255.66o in its orbit. The difference in positions would therefore be 255.66o - 180o, or 75.66o, with the Earth ahead of Mars by this value Mission to Mars Phasing and synodic period For the return trip, Mars must be ahead of the Earth by the same 75.66 degrees (the launch from Mars can be visualized at the 0o position with the Earth-arrival 180o from that launch position, which would begin -255.66o from that point, hence the angular difference between the Earth and Mars would be 255.66o -180o, or 75.66o) For an estimate of the time required for the Earth to catch up to this position, the angle needed is just 360o minus the current lead of 75.66o minus the lead angle of 75.66o This is a total of 208.68o (360o - 75.66o - 75.66o). At a closing rate of 0.462 deg/day, this makes the return realignment period = 208.68o/0.462o/day = 451.69 day = 1.24 years Mission to Mars The important number here is the turnaround or realignment period of 1.24 years. With a transfer period of 0.71 years (times two for return travel time), the total mission time would be 2.7 years for this 2dimensional, circular orbit, Hohmann transfer approximation Although the transfer orbit period can be decreased by increasing vehicle propulsion thrust, the realignment period is a critical element that can be as long or longer than the travel time to and from Mars Mission to Mars A nearly three year minimum mission period to Mars and return, of which nearly 1½ years is in zero-g transit, means that the critical human space flight exposure problems must be researched and solved before humans can reach Mars If the total mission duration were reduced to a minimum with an immediate return - a turnaround time of zero duration using a high-thrust, Type I direct transfer orbit - the crews would encounter reduced space exposure, but could contribute little, if anything, to a Mars exploration mission Finis