Space Flight: The Application of Orbital Mechanics

when ancient man looked to the heavens for guidance from the gods he noticed star patterns and began to document their movement across the heavens the ancients believed that the earth was flat but around 350 BC Aristotle proved that the earth was round later about 150 AD Ptolemy presented the geocentric theory the belief that the earth is stationary at the center of the universe with the Sun Moon stars and planets revolving around it in complex orbits in the 1500s Nicholas Copernicus of Poland presented the heliocentric theory the belief that the Earth revolves around the Sun as it rotates on its axis this aspect of astronomy evolved into an intricate study of planetary motion known as orbital mechanics today orbital mechanics is applied to spaceflight and satellites that orbit the Earth or travel beyond our solar system in the early 1600s Johann Kepler a German mathematician using the data on planetary observations collected by the Danish scientist Tycho Brahe he developed three laws of planetary motion Kepler's first law states all planets move in ellipses or 'but there's a Sun at one focus and the other focus empty applied to earth satellites the center of the earth becomes one focus with the other focus empty for circular orbits the two foci coincide Kepler's second law the law of areas states the line joining the planet to the Sun sweeps over equal areas in equal time intervals when a satellite orbits the line joining it to the earth sweeps over equal areas in equal periods of time if areas one two and three are equal times 1 2 & 3 are also equal therefore the speed of the satellite changes depending on its distance from the center of the earth speed is greatest at the point in the orbit closest to the earth called perigee and is slowest at the point farthest from the earth called Apogee it is important to note that the orbit followed by a satellite is not dependent on its mass a large heavy satellite could be in the same orbit with a small light one each sweeping out equal areas in equal periods of time Kepler's third law the law of periods relates the time required for a planet to make one complete trip around the Sun to its mean distance from the Sun for any planet the square of its period of revolution is directly proportional to the cube of its mean distance from the Sun applied to earth satellites Kepler's third law explains that the farther a satellite is from the earth the longer it will take to complete an orbit the greater the distance it will travel to complete an orbit and the slower its average speed will be Isaac Newton the father of classical mechanics laid the groundwork for orbital mechanics he combined the work of Kepler and others to formulate the law of universal gravitation and the three Newtonian laws of motion while Kepler's laws provided a conceptual model of orbital motion Newton's laws provided the foundation for the mathematical description of orbits they explain why a satellite stays in orbit Newton's law of universal gravitation any two objects in the universe such as the earth and the moon attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them stated more simply the more massive the objects are or the closer they are the greater the gravitational pull between them Newton's first law of motion a body in motion will keep moving in the same speed and in the same direction and let's exit upon by an external force a satellite moves in a curved path around the earth because the Earth's gravitational pull acts as an external force on it Newton's second law of motion if the sum of the forces acting on an object is not zero the object will have an acceleration proportional to the magnitude and in the direction of the net force newton's second law states that force equals mass times acceleration it is this mathematical equation and the equation for universal gravitation that forms the basis for calculating orbits Newton's third law of motion explains how a satellite gets into orbit for every action there is an equal and opposite reaction if you blow up a balloon and let it go the balloon is pushed forward by the action of the air rushing out of it a Rockets exhaust gases are like the air rushing out of the balloon the following illustrates how a satellite stays in orbit if a man stands on a mountain and fires a projectile horizontally gravity will cause the path of the projectile to curve downward and it will strike the earth however if the man fires the projectile fast enough at a specific speed the curvature of its path due to gravity will match the curvature of the earth under it the projectile will then fall around the earth becoming an earth orbiting satellite a projectile fired even faster will have a flight path away from the earth but gravity will act to slow the projectile down change its flight path and pull it back toward Earth if the projectiles velocity increased enough a velocity sufficient to escape the Earth's gravitational pull will be reached this velocity is known as the escape velocity it is equal to about seven miles per second at the Earth's surface the preceding description did not consider atmospheric drag and the Earth's rotation both of which will affect the trajectory of the projectile it Illustrated the principles governing a satellites orbit there are six numbers called the orbital elements which specify the size shape and orientation of an orbit in space as well as the location of the spacecraft in the orbit based on an orbit which is an ellipse the six orbital elements are length of the semi-major axis eccentricity inclination right ascension of the ascending node argument of perigee time of perigee passage the major axis of an elliptical orbit is the line joining the perigee and Apogee this line is also referred to as the line of APSA DS the first orbital element is the semi-major axis it is simply one-half the major axis circular orbits have no Apogee or perigee therefore the semi-major axis is simply 1/2 the diameter of the orbit the semi-major axis is used to define the size of the orbit from this the orbital period or time that it takes for the satellite to complete one orbit can be calculated the shape of an orbit is defined by the second orbital element called eccentricity for all ellipses the value of eccentricity lies between zero and one the larger the value the more elliptical the orbit a spacecraft in Earth orbit with an eccentricity equal to or greater than one will escape the Earth's gravitational field when orienting an orbit in space a three dimensional coordinate system must be defined the coordinate system commonly used is the geocentric equatorial coordinate system which has its origin at the Earth's center this coordinate system is a non-rotating reference system in which a satellites orbital plane tends to remain fixed relative to the stars while the Earth turns beneath it the XY plane is the Earth's equatorial plane the positive x-axis points to the vernal equinox this is the point where the Sun appears to cross the earth's equator on its way north on the first day of spring each year the z axis is along the Earth's spin axis toward the North Pole nodes are points in a satellites orbit which intersect the Earth's equatorial plane the ascending node is the point at which the spacecraft crosses the equator going from south to north the descending node is where the spacecraft crosses the equator going from north to south the line joining the two nodes is called the line of nodes the orientation of an orbit is determined by three orbital element angles the right ascension of the ascending node is the angle between the x-axis and the ascending node it is always measured eastward from the direction away from the vernal equinox in the Earth's equatorial plane the argument of perigee is the angle between the ascending node and the point of perigee it is measured in the orbital plane in the direction of spacecraft motion inclination is the angle between the equatorial plane and the orbital plane a satellite which has an eastward velocity component at the ascending node as an orbital inclination lying between 0 and 90 degrees such an orbit is called a pro-grade orbit a satellite which moves due north at the ascending node is in a polar orbit polar orbits have an orbital inclination of exactly 90 degrees a satellite with a westward velocity component at the ascending node is in a retrograde orbit and has an orbital inclination between 90 and 180 degrees the five orbital elements explained thus far described the size shape and orientation of the orbit in space the final element is a time value used to locate the satellite in its orbit a satellite moves in a very predictable manner it stays on schedule thus if the time at which a satellite passes a particular point is known the time when it will pass any other point can be determined the particular point chosen is perigee and the time of perigee passage is the last of the six orbital elements the six orbital elements depict a spacecrafts orbit in non rotating coordinates to visualize an orbit relative to the rotating earth a projection traces the spacecraft's position on the Earth's surface the projected path is called the ground track as a satellite orbits the earth the ground track shifts westward there are two causes for this first the primary contributor is the Earth's rotation toward the east under the orbital plane second because the earth is not a uniform sphere and bulges at the equator its gravity is greatest at the equator this causes the orbital plane to rotate slowly around the Earth's polar axis in a motion called precession precession is toward the west where pro-grade orbits and toward the east for retrograde orbits for low Earth orbits such as those of the space shuttle at 150 miles altitude the westward shift of the ground track due to the Earth's rotation is about 22 and a half degrees while the shift due to precession is only about a half degree the inclination of a satellite orbit determines the north and south latitude limits of its ground track the minimum orbital inclination is equal to the latitude of the launch site and is achieved by launching due east for example if a satellite is launched due east out of the Kennedy Space Center which is located at 28 and a half degrees north latitude its orbital inclination will be 28 and 1/2 degrees and the limits of its ground track will vary between 28 and 1/2 degrees north latitude and 28 and 1/2 degrees south latitude if launch azimuth or direction of flight at launch measured eastward from due north is increased from due east the orbital inclination angle increases as well as the maximum latitude of the north-south ground track therefore the latitude limits of the ground track equal the new launch inclination similarly if launched azimuth is decreased from due east orbital inclination once again increases as well as the latitude limits of the ground track the maximum practical inclination from a Kennedy Space Center launch is 57 degrees this limit is imposed for safety considerations in order to keep the spacecraft and its booster system from flying over land masses during the ascent phase to obtain an orbit with an inclination greater than 57 degrees the spacecraft is launched from Vandenberg Air Force Base in California Vandenberg offers the opportunity for southerly launches with orbit inclinations between approximately 70 degrees pro-grade through 138 degrees retrograde a significant advantage of launching from Vandenberg is the capability to economically achieve polar orbits with ground tracks covering all latitudes from the North Pole to the South Pole the earth is constantly turning and all points on its surface have an eastward velocity with the greatest velocity occurring at the equator the farther the launch site is from the equator or as launch azimuth is increased or decreased from due east less of the Earth's rotational velocity will be imparted to the launch vehicle this requires more fuel to get into orbit or payload weight will have to be decreased launches due east from a position on or near the equator such as the kuru launch site in French Guiana used by the European Space Agency acquire the advantage of a free velocity gain of about 1500 feet per second this compares to the approximate 1,300 feet per second gain available at the further north latitude of the Kennedy Space Center launching from an equatorial site offers a significant advantage in payload weight capability and minimizes the amount of fuel needed to achieve an equatorial orbit since many satellites operate in equatorial orbits these are important considerations spacecraft are launched within a specified time interval called the launch window some of the factors affecting the launch window are launched in orbit lighting conditions Sun angles payload orbit requirements rendezvous phasing if a rendezvous is planned tracking and communication requirements and collision avoidance with other orbiting objects to name a few one of the factors defining the launch window for the Space Shuttle is launched lighting conditions which can be illustrated by plotting time versus day of year on this plot we see daylight and darkness at the launch site the longer daylight hours occur in the middle of the year summer time if daylight conditions are required for a convenient emergency landing site for the space shuttle the launch window would now look like this during the winter months the available launch window for lighting conditions alone can be as little as three hours per day when combined with the many other launch factors the launch window becomes even more constrained the choice of a particular launch vehicle for a mission depends upon the weight and size of the payload and the desired orbit expendable rockets used to place spacecraft in orbit usually consist of several stages that may incorporate both solid and liquid propellants for propulsion when the fuel in each stage is depleted the spent stage is jettisoned staging offers the advantage of discarding weight when it is no longer needed the Space Shuttle is a two-stage system at liftoff the two solid rocket boosters and three Space Shuttle main engines are all producing thrust after approximately two minutes of flight at an altitude of 25 miles the fuel and the solid rocket boosters is depleted and they are jettisoned the three main engines fueled by liquid oxygen and liquid hydrogen carried in the external tank continue to burn for several minutes until the shuttle reaches its cutoff velocity at this time the main engines are shut down and the external tank is jettisoned to additional burns using the orbiters maneuvering system referred to as ohm's are required to place the orbiter in its final orbit the ohm's one burn occurs about two minutes after main engine shutdown and establishes the orbital Apogee point the ohms to burn takes place approximately 30 minutes later and circular eise's the orbit once satellites are launched and put into orbit it is often necessary to change the orbit with an on-orbit burn the common term used in describing on-orbit burns or engine firings is Delta V Delta V is the incremental change in spacecraft velocity measured in feet per second resulting from the burn the amount of fuel used during a burn depends on the desired Delta V change and the mass of the spacecraft because the amount of fuel carried is limited fuel consumption is one of the primary considerations in spacecraft mission planning and is critical to orbit lifetime on orbit a spacecraft can thrust in any direction burns along the flight path forward and backward are the most common a unique feature of any orbital burn is that if no other burns occur the spacecraft will later always pass again through the point of burn forward burns increase the spacecraft's velocity and are known as paws agreed burns with paws agreed burns the flight path of the vehicle will be raised at all points except the burn point burns opposite the direction of flight which slow the spacecraft down are called retrograde burns for retrograde burns the orbit will be lowered at all points except the burn point the greater the Delta V the greater the difference between the pre burn and post burn orbits burns can be combined into maneuver sequences to change orbits size shape or orientation one of the most common maneuver sequences is made up of two burns and is used to accomplish an orbit transfer between two circular orbits in the same orbital plane the most energy efficient transfer between two orbits of this type is the Hohmann transfer the Hohmann transfer is actually one half of an elliptical orbit with its perigee in one of the orbits at its Apogee in the other the burns occur at the perigee and Apogee of the transfer orbit the use of the Hohmann transfer minimizes the Delta V required thus having the advantage of using minimum fuel the disadvantage of the Hohmann transfer is that it takes longer than most other transfers the type of the transfer sequence depends on the mission and the amount of fuel available for example a space rescue where time is critical might use a fast transfer while a routine satellite deployment where fuel saved for later use is important would most likely use a Hohmann transfer the burns discussed so far have all been maneuvering in the original orbital plane and do not affect orbit inclination or node position there are situations which require an orbital plane change such as setting up a rendezvous or placing a satellite in an equatorial orbit to change the inclination the thrust vector must be directed at an angle to the orbital plane a thrust with a component that is perpendicular to the orbital plane at either the ascending or descending node will rotate the orbital plane about the line of nodes Northerly out of plane thrust at the ascending node will increase the inclination of a pro-grade orbit while a southerly thrust will decrease it out of plane thrusts require considerable amounts of fuel and are performed only when absolutely required the Space Shuttle for example using all of its onboard propellant is capable of an on-orbit plane change of less than three degrees satellite orbital planes and altitudes are determined by their design mission which very often includes a field of view requirement for optical or communications purposes the field of view of a satellite is defined as the area of the Earth's surface that is in view from the satellite at any given time satellites in high orbits have greater fields of view than those in lower orbits for example a satellite at an altitude of 800 nautical miles has a circular field of view with a diameter of about 40 100 nautical miles a satellite at 200 nautical miles has a circular field of view with a diameter of about 2,000 nautical miles low orbit satellites are often used for photography and other types of Earth observation a satellite placed in a low inclination circular orbit at an altitude of about 19,000 300 nautical miles will have an angular velocity exactly equal to that of the earth the satellite would seem to remain stationary in longitude as viewed from the ground such orbits are called geosynchronous and are used to provide a continuous communications capability among any system of ground stations within their field of view the geosynchronous orbit field of view is constant and is limited to a latitude zone of about 70 degrees north and south of the equator effective satellite communications from geosynchronous orbit is not possible at either Pole however because of their altitude their field of view covers nearly half the globe a special type of geosynchronous orbit with an inclination of 0 degrees is called a geostationary orbit it appears to hover over a fixed point on the Earth's surface at the equator most us communication satellites are in geosynchronous orbits providing near worldwide communications coverage for effective communications at high latitudes the molniya orbit is used mahlia is the Russian word for lightning and is an orbit used extensively by the Soviet Union for its communication satellites pneumoniae orbit is highly eccentric with an Apogee that is near the geosynchronous altitude and an inclination of about 63 degrees the satellite slows down at Apogee in the northern hemisphere and whips through perigee in the southern hemisphere this provides communications in the northern hemisphere for up to 75% of its orbital period several satellites properly spaced in molniya orbits can provide constant communications at the northern latitudes navigation satellites such as the US Navy's transit system and the joint service Navstar GPS Global Positioning System use lower orbits so that a user can receive signals from more than one satellite at any time another frequently used orbit is known as a sun-synchronous orbit these take advantage of the precession of the orbital plane caused by the earth not being a perfect sphere all Sun synchronous orbits are highly inclined retrograde orbits which precess eastward around the Earth's polar axis at the rate of one revolution per year since the Earth's Sun line also revolves eastward of the rate of one revolution per year the orbital plane will maintain a constant orientation relative to the Earth's Sun line if the satellites period is then synchronized with the rotation of the earth it will pass over the same point on the Earth's surface at the same local time at a regular interval a sun-synchronous satellite ensures that a constant sun angle and uniform lighting exist for the same field of view from past to pass satellites such as those in the defense meteorological satellite program and Landsat our Sun synchronous imaging the entire Earth on a regular schedule the gravitational attraction of the earth on a spacecraft causes it to move in its orbit around the Earth there are other much smaller forces which will cause a spacecraft to deviate from its desired orbit these forces cause what are known as orbital perturbations orbital precession which is used to obtain Sun synchronous orbits results from the perturbing effects of the Earth's non spherical shape other perturbing forces are the gravitational pull of the Sun the moon and planets and solar winds which are charged streams of protons and electrons that heat the Earth's atmosphere and increase atmospheric drag in most cases perturbing forces can be compensated for in the spacecraft and orbit design and present no major problems if the forces disturb the orbit too much thrusters can be fired to re-establish its desired orbital orientation or altitude this is particularly true for spacecraft orbiting at very low altitudes where the effects of atmospheric drag are greater and if not compensated for will eventually cause the spacecraft to deorbit a spacecrafts operational lifetime is frequently limited only by the amount of fuel available to maintain its desired orbit when its useful life is complete a satellite is left in orbit or is deorbited burning up when re-entering the Earth's atmosphere when the Space Shuttle completes its orbital mission it executes a precise retrograde burn to initiate its controlled return to earth this burn occurs nearly halfway around the Earth from the landing site the new orbit established by the retrograde burn causes the orbiter to enter the Earth's atmosphere about four thousand miles from the landing site during the period the orbiter descends from its orbital altitude to atmospheric reentry its attitude is maintained by the use of reaction control jets located in the nose and tail of the orbiter once the orbiter enters the Earth's atmosphere its wing and tail arrow surfaces begin to become effective and gradually replace the Jets for attitude control as the orbiter nears the landing field it maneuvers to a long straight in approach at an angle of 17 to 19 degrees nearing the runway it executes a flare maneuver to reduce its sink rate and glides to a touchdown at approximately 230 miles per hour as the orbiter rolls to a stop our journey into the world of orbital mechanics comes to an end for now this is only the basics of orbital mechanics an intricate study of planetary and satellite motion the next time you see a launch you will see it from a different somewhat knowledgeable perspective you will understand the fundamentals of spaceflight Oh you

Channel: NASA STI Program

Views: 500,514

Rating: 4.9338183 out of 5

Keywords: ORBITAL, MECHANICS;, SPACE, NAVIGATION;, EDUCATION;, STUDENTS

Id: Am7EwmxBAW8

Length: 36min 4sec (2164 seconds)

Published: Wed Sep 21 2011

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