nasa facts orbits and revolutions

8/8/2019 NASA Facts Orbits and Revolutions

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AN EDUCATIONAL PUBLICATION OF THENATIONAL AERONAUTICS AND SPACE ADMINISTRATION

SCIENCE SERIESSR. HIGH SCHOOLPHYSICS

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When a body in space is moving about a primarybody, such as the earth, under the influence of gravi-tational force alone, its path is called its orbit.(Figure 1) If a spacecraft is traveling along a closedpath with respect to the pr ima ry body, i ts orbit w il lbe a circle or an ellipse. Perfectly circular o rbits arenot achieved in practice. However the ellipse , as onelearns in analytical geometry, approaches a circlewhen the eccentricity becomes small. The orb its ofthe m anned satellites launched by NASA in the Mer-

cury and Gemini programs were nearly circular -they were ellipses with very small eccentricity. Theplanets, with th e sun as the primary body, also follownearly circular orbits. When a satellite makes a fulltr ip in orbit around its pr imary body, it is said tocomplete a revolution, and the t ime required istermed its period, or period of revolution.If an observer is me asuring the period, the meas-urement that he gets will depend upon his point ofreference. If he were located far out in space, he

Figure 1


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Figure 2. Relative positions of satellite and observer at beginning and end of synodic periodcould visualize the o rbit o f the spacecraft against thebackground of fixed stars, and de termine the pe riodby timin g th e interval between successive passagesover some point in that background. His measure-ment would then indicate the t ime needed to makeone complete transit of the ellipse and would becalled the sidereal period of revolution. The wordsidereal means of or relating to the stars. Thesidereal period is not affected in any way by therotation of the earth under the satellite.Let us suppose, however, that the observer is notout in space bu t is standing on the equator, with th esatellite in a low earth orbit moving directly eastabove the equator. He uses his own position as th ereference poin t for measuring the p eriod. Then whenthe satellite has passed through one completeellipse, it will be behind the observer because therotation of the earth will have carried h im a distanceeastward. The satellite will be over the observeragain only after it has traveled an additional dis-tance eastward. The perio d as now measured by theobserver will obviously be greater than the siderealperiod. It is called the synodic period.The word synodic refers to a m eeting or conjunc-tion. At the be ginning of the period, the posit ion of

the spa cecraft over the observer results in a certaingrouping or meeting of the earth, spacecraft, andsun. A t the end of the period the spacecraft wil l beover the observer again, and this same grouping ormeeting will be repeated.In practice, very few sa tellites are placed in equa -tor ial orbits. Most orbits are inclined at an angle tothe e quator, as shown in Figure 2. In the case of aninclined orbit, the spacecraft will not make succes-sive passes over th e observer. The ob server moveswith the earth on a circle in a plane parallel to th eplane of the equator, while the spacecraft movesthrough an ell ipse i n a plane inclined to the plane ofthe equator. Thus the p oint at which the spacecraftpasses over the observers longitude changes witheach pass. For an inclined orbit, the time elapsingbetween two consecutive passes over the referencelongitude is the synodic period.In day-to-day operations, th e p ractice arose ofreferr ing to the synodic periods simply as revolu-tions an d the sidereal periods as orbits. The readerwil l f ind this terminology used in news accounts.Thus, astrona uts Borman and Lovell completed 206revolutions and 220 orbits during the 14-day rnis-sion of Gemini VII.


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I f the spacecraft is orb it ing in an easterly direc-tion, the same as the earth's rotation, the orbit issaid to be posigrade, and the synodic period isgreater than th e sidereal period. I f the spacecraft isrevolving in a westerly direction, opposite to theearth's rotation, the orbit is said to be retrograde.In this case th e eastward motion of th e observercauses him to meet the satell i te before it has com-pleted a fu l l transit of th e ell ipse, and the synodicperiod is therefore less than the sidereal period.All manned space flights launched to date by theUnited States have been placed in posigrade orbits

th e synodic and sidereal

gure 3of a is found by averaging the apogee and perigeedistances as me asured from the center of the earth.The apogee distance is the farthest distance of thespacecraft from the earth, while the perigee dis-tance is the nearest. (Readers who have studiedanalytical geome try will recognize tha t a is the semi-major axis of the ellipse.) The reader can find adiscussion of G and M, and of how they are meas-ured, by referr ing to standard physics textbooks.If our un its of m easurement are the statute m ilefor distance and the second for t ime, the numericalvalue of GM for orbits about the earth is 9.56 (lo4).We shall use this value, since most news reports ofspacecraft orbits are given in English units. (If oneuses metric units, he will have other values for aand GM, but the computed length of the period wil lbe the sidereal period of a satellite withan average altitude above the earth of 100 miles.in an ell ipt ical orbit for which thepogee and perigee altitudes is 100e average radius of the earth is3960miles, a = 3960+ 100 = 4060. hen

4)The solution is easily completed through the use of


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able of powers and roots. P = 5256 secondsapproximately, which equals 87.6 minutes or 1.46hours.Computing the synodic period for an inclinedorbit is beyond the scope of this discussion. How-ever, for an equatorial orbit the synodic period canbe found fr om the sidereal period by a rather simp lecomputation. In Figure 4,et A be a position on theequator at which the satellite discussed above isdirectly over the observer. Then d urin g one synodicperiod, the rotation of the earth carries the observerto B, where the satellite overtakes him again.Our basic problem is to f ind th e angle 6.

If 6 is the angular distance traveled by the ob-server durin g one synodic period, the n 360 + 6 sthe angular distance traveled by the satellite. Theobserver travels 360 degrees in 24 hours, or onedegree in- hours, and is caught by the satellite4360

catch the observer is 1.46 (360 + 6). Thesetwo times are equal. Therefore,360

& -.46 360+ 6) =-4 e360 3601.46 (360+ e) = 24525.6+ 1.46 = 24

22.546= 525.60 = 23.32 egrees

The synodic period is (360+ 23.32)= 1.555hours or 93.3minutes. Thus the synodic period is93.3- 87.6 = 5.7minutes greater than the side-real period.ACTIV171ES

1. f ind the synodic period of the above space-craf t i f it is i n a retrograde equatorial orbit. (82.6min.)2.A similar situation would be presented by twoboys going around a circular track, one walking at

360

a speed of 200 eet per minute and the other ridinga bicycle at a speed of 1000 feet per minute.Assume tha t the circumference of the track i s 1000feet. Solve the following problems when the boysare going in the same direction. In opposite direc-tions.a. How long does it take each boy t o completeone lap? (5 min., 1 min.)

Figure 4b. How many laps does the riding boy makewhile the walking boy makes one? (5)c. If the boys start together, how lon g will ittake the r iding boy to catch the walking boy? (75sec., 50 ec.)d. How far does the walking boy go betweensuccessive passes of the riding boy? (250 ft.,166.7 t.)

3.Compute the sidereal periods of satellites ataverage altitudes above the surface o f the earth of500, 1000, and 22,300 miles. (101 min., 120rnin., 24 hr.)4.The mass of the moon is .0123 imes the massof the earth, or .0123M. Therefore the value ofGM when applied to the moon is (9.56) (-0123)(lo4).Use this value to f ind the sidereal period ofa lunar satellite whose average distance above thesurface of the moon is 60 miles. The radius of themoon is 1080miles. (118min.)5. Readers who have studied physics may wishto compute the sidereal period of a satellite 100statute miles above the earth by expressing thevalues of a, G, an d M i n t h e MKS system of units.The same length of period should be found.

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