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KRONOS Vol III, No. 3
Orbits And Their Measurements
|a||the semimajor axis of the planet's orbit, often called the planet's mean distance from the Sun|
|e||the eccentricity of the planet's orbit|
|P||the planet's orbital period (sidereal period)|
|rmax||the planet's distance from the Sun at aphelion (its farthest distance from the Sun)|
|rmin||the planet's distance from the Sun at perihelion (its closest approach to the Sun) Vmax the planet's maximum velocity, which occurs at perihelion|
|Vmin||the planet's minimum velocity, which occurs at aphelion|
|H/m||the planet's orbital energy per unit mass|
|l/m||the planet's orbital angular momentum per unit mass|
A somewhat different parameter, the mean synodic period S, is dealt with at the end of the paper. The other nine parameters depend only on a single orbit, but the mean synodic period depends on the orbits of two different planets. It remains constant as long as both orbits remain constant, but may change if either orbit changes. Like the other parameters, it can be determined from the map.KEPLERIAN ORBITS
The orbital motion of the planets is described very well by the three laws published by Kepler in 1609 and 1619. Slightly more accurate (and much more complex) ways of expressing orbital motion have been provided in turn by Newton and Einstein, but such improvements are beyond the accuracy intended here.(l) Hence, it will be assumed for the purposes of this paper that Kepler's three laws give a true and complete description of the orbital motion of the planets, and, more generally, of any body revolving in a periodic orbit around the Sun.
Kepler's laws are strictly kinematic: they make no reference to the mass of a planet. Since the orbital motion of a planet or other body does not depend on its mass,(2) or on any other property of the body, the body itself can be ignored in any consideration of the orbit. Likewise, any hypothetical orbit can be considered without regard to the properties of the body that might occupy it. It is thus possible to study and classify all orbits as if they were "empty" orbits, where various parameters commonly associated with the motion of an orbiting body are treated as characteristics of the orbit itself. Orbital characteristics and characteristics of the body can then be treated separately, or recombined, as is most convenient in each instance.
For example, orbital energy per unit mass (H/m) and angular momentum per unit mass (I/m) are parameters that depend on the orbit but not on the body. When a body is reintroduced to the "empty" orbit under consideration, the body's mass m multiplied by H/m will give the body's orbital energy H, and the body's mass m multiplied by l/m will give the body's orbital angular momentum 1.Kepler's laws are:
1. The orbits of the planets are ellipses. The Sun, in each case, is located at one focus of the ellipse.2. Each planet moves such that the line (i.e., the radius vector) from the Sun to the planet sweeps out equal areas in equal intervals of time.(3) 3. The square of each planet's orbital period is proportional to the cube of its mean distance from the Sun, i.e., proportional to the cube of the semimajor axis of its orbit. In other words, the ratio of a planet's period squared to its semimajor axis cubed is the same for all planets.
ORBITAL SIZE AND SHAPE
An ellipse is shown in Figures 1, 2, and 3. The ellipse itself is the outer curve enclosing the two points known as foci, while the other lines show some of the standard ways of measuring an ellipse. The dimension a is the semimajor axis; it is the most common measure of the size of an ellipse and is incorporated in all ellipses in two distinct ways, as seen in Figures I and 3. The lengths r and r' are the distances from the two foci to any point on the ellipse (so that one of them is equivalent to the radius vector mentioned in Kepler's second law); they are not fixed lines or lengths, in contrast to the other lines drawn. Their sum, however, is always a fixed length: r + r' = 2a (constant) for all points on the particular ellipse. This relationship can serve as the definition of an ellipse, and it allows an ellipse to be drawn using two tacks and a loop of string, as shown in Figure 4.The ratio of c to a is called the eccentricity, designated e; the limiting cases for an ellipse are e = 0 and e = 1. When e = 0, then c = 0, so that both foci coincide at the center point and the ellipse is a true circle of radius a and diameter 2a. As the parameter e increases from zero, two simultaneous effects can be seen: the circle gradually becomes flattened (more elliptical) and the foci diverge. The latter effect becomes evident more quickly than the former, so that an ellipse with e = 0.2 has its foci obviously off-center although its shape is still very nearly a true circle. (For clarity, the ellipse drawn in Figures 1-3 has a relatively high eccentricity: e = 0.75.) Ultimately, as the value of e approaches 1, the ellipse will become so narrow that it looks like a straight line of length 2a.
Four parameters (a, b, c, and e) have been used to describe an ellipse; however, only two of them are independent parameters. In other words, specified values for two of the four parameters (any two, although a and e are most often used) are sufficient to define a particular ellipse. The two other parameters are dependent: they can be calculated from the independent parameters with the two equations a2 = b2 + c2 and c = ae.VARIOUS ORBITAL PARAMETERS
In order to describe completely the motion of an orbiting body, six independent parameters are needed. The six that are traditionally used are called the elements of a body's orbit. Two of the elements (a and e) describe the orbit's size and eccentricity, three describe the orbit's orientation in space, and one establishes a reference between a point on our calendar and the position of the orbiting body (so that it can be located at any given time). However, the latter four elements will not be dealt with here. They can be left out, as a matter of convenience, since they do not effect a or e or any of the other parameters treated in this paper. Hence, in the terminology of this paper, all that is required to "specify" or "establish" an orbit is that a and e be established, regardless of the orientation of the orbit and regardless of the body that might occupy it.
The next step will be to examine the interrelationship that exists among a, e, and the other seven orbital parameters listed in the Introduction, and to portray this interrelationship on a diagram or map of some sort.(4) As mentioned previously, the nine parameters are not completely independent of one another; on the contrary, only two of the nine can be considered independent. The remaining seven will be dependent on those two and can be calculated from them by means of the seven equations in Table 2. The parameters a and e are often treated as the independent parameters and will generally be used here as such. Nevertheless, within certain limitations,(S) an alternative pair of parameters can be chosen from the list of nine, to serve as the two independent parameters which suffice to specify an orbit and from which the remaining seven can be calculated.No units of measurement are necessary for e, which is expressed as a dimensionless ratio. The units used for all other parameters except S (which is discussed later) are based on Earth's year, adopted here as the unit of time, and the astronomical unit, adopted as the unit of length. (One astronomical unit, defined as 1.496 x 108 km, is equivalent to Earth's mean distance from the Sun.) Additional in formation on units of measurement is given in Tables 2 and 3.
Two independent parameters can be represented on a two-coordinate grid or graph. Thus, for example, orbital size and shape can be represented as shown in Figure 5, using coordinates a and e. Each point on the grid represents a unique pair of values of a and e. The width of the grid extends from e = 0 to e = 1, with e = 0.5 Iying midway between. The height of the grid extends from a = 0.1 to a = 10 (with a = 1 lying midway between, since the scale for a is logarithmic). Any elliptical orbit between one-tenth and ten times the size of Earth's orbit thus corresponds to a unique point on the grid in Figure 5.
The points representing the orbits of Mercury, Venus, Earth, Mars, Jupiter, and Saturn are marked on the grid. It can be seen, for example, that Venus has the least eccentricity, Mercury has the greatest eccentricity and also the smallest orbit, and Saturn has the largest orbit, of the six planets shown. The orbits of asteroids and some short-period comets could also be represented on the grid. In order to include larger and smaller orbits, the grid may be continued beyond its arbitrary top and bottom margins. One upward continuation of the grid, in the same format, would extend from a = 10 to a = 1000, thus easily including the orbit of Pluto (a = 39.5) and also including various cometary orbits. One downward continuation would extend from a = 0.1 to a = 0.001, thus reaching the practical lower limit, since an ellipse with the latter value would not fit around the visible surface of the Sun.
The grid with coordinates a and e will henceforth be called a map of orbital parameters. Obviously, it is an abstract map, on which any real or hypothetical orbit is represented by a point.
PARAMETER CONTOURS ON THE MAP
At each point on the map, the values of the other seven parameters can be calculated algebraically by means of the equations in Table 2. If a system of labelling were available, it would be possible to indicate the values of the other seven parameters at every point on the map.
This is done in an abbreviated fashion in Figure 6. The underlying grid is identical to Figure 5, so that orbits are represented in exactly the same way; however, reference lines have been added which allow the value of each parameter to be determined at any point. These reference lines are analogous to contour lines on a topographic map or isobars on a weather map: they are lines along which the particular parameter has a particular constant value. These lines will be called contours. They were, of course, plotted using the equations in Table 2, but now that they are drawn on the map it is not necessary to be concerned with how they got there.
Two reference contours have generally been drawn for each parameter, and each is labelled with the parameter symbol and value. For example, one of the l/m contours is l/m = 2 and the other is l/m = 10. Both of them have exactly the same shape and orientation; they are thus equidistant from each other when measured along any vertical line (i.e., measured parallel to the a-axis on the map). Other contours can be drawn for other values of l/m (not only integers and fractions, but also irrational values such as l/m =sqrt(2) and l/m = 3.14159 . . .); they too will have the same shape and orientation, and any two will be equidistant from each other along any vertical line on the map. The contours always lie in numerical order on the map: e.g., I/m = 9.99 must lie between l/m = 9.98 and l/m = 10, although it will not be exactly midway between since the scale is logarithmic.
The same remarks hold true for the contours drawn for rmax, rmin, Vmax, Vmin, and H/m. It should be observed that contours also exist for a and e. This is a rather trivial observation, since it is implicit in any graph; nevertheless, to be consistent, it should be recognised that all horizontal straight lines on the map are contours of a, and all vertical straight lines on the map are contours of e.There are no contours drawn especially for P, since some of the integer-value P contours and a contours coincide (P = 1 is the same as a = 1, and P = 8 is the same as a = 4, etc., since the ratio P2/a3 is the same for all orbits around the Sun according to Kepler's third law). The scale of P is marked along the righthand edge of the map in Figure 6.
The two reference contours for H/m are drawn as dashed lines. All H/m values on the map are negative, as will be discussed later. Since the H/m contours are the same shape as the a and P contours (i.e., straight lines), any H/m contour is also a contour for a and for P: e.g., the three contours H/m = -10, a = 1.97392088 . . ., and P = 2.7732857 . . . are the same line on the map.
In summary, Figure 6 provides a map upon which orbits can be represented as points. Each of the nine parameters is indicated by at least two reference contours. Additional contours can be drawn for other values. All contours of the same parameter have exactly the same shape and orientation (but, except for a, P, and H/m, the contours of different parameters do not). Any two contours of the same parameter are equidistant from each other along any vertical line on the map (except for the e contours, which are obviously equidistant along any horizontal line). All contours for each parameter lie in numerical order on the map. The spacing of the contour values (i.e., the scale) is logarithmic for all the parameters except e.SCALING BETWEEN CONTOURS
In theory, the preceding paragraph provides enough information to construct additional contours for any given parameter values, or to identify the values of the contours that pass through any given point on the map.
In practice, the value of any parameter (except e) at a point that is not located on one of the existing reference contours can be determined by measuring vertically on the map, i.e., parallel to the a-axis, from one of the reference contours to that point with a logarithmic "ruler" of the appropriate scale.(6) Such a "ruler", with three logarithmic scales, is provided on pages 41 and 42. It may be cut out from the page on which it is printed. The logarithmic scales of the "ruler", like those of a slide rule, do not indicate the location of the decimal point: it must be inferred from the reference contours on the map.
As an example, the value of rmin at the point on the map where a = 0.4 and e = 0.5 would be measured in the following way:
The value for the point can be measured from one (or both) of the rmin reference contours; the contour rmin = I will be chosen arbitrarily. The rmin-scale on the "ruler" should be laid alongside the vertical line e = 0.5 on which the point lies, and the reference contour rmin = 1 (where it crosses e = 0.5) should be lined up with the value on the rmin-scale which is the same as the reference contour's value, which in this case is 1. There are three different marks on the rmin-scale that can represent the value 1, since the decimal point location is not indicated there; the mark in the middle will be chosen (arbitrarily) to be aligned with the contour. It may be noted that the "ruler" is now spanning both rmin contours; this is not actually necessary, but it should be done whenever possible, in order to double-check the alignment of the "ruler".
The "ruler" is now in position so that its rmin-scale shows the values of rmin for all points along the vertical line e = 0.5. At the point under consideration, the scale shows the value of rmin to be 2 (with the decimal point location not indicated). Since the point lies between the contours rmin = 0.1 and rmin = 1, the value must be rmin = 0.2 at the point. In other words, the map shows that any body in an orbit around the Sun, with a semimajor axis a = 0.4 astronomical units and an eccentricity e = 0.5, will be 0.2 astronomical units from the Sun at perihelion. Other parameters would be measured from their reference contours in the same way.
The preceding example shows how the value of a parameter can be determined for a given point on the map. Determining the location on the map of a given parameter value is similar; the appropriate scale of the "ruler" would be positioned to measure up or down from the appropriate reference contour. The location (locus) of a given parameter value on the map is not a single point, of course, but an entire contour, i.e., the series of points on the map through which the contour passes.
Suppose, however, that values are specified for two parameters, thus defining two contours on the map. If the two contours cross, then the point where they cross represents the orbit that has those two specified values. When no elliptical orbit is possible with both those values, the two contours will not cross.
Establishing a point on the map in this way, by the crossing of two contours, is equivalent to establishing an orbit algebraically from two independent parameter values. Whether two orbital parameters are independent can be seen, quite simply, by the ability of their contours to cross and thereby establish a point on the map.(7) The fact that three orbital parameters (from the list of nine) cannot be independent means that three specified contours cannot all meet at one point, except by chance.ORBITAL ENERGY AND ANGULAR MOMENTUM
The angular momentum I of a body in orbit can be determined by multiplying its mass m by the l/m-value of its orbit. The mass of each planet (expressed in terms of Earth's mass, which is defined as 1) is given in Table 1, along with values of a, e, and P for the orbit of each planet.
The orbital energy H, which is the total of the body's kinetic and potential energies, can be determined by multiplying the body's mass m by the H/m-value of its orbit. All values of H and H/m are negative, since, by definition, an orbital energy of zero or greater would cause a body to escape from the gravitational field of the Sun.(8) Since they are negative, both H and H/m increase as they approach zero; in other words, H and H/m both increase as a increases.
The motion of orbiting bodies is governed by the laws of conservation of energy and of angular momentum. Whenever an orbit changes, orbital energy and/or angular momentum must be transferred to or from another body (or, in the case of energy, converted to or from another form of energy). An orbit thus cannot change readily. As can be seen on the map, the point representing any given orbit lies on a contour for some value of l/m and also lies on a contour for some value of H/m. It is possible to move the point freely along its own l/m contour without violating conservation of angular momentum; i.e., if the orbit changes in that manner, its angular momentum will remain unchanged. Likewise, the point may move freely along its own H/m contour without violating conservation of energy; i.e., if the orbit changes in that manner, its orbital energy will remain unchanged. However, the l/m contours and the H/m contours do not have the same shape and thus cannot coincide, so that no point on the map can move without departing from either its own l/m contour or its own H/m contour or from both (i.e., no orbit can change without gaining or losing either angular momentum or energy or both).
Angular momentum can exist in a mechanical form only, and is always associated with the revolution (orbiting) or the rotation (spinning) of a body. Any change of orbital angular momentum is therefore a transfer, to or from another body's orbital angular momentum, another body's rotational angular momentum, its own rotational angular momentum, or some combination thereof. In any case, a suitable transfer process must be available.
It should be noted that angular momentum is a vector quantity, so that the angular momentum values of orbiting bodies cannot be added or subtracted arithmetically unless the orbits are exactly or approximately coplanar (yielding exact or approximate results, respectively), whereas energy is a scalar quantity and can be added or subtracted without regard to orbital orientation.
Energy exists in various forms and can be converted among those forms within the limitations imposed by entropy. Any change in a body's orbital energy is therefore a transfer (to or from another body's orbital energy, another body's rotational energy, its own rotational energy, or some combination thereof) or a conversion (to or from another form of energy) or both. In any case, a suitable transfer or conversion process must be available.MEAN SYNODIC PERIOD
The mean synodic period S is a measure of the motion of one planet as observed from another; it expresses their relative period of revolution, or the average time required for an inner, faster planet to gain one full revolution on an outer, slower planet. The mean synodic period will be measured here in terms of the "years" of the observer's planet, so that the mean synodic period of Saturn as observed from Earth will be measured in ordinary years, while the mean synodic period of Saturn relative to an imaginary observer on Mars would be expressed in sidereal periods of Mars. One of the scales on the "ruler" is for measuring values of S on the map. Measurements should be made vertically on the map (i.e., with the "ruler" parallel to the a-axis), with the symbol for infinity on the scale lined up with the horizontal line on the map on which the observer's orbit lies. When aligned in this fashion, the S-scale on the "ruler" shows the mean synodic period associated with any horizontal line on the map. For example, relative to an observer on Earth, the mean synodic period of Venus is almost exactly 1.6 years.
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NOTES1. The accuracy achieved here is within approximately 1-2%, limited by inaccuracy of measurement on the map and by the slight inaccuracies of Kepler's laws. Kepler's inaccuracies are: (A) The laws do not take into account perturbations (the influences of different orbiting bodies on one another) and the effects of relativity, both of which are negligible in this context. More severe perturbations would occur, of course, if the orbits of the planets were not so widely separated. (B) The third law does not take into account the mass of a planet, which actually has a very slight effect on the supposedly-constant ratio of p2 to a3. (When the most massive planet -- Jupiter -- is compared to the least massive body orbiting the Sun, the discrepancy in the supposedly-constant ratio amounts to less than one-tenth of one per cent.) The effect of inaccurate measuring on the map depends on the parameter. Assuming that the map and "ruler" are printed with complete accuracy, a measurement error of 0.01" would cause a Vmax, Vmin, or l/m value to be incorrect by 0.38%. An error of 0.01" would cause an a, rmax, rmin, or H/m value to be incorrect by 0.77%, or a P value to be incorrect by 1.16%. 2. See note 1 for the slight effect that an orbiting body's mass has on its orbital parameters. 3. Kepler's second law is often paraphrased in terms of the planets' having a constant areal velocity or sector velocity, i.e., a constant rate at which area is swept out by the radius vector. Areal velocity is an orbital parameter whose value is always one-half the planet's angular momentum per unit mass. In effect, Kepler's second law is a statement of the law of conservation of angular momentum. 4. The choice of nine parameters is somewhat arbitrary. Various other parameters, such as the ellipse parameters b and c, could also be included in the interrelationship that is shown on the map. There would still be only two independent parameters.
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5. There are three pairs of parameters (a and P; a and H/m; P and H/m) which do not consist of two independent parameters. On the other hand, there are four pairs of parameters (a and e; P and e; H/m and e; rmax and e) which are each completely independent, such that a unique orbit will be defined by any pair of specified values within each parameter's range of possible values (as given in Table 2). All other pairs of parameters are independent within certain mutual limits: e.g., a and rmin are independent as long as a is greater than or equal to rmin. Values specified with rmin greater than a are impossible; no elliptical orbit can have such values. The mutual conditions on other such pairs (e.g., l/m and Vmax) are less obvious than in the preceding example. One of the purposes of the map being developed here is to show what combinations of parameter values are possible.6. The "ruler", with three logarithmic scales and one other scale for S, can be either cut out from the magazine or made from a photocopy, as long as no distortion occurs in photocopying. The scale of a on the map is 2 logarithmic cycles per six inches. The scale for rmax and rmin is the same. The scale for H/m is the same, but inverted and negative. The scale for P is 3 cycles per six inches. The scale for l/m is one cycle per six inches. The scale for Vmax and Vmin is also one cycle per six inches, but inverted. 7. There are three possibilities: All possible contours of two orbital parameters cross each other; this is true for 4 parameter pairs. Some of the contours of two orbital parameters cross each other; this is true for 29 parameter pairs. None of the contours of two orbital parameters cross each other; this is true for 3 parameter pairs. This same information was given in a different form -- not in terms of the map -- in note 5. 8. The total mechanical energy (kinetic plus potential) of a motionless body infinitely far from any gravitational field is conventionally defined as zero. Kinetic energy is always positive, whereas potential energy can be considered negative (or captive) energy. A body whose kinetic energy exactly balances its potential energy has a net mechanical energy (H) of zero; such a body follows a parabolic trajectory around the Sun and never returns (unless its energy is changed by perturbation). A body with positive mechanical energy always has more than enough velocity (and thus kinetic energy) to escape from the negative potential energy of the Sun's gravitational field; such a body follows a hyperbolic trajectory around the Sun. A body with less than zero (i.e., negative) mechanical energy does not have enough velocity at any time to offset and escape from the Sun's gravitational field; the path of such a body is described by Kepler's laws.
For an interesting account of how the phenomena of planetary motion as seen from Earth are related to Ptolemy's, Copernicus', Tycho's, Kepler's, and Newton's explanations of planetary motion, see R.A.R. Tricker, Paths of the Planets (London: Mills & Boon, 1967). An elementary treatment of planetary motion according to Newton's laws, and the relationship to Kepler's laws, can be found in any college physics text; for a more advanced work on Newtonian orbital mechanics, see J.M.A. Danby, Fundamentals of Celestial Mechanics (New York: Macmillan, 1962). A diagram that is similar in some ways to the map presented in this paper can be found in M. Vertregt's Principles of Astronautics, 2nd edition (Amsterdam: Elsevier, 1965), Figs. 11.7-11.8 and 11.10-11.12. Vertregt uses an orbit's semilatus rectum, scaled linearly, for one coordinate of his diagram (instead of semimajor axis, scaled logarithmically); his diagram is thus able to show hyperbolic and parabolic orbits, but the nonuniformity of its contours ("iso-ergs", "isochrones", "isogons") makes the diagram incompatible with devices such as a "ruler" that would allow direct measurement of numerical values.
TABLE 1THE MASS (m) AND THREE ORBITAL PARAMETERS OF EACH PLANET
TABLE 2DEFINITIONS OF ORBITAL PARAMETERS IN TERMS OF a AND e;
|a||a >0||[L]||Semimajor axis|
|e||1 > e >= 0||(dimensionless)||Eccentricity|
|P = sqrt(a3)||P > 0||[T]||Orbital (sidereal) period|
|rmax = a ( 1 + e)||rmax > 0||[L]||Maximum distance from Sun|
|rmin =a (1- e)||rmin > 0||[L]||Minimum distance from Sun|
|Vmax = 2PI sqrt((1+e)/a(1-e))||Vmax > 0||[ LT-1 ]||Maxi mum velocity|
|Vmin = 2PI sqrt((1-e)/a(1+e))||Vmin > 0||[LT-1 ]||Minimum velocity|
|l/m=2PI sqrt(a(1-e2))||l/m > 0||[L2 T-l]||Angular momentum per unit mass|
|H/m =-2PI2/a||H/m < 0||[L2 T-2]||Total mechanical energy per unit mass|
The unit of length [L] here is the astronomical unit, which is the mean distance from Sun to Earth. The unit of time [T] here is the year, or orbital period of Earth. See also Tables 1 and 3.
TABLE 3CONVERSION OF UNITS OF MEASUREMENT On the map, in Tables 1 and 2, and elsewhere in this article, the parameter values have been expressed in what might be called "geobasic" units: astronomical units, years, Earth-masses, and combinations thereof. Factors are given below for the conversion of "geobasic" units to and from those of the metric system (where mks units are expressed in meters, kilograms, seconds, and combinations thereof; and cgs units are expressed in centimetres, grams, seconds, and combinations thereof).
|System of units||"geobasic" to mks||"geobasic" to cgs||mks to "geobasic"||cgs to "geobasic"|
Figure 3: r and r' are the distances from the two foci to any point on the ellipse: r + r' = 2a
Figure 4: Ellipse can be drawn by moving a pencil around inside a loop of string. The loop goes around two tacks and is kept tight with the pencil. The two tacks are the foci of the ellipse.
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PLANETARY SYMBOLS: Saturn, Jupiter, Mars, Earth, Venus, Mercury.
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