An Explanation of Aspect Ratio and how it affects sailingbyIvor G. Slater, P. Eng. |
WING SECTIONS
This time, to save network time and space, I won't offer a review of what has gone before. Let's just say that we have come (I hope) to understand and agree that Lift is caused by the addition of a (real) circulation to a (real) free stream.
Since posting AR-3, I have thought of another common example of the dynamics of circulation. Next time you see a "dust devil" (a miniature heat-tornado on a field or parking lot, whose spin is caused by coriolis), notice that it moves ACROSS the wind. Inward coriolis spins in the northern hemisphere are counter- clockwise. So, as you look upwind, the devil must move to your right. You now know enough to figure out why. Because winds in confined spaces in the conditions that produce devils are often gusty, you may have to look sharp to see just which way the local wind is blowing. Open fields are easier to observe.
In practical AHdynamics, the circulation is caused by the action of the free stream on certain sorts of bodies called foils, when they are presented to the stream at small "angles of attack" (to be defined.) [We look at the cause and nature of that "action" near the end of this Part.] The class includes wings, stabilizers, propellor and fan blades, (and oar blades, your hand, or a teaspoon, in some circumstances), as well as sails, keels, centerboards, bilgeboards, leeboards, skegs and rudders, and even flat plates.
The path to understanding (of sailboats in wind and sea) meanders through (or perhaps past) two linked fields of fluid dynamics. First is the character and performance of the cross-sectional shape of the element that will start the circulation. Study of this aspect is called "Section Theory." (I assure you that what we need from it is fairly easy to understand.
At a secondary level, the study of Sections involves "high-lift devices." They are not so easy. This basket of goodies includes an array of ways of increasing camber (to be described) and/or preventing "separation" of the flow (to be discussed) which leads to "stalling" (also to be discussed.) This class of device includes: Leading-edge flaps (like a mast "over-rotated" at the leading edge of the sail to increase the camber; Leading-edge slats (related to jib and mainsail); Plain flaps (related to rudders on skegs, or flaps on keels, or even sails "broken" to windward along the line of the forward end of their battens); Trailing-edge slotted flaps (commonly seen on airplanes, and what the after elements of a divided rig might be sometimes); multiply-slotted wings (like the separated and individually-rotated wing-tip feathers of a big bird like a goose coming in for a landing, and much like many multi-sail rigs.) High-lift devices are often used in combination in the search for effect. Unfortunately, there is no theory to account for high-lift sections, as there is for mono-sections. What we know of them is based on interpretation of experimental data (and practical experience.) Talking about them in detail without diagrams and tables of data is practically impossible, so we'll deal only in generalities and perhaps some reference to Abbott and von Doenhoff. (We can still estimate what they do, in a gross way, and talk about them in wing theory.)
How section data is applied to the performance of real wings is called "Wing Theory." (Likely to be AR-5.) As far as we will go with it, this too is easy to understand. It is in this latter field that we will see how Aspect Ratio applies to the performance of finite foils or wings.
After wing theory comes the ART of applying it to sailboat elements, and then putting them together into an understanding of sailboats and how they go. (Likely to be AR-6, and perhaps beyond if wanted.)
Imagine shapes that you think of as wing sections. Not a wing, just a cross section. It could be the thin line of the shape of a sail cut by a plane parallel to the water. Or a section of a bird wing. Or an airplane wing. Or a straight line. Let's keep it down to a reasonable length, say two or three feet. Now consider moving this shape sideways a few feet, following straight parallel lines. By this action we "sweep out" a rectangular wing shape. Now, so that it's not flailing around loose, let's put it in between parallel walls that come very close to its ends, but don't interfere. We mount it on balances, so we can measure any forces it may generate. Now, if we close the top and bottom of this space and arrange to blow air through it from our right, we have the basis of a wind tunnel. (We may extend the tunnel around in a closed loop, and put the fans internally on the other side, to save energy costs, and for other reasons to be discussed.) We'll assume to start that the section is more-or-less level before our eyes.
We need to be able to describe this section shape in terms of standard parameters. Most sailors will be familiar with at least some of the terminolgy, but I'll review it briefly for the benefit of those who are not. The upstream edge (to the right) is called the leading edge. The down-stream edge is called the trailing edge. The straight line joining the leading and trailing edges of the section is called the "chord." In a SECOND MEANING, chord also means the length of the chord (the fore-and-aft width of the section as we see it in our wing sample.) Distances along the chord are measured from the leading edge aft, and expressed as percent of chord. If the section has more than zero thickness, its maximum thickness is expressed as "percent of chord," and the location is specified. So we might say the section is ten percent thick at thirty percent chord. This is not a full specification of shape. There is a "thickness distribution" parameter. It differentiates shapes of the same thickness, even those with their maximum at the same chord location.
Thick or thin, the section has a "mean line." If the mean line is straight, and the section has zero thickness, the foil is a flat plate. If the "straight-meanline" foil has thickness, we say it is symmetrical. If the mean line is curved, we call the section "cambered." Positive sense of camber is convex upwards. The measure of camber is the offset of the mean line from the chord. It is expressed and located as a percent of chord, as for example, 3 percent camber at 40 percent chord. Again, within the limits of these parameters, the shape of the mean line can vary. It may be nearly straight in the middle, and very curvy at the ends. Or the opposite. It may combine a curvy leading end with a nearly-straight, or even "reflexed" trailing end. Or the opposite. So there is a need to define "camber distribution."
In summary, arbitrary thickness distribution may be added to an arbitrary distribution of camber to produce an airfoil. For a given application, some are much better than others. In other words, one that is best "here" may not be best "there." Please note that the addition of thickness to a cambered mean line may well produce positive (outward) curvatures on top and bottom (typical airplane wing). If the camber is large and thickness slight, the bottom may be concave (typical bird wing.)
The conventional symbol for chord is "c" (lower case.) The length of our piece of wing, from tip to tip between the walls (and of all wings from tip to tip) is called the "span," with symbol "b" (lower case - think of "beam"). The area of the wing is given the symbol "S" (upper case) representing "Surface." ("A" is reserved for Aspect Ratio.) So, in our rectangular sample, S = bc. [Here, and throughout, I use the old convention that symbols without signs between them represent their product.]
The vertical angle between the upstream direction and the chord is called the "angle of attack." There are many forms of this idea to deal with different conditions. All are given the primary symbol "alpha," (Greek lower case "a") each with its own subscript. The standard symbol for section angle of attack is "alpha sub nought" which you might read as "alpha nought" and I'll write (alpha-0).* An upward inclination of the leading edge of the foil from the upstream direction of the free stream is positive. Down is negative.
I use this computer-enforced corruption of the traditional symbol (rather than coining something like SAA) to relate to standard aerodynamic discussion. This will help readers who go looking for further theory and standard airfoil data. | |
Abbott and von Doenhoff (referred to in AR-2) is a fine source of both. This NACA-data-based book includes some theory, plus graphical and tabular presentations of many valuable mean lines, thickness distributions, and actual sections and high lift devices. [But I should warn non-mathematical readers that, while there are some parts of clear explanation in English, taking "data-bites" out A&D requires determination and some mathematical experience.] |
Now, if we switch the fans on, wind blows around our tunnel and past our section from right to left. The balances indicate Lift and Drag forces on the foil. But even if we put smoke tracers in the tunnel, we would see no trailing vortices to correspond to those we saw screwing themselves off our oar blade in Part 2. There are none. [There may be some turbulence at each end caused by the interaction of wing with wall, but that effect can be accomodated (by experts) in assessing test results.] Trailing vortices are associated with a component of fluid velocity along the length of the wing. Here there is no spanwise motion because of the constraint at the ends. So, the air can possibly go up, back, down, forwards around the foil (or straight past it in an orthogonal plane) but not sideways.
Because motion in the third spatial direction (sideways or "spanwise") is precluded, this is called "two-dimensional flow." It is the same as would be observed at every section near the middle of an indefinitely-long rectangular airfoil of constant sectional shape and angle of attack. When we draw a section on paper, it is as though the flow also was confined to the paper. That flat picture represents what is going on at every section along the span of our rectangular test wing.
Two-dimensional flow is inherent in any discussion of the behaviour of sections. Consequently, trying to apply Section Data to the three-dimensional flow of wings directly, without using Wing Theory, is a gross mistake.
Now let's record the Lift and Drag of our sample section as we change flow Velocity. Say we start at 100 feet per second, observing 100 pounds of Lift and 1 pound of Drag. (Numbers chosen to make arithmetic easy.) This Drag is called "profile Drag" because it is caused by the flow around the section. It has nothing to do with "wing" behaviour.) We double the Velocity to 200 feet per second (fps), and measure 400 pounds of Lift, and 4 pounds of Drag. At 300 fps we measure 900 pounds of Lift and 9 pounds of Drag. (It appears that we have a pretty good section here, with a Lift to Drag ratio of 100.)
Actually, at the higher velocities, the Drag may be a little less than the quoted figures, and the Lift a little higher, because of Reynolds Number effects, mentioned in AR-2. Later in this part, we'll have a "micro look" at the flow around the section. It will give us an intuitive sense of what causes the circulation that leads to Lift, and governs its strength, and what causes profile Drag, and a hint of how these things are affected by Reynolds Number. |
The Velocity-Force relationships we observed always holds. From this we conclude that Lift, Drag (and other forces in any other direction we might choose to look at) are "directly proportional to" the SQUARE Of Velocity, (V times V). (Here we are treating V only as the scalar value of the magnitude.) If we assign "F" to represent generic force (in any direction of interest) may write this as
F = (K1)V*2, (Kay-one V squared) (1) | |
where (K1) is a "multiplier," called a "constant of proportionality." It makes the equation true by representing the effect of all other (invariant) factors in the system while we vary Velocity alone. The star * indicates that V is "raised to the power" of the following 2. |
We are on the track of a number of these "proportional relationships." Having found all those that relate to AHdynamic force, we will put them together in one equation.
If we close our system and pressurize it to 2 atmospheres (so doubling the density of the air) we will find that all other factors being equal, the force is doubled. If density is cut in half, so is the force. From this we conclude that AHdynamic Force is directly proportional to fluid density. We write this as:
F = (K2)(rho) (2) | |
where (rho) Greek lower case letter "r" stands for mass density (K2) is a constant of proportionality that accounts for all other factors affecting force while we vary density alone. |
The operators of sophisticated wind tunnels commonly adjust the density of their test gases (including air.) They do it by pressurizing, or partially evacuating, a closed, tight system. It is a way of altering the ratio of density to viscosity, and hence Reynold's Number. (Viscosity does not vary with pressure in the same way as density.) This is useful in making small-scale tests of full-scale applications without encountering model "compressibility" effects. Let me try to bring that down to earth - to represent a full-size foil in standard air at 200 miles an hour would require a quarter-scale model to be tested at 800 miles an hour to match Reynold's Numbers. But 800 miles an hour is "supersonic," and flow around the model would be deformed by pressure-caused changes of air volume and hence shape. So the flows could not be scale models of one another, despite the equal Reynold's numbers. The answer is to use a bigger model, fiddle with the air density, or choose a different test gas.
At a practical level, cold-weather sailors may be heard talking of the "weight" of cold air. The density of gases is inversely proportional to the "Absolute temperature" [ =460 degrees + Fahrenheit temperature, or 273 + Celsius temperature.] To compare the effect of sailing in 80 with 40 degree (F) air, we do this: Density being "inversely proportional" to temperature, we put the high temperatures on the top, getting (460+80)/(460+40)=1.080. This is the ratio of densities of cool to warm air. To convert to percent, subtract the 1 and multiply the remaining 0.08 by 100, getting 8 percent. This tells us that, all other conditions being equal, the 40 degree air is eight percent more forceful in its dynamic effect (or weightier, in sailor's parlance.)
If I may be forgiven a personal aside:
Well, I'll take it anyway! Near the beginning of that two year's sailing aboard my old Nova Scotia sloop, which I rebuilt as NAOMI, we encountered some "cold air." We went down Chesapeake Bay in the coldest winter in history (early January, 1970.) Cut the bows through in sheet ice, and found Zahneiser's yard (while pumping madly) and drove onto the marine railway. Some days later, after repairs, NAOMI hopped her way southward, trying to escape snow, cold, and ice, sometimes finding anchorage in blinding snowstorms, while running low on heating oil. After days of this weather the US Coast Guard was dropping emergency supplies to islanders from heliocopters. We approached Norfolk in late afternoon in 35 knots of 16(F) degree (-8.9C) air dead on the nose. Spray flying high in a sharp chop! Sailing impossible, of course. I steered with jury-rigged tiller lines from the cabin, wearing a plastic face shield (which immediately iced up) when I had to stick my head out. So, limping along at two knots or less, under a tired 5 HP Volvo MD1 Diesel (with a fortunately large and effective three bladed prop), over several frigid hours NAOMI collected one and a half inches of salt-brackish(?) ice on every surface from water line to spreaders (estimated nearly two tons on a 6-ton boat.) Talk about "heavy air"! You could have cut that stuff into blocks and used it for ballast! Temperature that night went down to all-time Norfolk record 10 degrees F = (-)12.2C. Got safely secured after being run aground in more sheltered water (by a *&^%$#@! fish boat - I learned the language from Bill Heinlen) and carrying out a kedge and pulling off. Hot toddy that night! |
A last comment about density might help prevent thinking from going astray: At subsonic speeds, the changing pressures around a foil (caused by velocity changes) do not affect the fluid density. The free air or water about a sailboat is able to tranmit pressure changes across space. But it is not compressed by increased pressure, nor expanded by reduced pressure. The "Pressure-times-Volume-equals-a-constant idea" (scientifically expressed as PV=nRT, where n is the number of molecules, R is a "universal gas constant," and T is absolute temperature), applies to closed systems. Our subsonic systems are "open." Effectively, that means fluid molecules "can get out of the way," or "fill in behind," without suffering volume, and hence density, changes. This is true until velocities begin to approach the speed of sound, which of course we don't encounter in sailing. The speed of sound in a fluid is closely related to the mean velocity of its molecules. That means that a body approaching or exceeding the speed of sound overtakes and "packs together" molecules that can't escape, at least not until the body passes them. Then the sudden "explosive" release of the local compression is heard as the "sonic boom" of a supersonic aircraft, or the Crack! of a whip in the air.
The idea that fluids are incompressible (in their motion about the foil) underlies subsonic AHdynamics.
Everybody knows that, other factors being equal, Force is directly proportional to Area (S), which we may express as
Force = (K3)S (3) | |
where (K3) is a constant of proportionality that accounts for all other elements of the system while we vary Area. |
Let's start with a symmetrical foil. Suppose that (in some set of conditions that we will hold steady) at zero degrees angle of attack [(alpha-0) = 0] the Lift is zero pounds. There is a slight drag. Tilting the foil up to (alpha-0) = 1 degree, we measure Lift = 10 pounds, with a little more drag. At 2 degrees, Lift is 20 pounds, and at 3 degrees 30 pounds. At 10 degrees, the Lift is 100 pounds. Throughout, the Drag has been increasing slightly. We tilt the foil down, and find that Lift (now negative or downward force) still varies directly as the (now negative) angle of attack. This "linear" relationship between Lift and (alpha-0) extends over a range from perhaps -10 to +10 degrees, or a little more or less, depending on the thickness ratio and thickness distribution of our symmetrical section.
Now, you will help yourself greatly by making a simple sketch. If you were here, I would do it for you. But cyberspace sits betwixt us. It is so important that I urge you, please, to start it, right now, rather than just thinking about it. It will strengthen your understanding. We will develop this sketch into an interesting insight into the behaviour of sections. If you don't do it, you'll likely miss something before we're done. If you "can't draw a straight line" don't worry. It doesn't have to be fancy. You don't need rulers or straight edge. Just slash away. Slight curvatures don't hurt. If you can't slash, "squiggle." Squiggles don't hurt either. Not trying will. If you have squared paper use it. OK, is everybody ready?
Start by drawing a square cross, in the middle of a sheet of paper, about parallel to the edges (without worrying about it.) Extend it nearly to the edges. Label the sheet "LIFT CURVE." Level one leg of the cross before your eyes, and label it "section angle of attack, (alpha-0) degrees." Put a (+) sign on the right end, and a (-) sign to the left. Label the vertical axis "Lift" and put a (+) sign at the top and a (-) at the bottom. Draw "O" around the crossing to indicate "origin," and think of it as the zero value of both angle of attack and Lift. Great beginning!
Now, on the "alpha" line, about half way between the origin and the sides of the sheet, on both sides, make a mark, labelled "-10" to the left and "+10" to the right. This establishes a "scale" for angle of attack. Do your best to divide the scale into degrees between 0 and 10, both sides. Accuracy is not necessary. We don't need a scale for the Lift axis.
Now, we need a straight line through the origin, slanting upwards to the right. About half way between the axes would be nice, or a little steeper. Carry it along until its end is about over the +10 on your positive (alpha-0) axis. Now continue the straight line downwards to the left from the origin until its end is about under -10 on the alpha axis. You have drawn a picture, or graph, of the "linear" part of what is called, for short, the "Lift Curve" by aerodynamicists. It represents the part of the experiment we have already done with a symmetrical foil.
Now, coming back to our section test, if we continue, the Lift will become less responsive to increases of angle of attack. We represent this on the positive part of the Lift Curve by drooping it off to the right, approaching "paper horizontal," at say, somewhere about 12 to 15 degrees. [This demonstrates graphically that increasing (alpha-0) does not increase the Lift.] Draw this suggestion in lightly, because it has two versions. A similar thing happens at negative values, so our Lift Curve looks somewhat like a shallow letter S (with a straight middle) leaning to the right. At some point, Lift stops altogether. It may stop quite suddenly. Or it may "mush down" through declining values as (alpha-0) increases for a few more degrees.
Let's draw these two generic possibilities of Lift failure on your graph, and give them names. That will help us to understand what sort of sections tend to cause each kind of failure. In your lightly-drawn curved part of the Lift Curve, stop at an early peak, with a short, quick drop and ending. This form of Lift failure is called "sudden stall." Label it so. The likely cause is what is called "leading edge separation." It represents an early failure of the flow to follow the upper surface of the section shape. Air peels off downstream, destroying circulation, and hence Lift, and leaving a broad wake, so causing a jump in Drag. This is characteristic of the performance of thin sections with small leading edge radii (which may have merit for other reasons.)
Separation may also rise from low Reynold's number (through small size or low Velocity.) It is a viscosity effect. So, perhaps counter-intuitively, by increasing the ratio of kinetic (mass) forces to viscous forces, a faster flow is better able to follow the upper Section surface. Where you might think that it would "schuss" off into space earlier, it doesn't.
Now let's draw a "gradual stall," and label it. Extending a little higher and further to the right than the "sudden stall" begin a short but gradual decline to an end (not too far away.) This is caused by a separation of the flow from the foil further back, nearer the trailing edge. This type is generically associated with thicker sections with fuller leading edges. Subtleties of shape can mitigate and merge the two different kinds of stalling, however.
You might extend these curves down in the negative range if you wanted to. Essentially, they are just mirror images. But generally, we're more interested in positive, rather than negative Lift. So we won't bother with lower end details.
For completeness, I'll mention that different sorts of stall are also associated with "laminar" and "turbulent" flows. Fortunately, we don't need a full discussion of them to get ahead. In fact, we can't even talk about what they are, and where they apply, until we've looked into the "boundary layer" a little later. Then I'll come back and try briefly to cast a liitle more light on separation and stalling.
Now let's test a positively cambered foil (of some unspecified percentage and form of camber) with the same thickness distribution as our first symmetrical foil. The first thing we notice is that, at zero degrees angle of attack [i.e. (alpha-0) = 0 ], there is a positive Lift. We may rotate the section clockwise (so reducing alpha-0) for several degrees before Lift vanishes. Only at more negative angles will Lift go negative. This point is called the ANGLE Of ZERO LIFT, a very valuable REALITY (as well as idea, guys) in AHdynamics. It is given the symbol (alpha-lo) read as "alpha l nought"* and meaning "the angle at which lift is zero. So, if our cambered section loses lift at (alpha-0) = -2 degrees) we may say that its "alpha l nought is -2".
To avoid possible confusion, let me emphasize that (alpha-lo) does not mean that the angle of attack is zero [properly indicated by (alpha-0)=0.] The symbol (alpha-lo) is an indication of the Section's ability, by virtue of its shape, to contribute to Lift, independent of angle of attack. Incidentally I am using lower case case l (not L) because, by convention, SECTION subscripts are lower case. Upper case is reserved for WING descriptions. |
The happy good fortune in all of this is that the "shape-lifting" capacity, measured by (alpha-lo), and the Lift contributed by angle of attack, are simply additive.
Don't think that these ideas apply only to hard-surface foils. Once circulation and lift have been established, the forces on a soft sail will keep it filled and drawing even at negative angles of attack, particularly in steady winds and moderate seas. Anyone who has ever gained advantage by "feathering" sails to windward has some intuitive prescience of points we will make in the later part of the series on applications to sailing. There, among other things, we will show that the search for "High Lift" as a Holy Grail for getting to windward may be something of a fallacy at times. |
Let's put this in the framework of our tests. Our original symmetrical foil lifted 40 pounds at 4 degrees angle of attack. If our cambered foil has (alpha-lo) = -2 degrees, then, other conditions being equal, it will lift forty pounds at 2 degrees angle of attack. To get the "total Lifting effect" you reverse the sign of (alpha-lo) and add it to (alpha-0) and then consider the effect as being effectively that of a "generalized (alpha-0)" alone. Again, this is one of the nice bits of serendipity in AHdynamics.
Another way of looking at this is to say: the "slope" of the straight part of the Lift curve of all sections is the same. (This is very nearly true, except for some very refined sections which may have an almost-indescernible slightly steeper slope.) You can represent this idea, on your Lift Curve, by going left on the alpha axis to (-2) degrees, and drawing a line parallel to your symmetrical-foil curve. If you have drawn more-or-less carefully, you will see that "cambered LIFT" [at (alpha-0) = (+)2 ] equals "symmetrical Lift" [at (alpha-0) = (+)4 degrees.]
To complete your understanding of the effect of camber, you should take note that, as well as moving left, the camered-foil Lift Curve also moves bodily upwards somewhat. This means that the stall representations (in both positive and negative fields) have been raised. That suggests an increase in maximum (positive) Lift. (That is true, but there are limits to how much camber can contribute to lift without stalling.) You may now draw any number of additional Lift curves, extending to the left, in a rising array, realizing that, as you go left, you are representing (principally) the effects of increasing percentage camber.
Sadly, there is no simple relationship between camber (and its refinements of shape) and (alpha-0). We can't say, "6 percent camber will produce a 6 degree angle of zero lift (or any simple ratio of it.) The effects of camber and other shape details can be calculated, but only in an aero-mathematician's special world. Even for them, the angle at which stalling occurs, and how it will occur, is as much a matter of test observation as mathematical prediction. (Or so it was back in the days when I took a few jumps over the fence of the aerodynamic compound and took a bite or two out of the contents and tried to chew them without choking.)
Force (either Lift or Drag) is not directly proportional to angle of attack beyond the linear range, nor is it proportional at any time to its sum with "section shape factor." But Force depends on them both together. To reflect this, rather than use a CONSTANT of proportionality (KX), as we did for density, area, and velocity, we might use a VARIABLE COEFFICIENT (let's call it C1) to represent their combined effect and "hold invariant" the other factors in Force. Then we might write:
Force = (C1)[ (alpha-0) + |(alpha-lo)| ] (4) | |
where (C1) has the meaning just defined, to be multiplied by the term [(alpha-0) + |(alpha-lo)|] representing a "flow shape factor" controlled by the combined effect of angle of attack and the positive lifting effect of the foil shape. [i.e. its "absolute value", ignoring or reversing its negative sign.] |
All the effects represented by equations (1),(2), (3), and (4) are "mutiplicative." That means that if we double density, the area, the velocity factor (by a 41 percent increase in velocity, to be "Squared" to 2), and the effect of the "shape factor," the force increases by 2 times 2 times 2 times 2, or a factor of 8. We can show this by putting all the equations into one.
We can tidy things up even more by rewriting (4) in the form
F = (C1)(FlowShapeFactor) |
F = (C1)(FSF) (5) |
F = (C1)(K1)(K2)(K3)(FSF)(rho)(S)(V*2) (6) |
F = Q(FSF)(rho)(S)(V*2) (7) |
F = (Cfo)(rho)(S)(V*2) (old form) (8) |
F = 1/2 (Cf)(rho)(S)(V*2) (9) |
Please indulge me in another personal
aside here. Apparently, the change was made to make these Force equations "similar" to dynamic energy equations. [The latter take the form E equals something times something else squared.] The idea that any expression with a "V*2" in it ought to have a "half" in front is an example of "parallelism gone mad," in my view. Mass and density are quite different, having different dimensions, as do Force and Energy. There are examples of thoughtless or ignorant parallelism in other fields, just about as silly as this. I am reminded of a discussion of astronomy in Bowditch. The author gives the pronunciation of aphelion (a position of the sun) springing from the Greek adverb of position "ap," and "helios" (sun), as AFFelion, (rather than AP-helion), presumably out of ignorance of the roots and in fancied parallelism with eleFFant. Following this train , I suppose we might soon expect the President of the United States, in his inaugural address, to swear to "uFFold the Constitution." |
However, Equation (9) is the form to which all modern Section data apply. Just be aware that these are Force, and not energy, equations.
AR-4 is a single Chapter of the AR-X series, (Continued in AR-4B ) which will appear on the Yacht-L filelist which you may obtain by addressing the command GET AR-4B DOCUMENT YACHT-L to Listserv@Listserv.SURFnet.NL
Editor: Eppo R. Kooi; email: E.R.Kooi@XS4all.NL Last update: 010716 |