the earth as a threepenny planet

Gravitational length is a concept closely allied with Planck quantities and can be used to define the Planck set. At this point what is offered is a condensed introduction for readers who already have some preparation. The same material will receive a leisurely treatment later.

The gravitational length is essentially a way of looking at the mass of a body. In the earth's case, this length is roughly the thickness of three pennies. Here it is depicted rather approximately: . The gravitational length of the sun is roughly a mile. By "roughly" I mean within ten percent--if more accuracy were needed at this point I would say nine tenths of a mile. Both lengths will be expressed more exactly as soon as there is need for it. Referring to the earth as a three pennywidth planet and the sun as a mile star is intended to evoke the notion of gravitational length in a way that implicates the Planck length. Where it seems cumbersome to write out "gravitational length" in full, I am prepared to write gravit for short. It will sometimes be convenient to write the gravitational length symbolically either as L or using some more specific notation such as L and L.

The next table lists a few of the things which gravitational lengths tell us. More conventional formulations depending on the Newtonian constant G and on inertia, denoted M, are shown for comparison.

application	version using gravit	version using inertia
light bending angle	4L/r	4GM/cr
orbit speed	(L/r)	(GM/cr)
escape speed	(2L/r)	(2GM/cr)
acceleration	L(c/r)	GM/r
black hole radius	2L	2GM/c
once-around pendulum	L	GM/c
stationary orbit speed	(L/c)	(GM/c)
earth-sun pull	LL/r	GMM/r
rest energy	L	Mc

In this table speeds are given as fractions of the speed of light. The light bending angle is given in radians and is the angle by which light is bent while passing within a distance r of center. Escape speed and acceleration due to gravity are for small bodies located at a distance r from center. Orbital speed is for a small body in a circular orbit, r being again the distance from center. The ratio L/r is basic to several of these applications and one instance of it will soon be evaluated: the ratio of the earth's gravitational length to its radius. The outline letter which you see in the table stands for the Planck force--the fact that a body with gravitational length L has rest energy equal to L is one connecting gravitational length with Planck quantities. "Black hole radius" refers to the normal model of a non-rotating black hole. One with mass equal to the sun's would have a 1.8 mile radius (twice the nine tenths mile mentioned earlier). A black hole with mass equal to the earth's would have a radius twice the earth's threepenny gravitational length. A stationary orbit, such as a communication satellite might follow, is one matching the rate (radians per unit time) at which the central body turns.

measuring gravitational length

The combined speed and separation of two things in circular orbit around each other reveals their combined gravitational length. If one of the two is small enough for its contribution to the combined length to be neglected this immediately indicates the gravitational length of the central body. The way this approach works is that if the orbiter is a distance r from center and traveling at a fraction of the speed of light, then the central body gravit is r. This is particularly easy to apply to the earth's nearly circular orbit around the sun. The speed fraction is , and consequently its square is , while the distance is some 90 million miles. Multiplying this distance by the square of the speed fraction gives 0.9 mile, indicating the sun's gravitational length. By taking a little extra trouble with the definitions, this approach can be generalized to cases where the orbit is elliptical and also where the two bodies are comparable, instead of one dwarfing the other. By arranging to measure both orbital speeds separately it can also be extended to determine the two gravitational lengths separately as well as their combination.

In normal sealevel gravity a pendulum whose length is L goes back and forth in the time it would take light to go once around the earth at sealevel--it is a "once-around pendulum" in this sense. When located a greater distance r from earth center, its period would be the time light needs to travel in a circle with that radius. If we are prepared either to adjust for or ignore some minor effects of the earth's rotation and irregular shape, this offers an elementary means of determining the earth's gravitational length. Observing satellites in orbit around the earth is clearly another option.

connecting gravitational length and Planck quantities

One connection between the gravitational length and the Planck quantities has been already suggested: the fact that the Planck force is the proportion relating gravitational length to rest energy. A body's gravitational length L and its rest energy E are related by E = L. Indeed this relation can serve to generalize the idea of gravitational length and define it for bodies whose rest energies are known but which are too small to allow their gravitational lengths to be measured in the ordinary way. The equation = is a special case of the foregoing: the Planck length is the gravitational length of a body whose rest energy is the Planck energy .

A further connection between gravitational length and Planck quantities occurs in the definition of the Planck energy . The Planck energy is uniquely determined by the fact that its associated gravitational length and its associated wavelength are equal. Every quantity of energy has such a pair of lengths associated with it but is the only energy for which the two lengths coincide. Moreover the Planck length is likewise determined as this common length--a meeting of wavelength with gravity. There will be more discussion later, for now we simply note a connection between gravitational length and the Planck quantities. The version of wavelength used is cyclelength divided by 2.

where gravits are more accurate than inertias

The gravitational length (whether as L or as Lc) is what astronomers measure when they measure mass. It is known with considerable precision and is the form of mass appropriate for subsequent calculation if there is a need for accuracy. Once gravitational length has been measured it may be converted to the less certain form of inertia by means of the Newtonian constant G, and expressed in grams or kilograms. Because of the uncertainty with which G is known, the earth's inertia in grams is provided with considerably less precision than the gravitational length. If inertias are to be used subsequently for calculating orbits they must first be multiplied by the Newtonian G (a second opportunity for error) to get the information back into relevant form.

In Astrophysical QuantitiesLc: the gravit multiplied by a factor of c. Because the speed of light is exact in modern units, this c factor is itself exact and does not effect the precision with which the length is known. In other words the c factor can be regarded as a formality and excised without loss of information.

Evidently it makes no logical difference whether one uses L or Lc--one being an exact multiple of the other makes them equivalent as scientific data--but the two quantities are of different types. One is a length and the other is of a type without conventional name or commonly understood meaning. For this reason it seems to me more intuitive to use the length consistently.

sample calculations with gravitational length

The length of a pendulum varies as the square of its period, and that of a "once-around" pendulum is the earth's gravitational length L. So the length of a "four-times-round" pendulum is 16L. This length is a practical one to measure experimentally. Light travels four times around the earth in a tick--the approximation is good to within one percent--so once you have adjusted a pendulum to have a period of one tick you can consider its length to be 16 times the gravitational length of earth.

A metronome set to 111 or thereabouts will beat approximately once per tick--on our metronome this setting is in a range labeled tempo moderato. To measure the gravitational length of earth using a small weight tied on the end of a thread, adjust the length so that the pendulum motion matches the metronome and divide the length by 16.

Because units like the pennywidth and mile have exact metric equivalents, we may use any desired degree of precision available in metric contexts. To the extent that a couple of extra digits of precision might now be appropriate, the earth's gravitational length L is 2.74 pennywidths. Stacking three coins does give a rough idea of the actual length, but for some applications this further accuracy is helpful.

The ratio L/r appeared basic to several of the items listed in the earlier table and one instance of it is worth calculating: the ratio of the earth's gravitational length to its radius. As a temporary convenience I refer to 1000 miles as a "grand"--it happens that the earth's equatorial radius is 3.94 grand. That makes the ratio L/r equal to . Considering that a pennywidth is a billionth of a grand, and that 2.74 divided by 3.94 is 0.70, we conclude that L/r is 0.70 billionths.

Multiplying L/r by four immediately tells the angle by which grazing light is bent--2.8 billionths of a radian. Furthermore, 2L/r is 1.4 billionths, which has 37 millionths as its square root. This means escape speed from the earth's surface is 37 millionths. A millionth of the speed of light corresponds approximately to the speed of sound in the cold air at ordinary cruising altitudes. So escape speed can be thought of as 37 times this cold air sonic reference speed: as "Mach 37" if you wish to interpret it that way. As a further illustration, L/r divided by 7 is 0.10 billionths, which has 10 millionths as its square root. This means that circular orbit speed at 7 earth radii from center is 10 millionths--ten-fold faster than cold air sonic. This "Mach 10" speed is roughly that of communication satellites in stationary orbit.

The ratio L/r has a physical meaning which has not been mentioned. In the case of a small body located a distance r from main body center, it expresses the kinetic energy needed for escape as a fraction of rest energy. For something on the earth's surface to get loose, it needs to be given 0.70 billionths of its rest energy. L/r is a kind of binding energy fraction, and for all of us here on the earth's surface this fraction is 0.70 billionths.

Another sample calculation: the acceleration due to gravity at sealevel. L(c/r) is the same as L divided by the square of r/c (the time light would take to dive down a well to the center of the earth). As for the dive time, our distance to center is 3.94 grand and light goes 1 grand in 0.01 tick, so r/c comes out to be 0.0394 tick. After writing L as 0.0274 span and dividing by 0.0394 we obtain that L/(r/c), namely the acceleration due to gravity at the earth's surface, is 17 span per tick.

gravitational length and baryon number

The protons and neutrons that provide most of the weight in ordinary objects are referred to as baryons (from the Greek word for "heavy"). There is an interesting number 13 billion billion = 1310 which comes into play here. The gravitational length of a proton is much shorter than the Planck length--in fact it is L. As long as we do not want too much precision the same number will do for the neutron--that being about the same mass. By contrast, the associated wavelength of either particle (the so-called Compton wavelength) is much longer, namely 13 billion billion . Evidently an ordinary object which happens to have for its gravitational length must comprise 13 billion billion baryons, and a seven-year-old whose gravitational length is a billion times the Planck length must comprise 1310. As a rule of thumb, to find the number of baryons in a thing one multiplies its gravitational length, expressed in Planck terms, by 13 billion billion. This can just as well apply to the sun, or to a mile star (one whose gravitational length is 10 ), as to the billion youngster featured earlier. The number of baryons in a mile star--alpha Centauri is an example of one--is 13 billion billion multiplied by 10. The baryon number of the sun--with its 0.9 mile gravit--is 90% of this.

reciprocal density and displacement

In seawater of normal salinity, namely 3.5% by weight, measured at a temperature of 20 Celsius (68 Fahrenheit), this form of the reciprocal density is 5.0010 span. It corresponds to 1.025 kilogram per liter and closely approximates the traditional 64 pounds per cubic foot, long used by ship designers. There is an alternative metric norm at 4 Celsius and 0% salinity, but those conditions are neither comfortable for swimming nor appropriate for the design of ocean-going ships.

Obviously multiplying the youngster's gravitational length of 10 span by the reciprocal density 5.0010 span gives 5.00 span, which is the same as 5.00 gallons. So that is the youngster's displacement in seawater at ordinary temperature. Swimmers can adjust their buoyancy using the air in their lungs. If you exhale a substantial amount of air while swimming you may achieve neutral buoyancy. In that case, whatever your gravitational length happens to be, multiplying it by 5.0010 span will give your actual volume in gallons.

perceptual validity of gravitational length

Material near the edge of our galaxy (r = 50,000 light years) is purported to be moving at a speed = 0.0007-0.0008. This leads to a straightforward r estimate of 9-12 light days for the galaxy's gravitational length.

The thickness of three pennies, a pace, a mile, ten days--it is easy to become acquainted with the masses of earth, Jupiter, the sun, and the Milky Way galaxy in those terms. Jupiter's mass, nearly a thousandth of the sun's, can be pictured as a pace, which has meant one thousandth of a mile ever since classical times. In the case of the galaxy, to the extent that a distance on the order of ten light days may be hard to manage, the time period of ten days can itself serve as a substitute handle.

Were more precision appropriate, these gravitational lengths could be expressed as 2.74 pennywidth, 0.87 pace, 0.91 mile, and 9-12 light days. For many purposes all one needs to know in order to use such lengths is the one equation L = r and its obvious corollary = . By contrast, the corresponding inertias in kilograms are not particularly easy to remember or to use in calculation.

If you have not already experienced the awkwardness of inertia as an index of planetary mass, and would like to do so, try to estimate the orbital speed of Callisto using kilograms. I suggest calculating this because it is something you probably do not know offhand. Callisto is about one million miles from Jupiter, but if you prefer metric units you can take the distance to be two million kilometers. The process of arriving at a rough mental estimate, based what you already know, will suffice.

Along metric lines, you might begin by recalling or reckoning the sun's inertia, expressed in kilograms, and divide that by 1000 to get a figure for Jupiter's inertia. This approach will then require that you recall the metric value of Newton's G. It would not be surprising to experience difficulty at three points: (1) reckoning Jupiter's inertia in kilograms, (2) recalling Newton's G, and (3) using the two in mental calculation.

To show the ease of using gravitational length, here is an approximate calculation of Callisto's speed. To allow calculation by inspection, Jupiter's gravitational length will be approximated as 0.001 mile and Callisto's orbital radius as a million miles:

= = 0.03 thousandths = 30 millionths = Mach 30.

It may be worth emphasizing that the gravitational length is not merely an aid to calculation. It is a perceptual means that can help us become acquainted with, visualize, and mentally compare various significant masses.

additional precision in gravitational length, samples

Although at this point we have no need for additional precision in gravitational length, I will illustrate how it might be obtained in the sun's case. We will use r. It was already mentioned that miles and kilometers are exactly interconvertible--the standard handbook figure for the earth's average distance from the sun corresponds to 92.408 million miles. The orbital period, or sidereal year, is listed as 365.25636 days. In a mantissa-calculation like this one, where the correct powers of ten are already known, we can use a factor of 1.6 to convert days to tick (because there are exactly 160 thousand tick in a day). Here is the circularized speed and the resulting gravitational length L:

= 10 = 0.993510
L = r = 0.92408 0.99352 mile = 0.912 mile

The calculation highlights how close the earth's orbital speed is to 0.0001, a coincidence providing the scale of speed with a useful point of reference.

For our purposes 2.74 pennywidth is sufficiently accurate for the earth's gravitational length, although Astrophysical Quantities gives a six-figure value for Lc corresponding in our terms to 2.73958 pennywidth. The earth's period of rotation (with respect to the stars) is shorter, by a factor of about , than the solar day from noon to noon. In comparable six figure accuracy its rotational period is 159,563 tick, but for approximate calculations the figure for the day, namely 160,000 tick, is often accurate enough. Having mentioned rotational period, we have an opportunity to use the formula listed earlier to reckon the speed of a satellite in stationary orbit:

stationary orbit speed (L/c) (GM/c)

Evidently the angular frequency of the earth's rotation is 10 radians per tick. As a result of a shakedown in the powers of ten, in the mantissacalculation shown here the angular frequency was represented by and the earth's gravit is represented by 274. The speed traveled by communications satellites turns out to be:

= (L/c) = 10 = 1010 = Mach 10

So the thing my neighbor Hugo has his dish aimed at is traveling Mach 10. Making this more precise would merely involve replacing with , resulting in a speed of 10.2610. Again the calculation highlights a coincidence providing a useful point of reference on the speed scale. The earth's orbital speed, the speed of communications satellites, and the speed of sound at ordinary cruising altitude are all close decimal relatives of Planck speed.

exotic uses of gravitational length: the fate of the universe

Primarily to exercise the concept, I am going to reformulate some material from standard textbooks of astronomy in terms of gravitational length. According to the usual (non-rotating) model, a black hole with radius R has gravitational length . Because its volume is R, its reciprocal density must be the area obtained by dividing that volume by , which comes to R, providing one possible indicator of how voluminous the black hole is in proportion to its mass.

An interesting encounter between gravitational length and the destiny of the universe appears if we use the same formula but take R to be the Hubble distance instead of the radius of a black hole.

The Hubble distance, currently judged to be 15-18 billion light years, is essentially the ratio of the estimated distance light has traveled to the fractional amount it is observed to have been stretched in transit. A few tenths of a percent stretch is seen in wavelengths of light which has traveled a few tenths of a percent of the Hubble distance. The distance is long because spatial expansion is slow--to be noticeably stretched by the expansion of the space through which it has been traveling, the light must have come a long ways. The Hubble distance indicates the current slowness of spatial expansion and it lengthens as the expansion slows, a fact adjusted for in observing the most distant objects.

If, as was just suggested, we take R to be this measure of the slowness of spatial expansion, then it turns out that R is the critical value for the universe's reciprocal density, indicating how dispersed or spread out matter must be in order for the universe to avoid eventual collapse. Intuitively, the slower the current rate of spatial expansion the more spread out things must be. The actual reciprocal density, which does seem to exceed the critical value, describes the spaciousness or degree of emptiness of the universe, which from our standpoint may be viewed in a favorable light. With insufficient emptiness, the universe might have collapsed before we evolved, or it might now be in the process of collapsing.

the forever test

Because the observed reciprocal density is a very large area it can simplify discussion if we use the length which is its square root as our index of spaciousness--the overall sparsity or spread-out-ness of material in the universe. This observed measure of spaciousness can be compared with the critical value R, the square root of R. Because is roughly three, the critical amount of spaciousness is approximately three times the Hubble distance, a length indicating the slowness of spatial expansion. We can summarize this as a test of forever: a criterion which a universe not destined to collapse (an "open" universe) must meet. No matter how slow expansion is, the spaciousness should be at least three times the slowness.

Putting the Hubble distance at 18 billion light years results in a critical spaciousness of 50 billion light years. To judge whether the universe is destined for continued expansion or for collapse we need to be able to estimate the actual reciprocal density and compare its square root, the observed spaciousness, with this critical 50 billion light year length. The observed figure, highly uncertain, is reckoned to be some 100 billion light years--twice what it needs to be, suggesting that unlimited expansion is in store.

The spaciousness surrounding our local group of galaxies can be estimated in a rudimentary fashion and gives some notion of the overall situation. The Local Group to which our galaxy belongs is judged to have a mass some three times that of our galaxy (the group's principal members are Andromeda and the Milky Way itself) and appears to represent the only significant mass in a cube some 10 million light years on a side. The observation of material out near the edge of our galaxy (r = 50,000 light years, = 0.0007-0.0008) leads to an r estimate of 9-12 light days for the galaxy's gravitational length. Three times that--an estimate for the Local Group--is roughly a tenth of a light year. Merely by cubing 10 million, dividing by 0.1, and taking the square root we arrive at 100 billion light years as an estimate of the local spaciousness. This does not mean a great deal--it is the overall figure that matters--but it is moderately reassuring since the neighborhood of the Local Group does not seem unusually empty and 100 billion light years is twice the critical value.