physicists have quite presumed that gravity is a force |

CLASSICAL BACKGROUND:

*It has come to my attention that contemporary college course discussions of gravity still use calculations in excess
of the mass-energy available--in violation of mass-energy-conservation: The discrepancy is 0.03% at the surface of a
solar-mass carbon-dwarf star, 10% at a neutron star, 100% at the Schwarzschild radius of a mass-hole, and 'infinity'
at its center 'singularity'... I shall resolve that here, then resume my discussion--*

PART I: Extracting mass-energy from a mass-hole:

When a test-mass falls towards a gravitational mass e.g. a planet, it converts 'gravitational potential energy' at the larger radius, to 'kinetic energy' or 'structural potential energy' at the descending radius (continuum-plural takes the singular form), or radiates energy in some 'neutral form' analogous to polarized photons-- where both mean mass-energy, but potential means internal 'standing' and kinetic means external 'relative velocity'--So, in 'Newton's equations revised for mass-energy, their combined system-mass is losing rest-mass by converting it to extractable-or-radiatable kinetic-or-thermal energy--albeit in the case of collapsing stars, much of the kinetic energy gets trapped kilometers-beneath the surface layer and cannot be immediately radiated: so the star long maintains most of its mass-energy taking millions of years to 'chill it out'... In a dense star like a dwarf star it's a mere 0.2% difference, and much of that drives degenerate electrons to permanent high-energy orbits effectually retaining the mass-energy and gravity, and the remainder is just unnoticed by astronomical techniques of today, But, in neutron stars it's 8%-40% and increases singificantly as the neutron star further condenses, as photon radiation pressure escapes... For dwarf stars at their limit consuming their debris rings, it means a slight delay before the next stage of supernova, but for neutron stars it can mean permanence.... Altogether, it is similar to electrons dropping into atomic orbitals and radiating mass-energy as bremsstrahlung photons....

(Note that mass-energy-equivalence-and-conservation, is demonstrated in weighing radioactive elements and their 'daughter products' and measuring heat generated and photons radiated, as well as a formulaic outcome of 'Einsteinian' spacetime relativity theory--wherein mass-energy is the compound-sum of a nonrelativistic rest-mass plus its relativistic momentum, E² = m² + p², or written-out = (E)²=(mc²)²+(pc)² as p=γmv² but γ=1/√(1-(v/c)²), or simply E=γmc² where γ includes v²....)

PART II: Correcting the inverse-square-law equation:

The original basic 'Newton'ian gravity equation has--anomalistically still has--an inverse-square-law calculation for its force, GMm/r², and inverse-law for its energy, GMm/r--as every massive portion attracts every other... so, scaling-up for example, two masses, to a million-times each, the equations have a trillion-times as much gravitational force and energy, gaining rapidly against the conservation of system mass-energy at two-million, and continuing up by-the-numbers must eventually exceed the total available... Yet that energy can only come from, the original, conserved, system mass, (And as-yet there is no loss to radiation)... (The 4th-grader version simply dropped a particle into a singularity, gaining infinite-energy as it reached bottom; however, mass-energy is conserved, so there must be a correction-factor to gravity...and to other particle-forces too...)

Moreover, by #1 above it'd take on the order of a dwarf star mass to noticeably affect the equation, maybe a neutron star (depending on the exaction of corrected theory), But it'd be definitely noticeably a problem for mass-hole energies: As a stellar core collapses into a mass-hole, the gravitational energy ramps-up by the inverse-radius and yet by conservation there is not enough to supply it: The equation for gravity has been wrong--the outer mass must be drawing the inner mass-energy-field apart, or pulling itself back up out on its outer fringe: so that it 'feels' less gravity as it falls into the mass-hole...

Consider the demonstration in the Theory of Inflation (cosmic big bang), from the 'late-1970's: If two concentric massive spheres are allowed to collapse toward their common center, the inner sphere 'feels' only its own self-gravity while the outer sphere 'feels' both, itself and, the inner--but only so long as the inner precedes it... so, the inner sphere must be leaving some of its mass-energy-field far outside itself--for the outer to 'feel'...

In other words, nothing really falls very far into a mass-hole, as it leaves its mass-energy (not matter but its field) behind-outside as a 'potential field' for other later masses falling-in... And the outer masses must be trimming the fringe of the inner field (which may still be symmetric and not 'feel' the outer)... What enters a mass-hole is a minuscule mass-energy-portion of its original mass-energy-field left 'parked' outside...

(Note that 'one-bit-corrections' have been tried on subatomic-electronic, inverse-law equations, but the results were undesirably confused--albeit interesting or 'inspiring'--yielding stranger-still properties such as violate continuity-equations in "orbit-spheres", though classic-infinities-still-accepted were really no better...

PART III: Theory Development:

As a sphere of mass shrinks, it must leave-behind 'outside' either an instruction, for conversion of mass to energy, for the next in-falling mass, or it leaves-behind some of its own mass-energy as potential-energy, (else there'd have to be communication-at-their-distance to convert mass to kinetic), which then entangles the next mass--and that conversion has a recoil in the gravitational field which amounts to a static field... (static probably in the sense of radial, not circumferential)...

RESUMING THE DISCUSSION:

Gravity is usually notioned as a static phenomenon: eg. you weigh an apple on a scale, but jump up, and gravity dissembles momentarily. Yet gravity is as amazing as the neutrino passing invisibly through famed light-years of lead: Gravity does, too!, But it is questionable whether gravity passes through cold neutron stars, -neutrinos do not,- or even through supraneutronium.

If we consider that photons slow as they travel through media such as glass, by mild elastic interactions with large electro-charges overall neutral, within the glass, then we recognize the necessity of handling the general case of interaction with all electro-charge phenomenon, including tiny cumulative, large-scale-neutral nano-ripple in the space around massive objects. (This proves extant the cosmic aether and as similar in substance as electro-charge; but which also proposes aether-ripple as energy-itself causing aether-string multiplication -aka aether base energy,- and charge is just a transport property of energy as we know it atop the aether.)

The gravity-force strength is 10^{-43} the attractive/repulsive force strength of
the electromagnetic, -for a small wavicle such as the electron (a charged lepton);- this
in itself suggests that gravity may be a left-over of larger forces in play and place:
In particular, we note that electromagnetic activity while broadly symmetric has no
particular definition which requires complete three-dimensional sphere-radial symmetry.
(The electron may be a doublet of intertwined charge flow,
or figure-eight threaded through a figure-zero, which would have a small polal anomaly
in addition to its magnetic anomaly polarity; and might definitively account for
quark-charges of 2/3rd and 1/3rd electron charge, being effectually directional
resonances within; ... and for negative temperature.)

An analogy, is the electromagnetic activity in atoms having overall null charge, but as
we know from its quantum mechanics, the charge field distribution is not perfectly
spherical, but partially-filled six-lobed, with under-layered 10-sided, 14-sided,
18-sided, 22-sided, lobations ... Consider all the lobes of 10^{50} atoms in
a planet-size mass, and conceivably there is a charge-lobe-field ripple distribution
inversely intense reaching out from the planet: This may count as a significant fraction
of the aether ripple; the rest being due to sub-wavicle ripple. Photons and wavicles
convolved of the same electromagnetic energy, passing through such a field, cross the
ripple slowing its path scopically, entropizing its path infrascopically; and steering
them differentially centerward.

So, while A. Einstein showed the consistency of his theory of photon-slowing in a gravitational field, the -predicate- cause may be the generalized wavicle-slowing-steering in an electromagnetic ripple field being observed and called, gravity ... Because the gravitational field of a compact object is radial, the density of aether ripple is inverse-law, maybe inverse-square-law proportional to the radius and elliptic orbits are the effectually "straight paths" near such. [It also suggests that an ordinary photon inside a mass-hole singularity, entering, slows so much it deforms and shatters to nano-wavicles ... and likewise the mass of its original star did too]

But, aether-ripple, nano-ripple, zero-point-energy gradient, attendent-proximate electromagnetic activity ... is in summary reality just the cosmos base aether activity business: more top-energy, is simplistically just, more aether-splitting--

COSMIC AETHER BASE FORCES: Factors in proximity to top-mass-energy--

- Diversion of aether-"noodles" by mass-energy hotspot presence is "space curvature":- the localized wavicle presence deflects aether-strings more perpendicular to that locale, whence other wavicles entering or passing through the region, get more locale-radial deflection-angle pressure, -called, "gravity";
- Conflict-resolution splitting of aether-flow increasing aether "noodle"-count nearer and within top-mass-energy, produces angular deflection at the Y-split backed by thicker stiffer aether-"noodles" outside and beyond: and the transport-loading steers countward; This string-splitting is measured as a stronger nuclear force, as it occurs right where the wavicle is most prominent "inside the shell radius," rather than around it; (*)
- Also, the scalability of thinner-softer aether movement proportionally slows top-energy passing, and so deflects it energy-ward.
- The aether "noodle"-recombinatorics rate less than unity means aether-base entropy and zero-point energy totals are gradually increasing ... accounting for redshift and redshift-acceleration: The effectual splitting-rate is proportional to "noodle"-density and so apparently compoundedly, increasing (cf the "Supernova Ia Candle" for the cosmic-expansion coefficient "unconstant").

Gravity itself (as electromagnetic standing-ripple) is gravity-lensed by massive objects, and greater lensed or scattered by large-scale refraction. Eg. During solar eclipses the moon slightly re-focuses the sun's gravity, thus increasing its field intensity in the lunar-Earth-line vicinity and extending its focal length: making the sun appear more massive (a time-slowing effect not yet measured in reports *) and further away; and ultimately round-about controlling the Allais-anomaly in the leverage-advantaged Foucault pendulum path (**) during solar eclipses. [We may also postulate that a significant re-focusing of a stellar object gravity might make it appear beyond the refocuser: a negative gravity, or anti-gravity effect, in the directional vector sense]

* (Reported measurements made on anomalous motion in the [Lageos] satellites during
solar eclipses by the sun, amount to a less-than 3*10^{-10}
cm/sec², gravity reduction.)

** (The Allais Foucault-Pendulum experiment effect takes place almost abruptly from approximately the moment the moon shadow touches the Earth horizon "first contact," continuing, until it leaves off as equally almost abruptly,-- a much longer effect than an observatory point-of-view lasting a few minutes of totality; But more importantly, that is the same moment the moon-focused solar gravity touches the Earth ... Allais' effect needs be not gravity directly but the whipping-action of gravity changing focus as well, as the gravity focal-center jumps to and fro, momentarily advancing the Foucault Pendulum swing, and restoring its parameters hours later ... Other scientists using different equipments observed results at times the moon merely passed, near, the Earth.)

FRAME-DRAGGING:

Muchly a misnomer despite 'aether-ripple-drag' theory, as the portion of slowing due to sun-spin back-passing was overlooked by both Newton and Einstein (See the larger paragraph on Mercury).

DECOUPLING FROM THE EXPANSION OF COSMIC SPACE: (Big Bang)

There are better corrections to the problem of localized-decoupling from the expansion of space, in the expanding-space version of the Big Bang, but, more-notably this-one is a 'blunderer'--for if gravity decouples from the expansion of space, then photons traveling the star lanes (galaxy filaments remnant from the early cosmic 'voiding' processes) must be less-expanded, warmer, and arrive sooner, distance less-expanded, and thereby younger, showing deeper into the CMB time-zone (but cooler if there are wear-spots), and, photons via decoupling-boundaries must be deflected and vanish worse than gravity-lensing, and sharp-decoupling-boundaries implicate 'vacuum-plate-tectonics'...

(It should be recalled that gravity is accelerative while cosmic-space-expansion is 'velocitive', and the two actions do-not, co-compensate... More likely is the rest-mass-gain in climbing out of gravitational wells, since the Big Bang, thereby increasing gravity and so pulling-back-in... the effect is already notable in the orbit of the Earth.)

EQUATIONAL 'COMEBACK' IN THE BEYOND:

If gravity is directional, as is commonly assumed among its 'properties' (as distinct from 'effect' by where-the-mass-is), then the field-beyond-both may be 'pulling' back, pushing the mass-center away...this may be either 'theory', or usable calculatory correction... (This is elsewhere considered for electrostatic forces too.)

SPEED OF GRAVITY, as if instantaneously fast: Gravity getting ahead of itself:

A massive object in motion exhibits a "headlight-effect" focusing its gravity forward (similarly to the same-named effect in space-time relativity theory);- but aether-ripple gravity is a confined field rather than exo-or endo-radiant (flowing; outward or inward): its velocity produces longitudinal bow-compression and wake-suppression (1/1-v¢ Doppler-like shift;- not latitudinal compression which spreads), thus appearing to be fore-centered, ahead of its central mass-definition, and cancelling the first-order aberration effect usual in radiant (stellar) objects: Thus the speed of gravity is a coincidence-appearance faster than light-speed, a paradox in the sense of "scissors effect" where the point of closure traverses the length at any speed (*). Gravity is the field, photons are vectors with momentum -of the same light-relative aberration:- thus cancelling with gravity an apparent pseudo-infinite speed. This is -classical- relativity, and the similar results among relativities is akin to Galileo's remark that any finite system has a dual definition.

* (And even as the "scissors effect" can run simultaneous or backward, gravity might be projectable; and on large scale, by galaxies.)

(Potentially also, the larger planets biting gravity may contribute to the deceleration anomaly discovered since 1980 in distant spacecraft paths,--or, which may be a directional anomaly, radial versus angular, as it was noticed in highly elliptical and ejectory orbits but not in more-circular planetary orbits; N.B. NASA says it's wind.)

But we understand: Objects do not act on each other gravitationally, but rather their gravitation fields interact on each other;- Gravitational proximity also involves the overhang retro-tractivity effect similar to the case for electron fields.

* (Similarly a spinning ring mass-hole might exhibit anti-gravity, and similarly self-centering around its host planet or star.)

COSMIC-WIDE NEGATIVE-GRAVITY, redshift:

Observed gravity also exhibits cosmos-distance-scale aging as the aether has cooled: Farther away was warmer long ago, -relative to now-here,- with slower wave-transport then-there effectually similar to gravity being equivalent to slower time: redshifting cosmic distances without inflation. [We're likely fairly near center, as are we on most mensurate scales] Reports from decades ago indicated redshifts striated at repetitions of 72 km/s - expectably concentric within the cosmos, possibly due undiscovered phase-states of early cosmic-aether development. (See above also: the "unconstant" increasing "noodle"-count over time.)THE BALANCE OF GRAVITY:

An additional theory-factor must be included for measurements: Consider the velocity of
an atomic orbital electron is order-of 10^{8} m/s, and Earth-escape velocity is
order-of 10^{4} m/s, and the deuteronic-average atomic mass (approximation for
mass-to-valence ratio up to about carbon) is order-of 10^{3.6} electron mass:
By calculus on a ring around the Earth, or surely around the moon, the electron fly-away
should overwhelm atomic gravity-weight, were the so-called constant of gravity not
empirical.... We can carry this one step further by noting that basically everything of
subatomic [stable] wavicles consists of 'photon's self-entwined, moving at the
[wavicle-local] speed of light - the actual constant of gravity is an order of magnitude
larger, and the empirical constant may be adjustable by constraining the motions of
electrons [to vertical axis dominant] ... and we may sooner achieve
geometric antigravity (*).

* (from a correction to Special Relativity--'the other end is running late'--preventing forward-observers
from 'seeing the future of the other frame' T_{grid} due to T_{bias}.)

Sit in a house, on Earth, feeling 1.0 g acceleration, the roof remains above, year after year; But sit in a rocket and feel that 1.0 g acceleration, the roof being a distance above you also feels that acceleration but slightly late from your perspective and so gradually moves-down year after year... Gravity is not so Equivalent or Principled...

NOTA BENE: In classical gravitation theory, a mass-hole singularity collapses to infinitesmal, and there shreds into a self-contained wave-flux event (a cosmic "big-bang") which having no width cannot differentially slow like a 'wide' photon passing through gravity: it may lose its gravity,- or that is to say, its gravity cannot escape: it may swallow its own gravity. A similar result follows the aether-ripple theory in that the ripple-field folds-down upon itself, and loses its entropic churning components of gravity: Its zero-frequency aether-rippled gravity, remnant subsequent to superfluid coherency, may exhibit less symmetry-regular distribution (in some cases permanently lacking at the poles) and may reduce its apparent mass to a tiny but nonzero fraction: as space itself tries to fill the black-hole.

A last theory note: Other theories of gravity also suggest it travels much faster than light, even 20x faster: read from solar eclipse timings between gravity alignment and visible alignment, in addition to a standard presumption-argument that delayed gravity would perpetually co-accelerate corotating stars, even planets, toward their mutually old visual positions (enough to be have discovered).Also note: Hyper-fast wave-signals are also derivable by semi-alignment in any, aether: Indeed the strength of gravity at 10

^{-43}electro-charge-force is small enough to suggest it is a neutral leak-forward race-ahead of charge-force fields in aggregate, that is therefor nonreflective,... and may lead to an estimate of aether quantum size; and could then also be the motion-redshifter slowly leaking-away charge-energy, and even slowly reducing the size of objects in the cosmos (another view of redshift).

POWER-LIMITED GRAVITY and GRAVITO-MAGNETIC INDUCTION: [2014/11/05]

A 'scientific phenomenon' is found in answering the question of how much mass-energy is in the potential-energy field around a gravitational mass:

Briefly, the question arises in ascertaining that
'potential energy' is, mass-energy: it is where an in-falling 'test particle' gets its kinetic energy by
immediate-direct 'commandeering' and conversion of potential energy, ΔE = ∫GMm/R² dr,
and mostly not by conversion of its own, mass, to energy, (albeit even a 'test photon' loses some
to accelerating a stellar mass likewise), based partly on the notion that its source depends on the
'attraction', and proportionally thereto; The point is then made 'how much', and while the potential
force may be large and constant holding satellites in orbits, as soon as something starts to 'fall' in
and convert the local-field mass-energy potential to kinetic, the near-field becomes diminished, its
force reduced, and near-field objects follow partially-released paths 'backward', appearing as 'like
magnetic induction' to the simple-observer, (we are more familiar with electromagnetic, induction,
because its effect is 10^{42.6} more-noticeable), and, deep inside collapsing
supernova cores, around neutron stars and black holes, the effect should be calculably significant as forces
and energies slowing the in-fall, amplifying convections, sucking the mass-energy out of the center...

Simply put, potential-energy is power-limited and the draw on mass-energy by a librating particle is omnidirectional inverse-square-law 1/r² at the speed of light c, (which in columns ascending or descending compensates linearly and manifests more horizontally as inverse-law 1/r), and, this scientific-phenomenon is commonly labeled as 'another fundamental force...'

(Aside-note that as an infalling test-particle gains kinetic energy its mass-energy increases and its gain compounds, However, the kinetic component of its mass-energy is a deBroglie-wave, comparable to a photon, and may therefor be independent of 'particulate' rest mass or only 'semi-entangled' dependent on total mass-energy; And may lead to orbit precession.)

In this way there is no point mass (cf nor point charge: see electron, harmonic magnetism), because it is distributed as mass-energy in its potential field: the equational balance is in the 'static' force vs. 'extractable' energy of a potential-energy field and the local density of objects: A particular-given potential-field source can only provide up to its own total mass-energy, and that's distributed 4π around and by inverse distance uniformly, to the first-order Galilean-Newtonian approximation, (rotation may affect this by space-frame-drag), and accessible at the speed of light (or gravity), and objects moving inward or outward, even in orbit, upset their own near-field...it's an equation to be resolved...

(We might use Einstein's General Relativity to estimate mass-energy-per-potential-energy, even as his Special Relativity was used to estimate mass-energy E = mc².)

But note that potential-energy equations also always lead to 'external infinities' and interpretations of cosmic
scale interactions perceived locally as mass and gravity with negligible tails that are now not-so negligible:
Compare, of two such infinities, the potential-energy pick-up at a given radius-distance, under
inverse-square-force but over square-affected-area by particle-length travel, the total available energy,
were it entirely *in situ* 'test-point'-local-patch-integrable exactly-where-each-test-particle is,
unaffected by adjacency to other test particles, would've amounted by Newton, Einstein,
to a fixed-amount-per-radius independent of the point-particulate reduction by distance, i.e.
ΔE = ∫GMm_{test}/R²*4πR²/particle-size dr →∞
(over R →∞ distance)... which would not be the actuality but a necessary rethink (a 'gegendanken'
experiment) and recognition of its power-limitation: loosely analogous to a 'solid under pressure' maintains
until something moves and then equates to very-little-energy for plenty-loss of local-regional attraction...
And yet not far beyond say-an-enclosing-test-sphere the gravitational attraction must resume 'normal' as its
total enclosed energy includes the infalling mass kinetic energy power-limited-to-whatever, meaning
total-mass-energy is conserved and so the-farther-beyond remains undiminished...
(and we don't-need a 4th-graders' singularity to 'gedanken' that, infinity)...

And, the collapse of a neutron star to a black hole, at first crushes its center to primordial energy densities, but must ultimately 'demassify' and invert the center to explode as a 'deBroglie detonation' ('big bang') wherein the reduction in mass lengthens its deBroglie wavelength to exceed its retaining radius...meanwhile its outer regions increasingly convert in-falling mass-energy to potential energy field residue--though a remainder likely does fall all-the-way-in for a future supercosmic-scale 'inside-armageddon' (and survivors will have to hide)...

GRAVITY POSSIBLY NOT, A FORCE:

The mentioned Einstein-bending of light slowing-massward in passing a star, would affect particles generally, being mass-energy convolved and so following the same bending but without escape from being convolved, (loosely analogous to an orbital-rotation-speed gyroscope not-escaping Earth, because it's self-convolved); a freefalling-particle, wavicle, or photon, will gain kinetic-energy ultimately equal to its own mass-energy upon reaching a Schwarzschild radius...but this may misinterpret the effect: the action may simply be conversion of self-convolved-eddy-motion-inertial-drifting, to tidal-drawing-co-involved-linear-motion, with particles becoming more-photon-like 'deBroglie' waves and entering the event-horizon, blue-shifted, but for no further gravitational energy pickup...

[under further construction for equations]

THINKABOUTS: (in-the-works)

Consider a heavy-particle (cf Z that spontaneously decays dominantly into a relativistic electron-positron pair escaping immediate recapture though sometimes the high-mass-energy 91.2 GeV nucleates y-rays too) and its 'frames' of motion, initially, and consequently,-upon-decay-to-subparticles: In its original heavy-particle-frame the subparticles (let these be equal-mass and charge-neutral for this 'gedanken') exit oppositely and perpendicularly-to such-that a test-particle 'feels' only their barycenter (center-of-mass)... But, in the frame of either exiting subparticle, the barycenter is straddling the middle between the low-energy-non-moving subparticle, local-frame, and the 'doubly-fast'-high-energy-moving subparticle other-local-frame carrying all the kinetic energy and shortened 'deBroglie wavelength' and therefor mass-energy...and thereby bending the common-center-angle at-least-paradoxically to solve how-local is kinetic-energy, how-soon does stellar-aberration set-in to 'sight' the escaping particle as closer... (A perhap simpler question might be to consider a moving-'Young's Double Slit Experiment' passing-through an electron standing in its own frame: How does the electron get its 'deBroglie-wave'-action-length... This is possibly simple-enough experimentally to fully-discover relativity...)

Where, is the electromagnetic term (Maxwellian equation) for the gravity of photons: A thermal mass of photons in a place is like a mass of self-convolved particles and should exhibit gravity; And too, linear (directional) photons bent through a mass-gravitational field impart momentum-away (as well as rotational gyre on the field), pulling-away like another-gravity... If we have a solar-mass of photons traversing the universe, it must have a solar-mass of gravity, Yet, as photons escape (diverge-from) that solar-mass, they lose energy, but to what--? The remaining photons have no rest-mass to increase potential, and space-itself is not-yet attributed with tenuous-mass-energy, (except ΛCDM)... So, photons need a gravity term, slow-down and blueshift as they approach each-other... maybe it has to do with overlapping sinewaves being taller ergo slower-via... (Note also that photons are, notably, electromagnetically 'couple'able...)

SIDEBAR: ORBIT vs. TIME:

It appears online that the formula for time vs. true anomaly (angle) and vice versus is not known since Kepler, But high school mathematicians have long known the trivial solution from the elliptical orbit elements: a,b,ν ⇒ compute f, Ac (central wedge) - At (triangle) = A (focal wedge), because the ellipse is a stretched circle; Conversely from the area but there's a reverse step (*) to get the true anomaly ν.

- t(ν):
- f² + b² = a²
- tan(ν) = y / (x-f)

cot(ν) = (x-f) / y - (x/a)² + (y/b)² = 1
- Ac = (a/b) arctan(y/(x b/a))

Ac = (a/b) arccot((x b/a)/y) - At = f y / 2
- A = Ac - At
- t = A / (π a b/P)

- ν(t):
- f² + b² = a²
- A = (π a b) (t/P)
- Ac = At + A (by successive-approximation * on x,y)
- At = f y / 2
- (x/a)² + (y/b)² = 1
- Ac = (a/b) arctan(y/(x b/a)) ,= (f y/2) + A

Ac = (a/b) arccot((x b/a)/y) ,= (f y/2) + A - ∂Ac/∂y = (a/b) / (1+(y/(x b/a))²) ∂(y/(x b/a))/∂y ,= f / 2

∂Ac/∂x = (a/b) / (-1-((x b/a)/y)²) ∂((x b/a)/y)/∂x ,= (f/2) ∂y/∂x - ν = arctan(y/(x-f))

ν = arccot((x-f)/y)

- ⇛ Ac = (a/b) arctan(sgn(x)(y/b)/√(1-(y/b)²)) ,= (f y/2) + A

⇛ Ac = (a/b) arccot(sgn(y)(x/a)/√(1-(x/a)²)) ,= sgn(y) b (f √(1-(x/a)²)/2) + A - ⇛ ∂Ac/∂y = (a/b²) / √(1-(y/b)²) ,= f / 2 [needs sgn(x,y)]

⇛ ∂Ac/∂x = (a/b²) / √(1-(x/a)²) ,= (f/2) (x/a) / √(1-(x/a)²) [needs sgn(x,y)] - ⇛ (y-y') ~ (A-A') / [((a/b²)/√(1-(y/b)²))-(√(a²-b²)/2)]
[needs sgn(x,y) and check arithmetic and convergence]

⇛ (x-x') ~ (A-A') / [((a/b²)-(√(a²-b²)/2)(x/a))/√(1-(x/a)²)] [needs sgn(x,y) and check arithmetic and convergence]

* (Note that successive-[digit]-approximation is trivial since elementary school long-division,
squarerooting, Newton-Raphson squareroots...)

* (Note that on computers and calculators the combined triangle-less-arccot could be a
high-precision-function-call comparable to arccot...)

* (Note that computation of the focal wedge area A can also be done by computing the areas
of the ellipse subtent ⩍a y and triangle △a f y , which a circle also has
but does not have the triangle behind, the focal wedge being the same as its central wedge,
and analysis might discover an equational improvement for the ellipse...)

But note also that the 'area' solution is not entire because a zero-angular-momentum-limit solution has, no, area, but does have distance... so, let's try--

A premise discovery under the title,

'Majestic Service in a Solar System'

Nuclear Emergency Management