If f(r) satisfies f"(r) - (N-1)r^-3f(r) + (N-1)r^-1f'(r) = lf(r) then f(|x|)x^~ satisfies Ñ_x² f(|x|)x^~ = l f(|x|)x^~ .

Newton Potential
    To understand relativistic gravitation, it is helpful to understand the model of which it is essentially a refinement, Newtonian gravitation and, in particular, gravitational potential theory.     In the Newtonian model, we can represent gravity by a 1-vector "acceleration" field G_N(p) [ where p is a 4D Euclidean spacetime coordinate ] representating the instantaneous acceleration exerted on a small test mass placed at event p. Time is global in Newtonian physics so for a given t, we can regard G as a function of 3D spacial position r. G_N is assumed to be irrotational (Ñ_rÙG_N(r)) = 0 and so expressable as the gradient of a scalar field -f(r) known as the gravitational 0-potential. The minus sign is incorporated so that the force ¦(r) = -Ñ_rf(r) tends to move matter from high to lower potential areas .
    We can add an arbitary point-independant scalar constant to such a field without effecting its gradient and we typically seek to choose this constant so that f(r) ® 0 as |r|®¥ (in any direction).
    A single point mass m at the origin generating an attarctive gravitational field G_N(r) = - gmr^~(r^-2) can be regarded as generating a 0-potential f(r) = -gm(r²)^-½ where point-independant scalar g is the universal gravitational constant. Let V be an arbitary volume (3-curve) having boundary surface (2-curve) dV. By Gauss's law, the integral of G_N(r) over dV is zero unless 0 Î V in which case it is -4pgm (regardless of the geommetry of V) . In general, if scalar m_r is the local "mass density" at r, assumed to be a continuous function of r, we have
    ò_dV G_N(r)¿dr² = -4pgò_V m_r dr³
    Now ò_dV G_N(r)¿dr² = ò_V (Ñ¿G_N(r))|dr³| by the divergence theorem so for arbitary small V we have
    ò_V (Ñ¿G_N(r)-4pm_r)|dr³| = 0 whence Ñ¿G_N(r) = -4pgm_r and so
    Ñ_r²f(r) = Ñ_r¿(Ñ_rf(r)) = -4pgm_r       which (for scalar f(r)) is known as Poisson's equation , or Laplace's equation if m_r=0 .

    If f(r) = f(|r|) = f(r) we have Ñ_rf(r) = f'(r)r^-1r = f'(r)r^~ and Ñ_r²f(r) = r^-1f'(r) + f"(r).

    This then is the Newtonian model. The 3D gravitational force field G(r) is characterised by it's scalar "integral" f(r) = -Ñ_r^-1 G(r) ; a 0-potential scalar field which satisfies Poissons equation everywhere and provides the acceleration on a test mass at r as G_N(r)=-Ñ_rf(r) . We can in principle compute f(r₀) as the infinite volume integral
    f(r₀) = gò_³ m_r(r-r₀)^-½ |dr³|
    The scalar field m_r fully determines G_N so the "flow" of matter is relevant in determining the instantanous gravitational potential only by its magnitude. It's direction effects only how G_N changes over time.

    Gravity is such that the force exerted on a test particle is proportional to the mass of the test particle so that the acceleration experieneced is independent of the mass, More generally we have a force field F_p = Ñ_pf_p .

Effective Potential
    Under a central force with classical 0-potential f(r)=f(r) , the 2-blade angular momentum L=mrÙr' and scalar energy E=½m(rdt)² + V(r) are both conserved and the radial energy equation ½m(rdt)² + V(r) - ½m^-1r^-2L² = E for r³0 implies that the shape of the 1D scalar effective potential f(r)   - ½m^-1r^-2L² = f(r) + ½m^-1r^-2L² determine the possible orbits for a given L=|L²|^½ and E combination. If the particle is released from rest L=0 , but otherwise the effective potential contains a term equivalent to that contributed by a repulsive inverse cube force m^-1L² r^-3

Orbital Stability Indicator

      Suppose a particle of inertia m is subject to a time t-independant central force ¦(r,t)r^~ =  ¦(r)r^~.
     If x=r-r₀ denotes the deviation of r from cirular orbit value r₀ then mx^.. » (r₀^-13¦(r₀)+¦'(r₀))x to first order in r^-1x where ^.. denotes second time derivative. For gravitationl forces ¦(r) is proportionate to the inertia m.

    mr^.. = ¦(r)+L²r^-3 where angular momentum L=mr²q^. is conserved.
    Let x=r-r₀. We have mx^.. = ¦(r₀+x) + L²(r₀+x)^-3 » ¦(r₀)+x¦'(r₀)+½x²¦"(r₀) + 3!^-1¦"'(r₀)x³ + ... + r₀^-3L²(1-3r₀^-1x + 6r₀^-2x² - 10r₀^-3x³ +...
    For circular orbit at r₀ we require L² = -mr₀³¦(r₀) , w = (-r₀^-1m^-1¦(r₀))^½ so
    mx^.. » ¦(r₀)+x¦'(r₀)+½x²¦"(r₀) + 3!^-1¦"'(r₀)x³ + ... - ¦(r₀)(1-3r₀^-1x + 6r₀^-2x² - 10r₀^-3x³ +...)   
    = x¦'(r₀)+½x²¦"(r₀) + 3!^-1¦"'(r₀)x³ + ... + ¦(r₀)(3r₀^-1x - 6r₀^-2x² + 10r₀^-3x³ +...)
    = x(¦'(r₀) + 3r₀^-1¦(r₀)) + x²( ½¦"(r₀) - 6r₀^-2) + x³(3!^-1¦"'(r₀) + 10r₀^-3) + _O(x⁴)

    Thus if orbital stability indicator O_SI(r₀) º -3r₀^-1¦(r₀) - ¦'(r₀)   =   3r₀^-1f'(r₀) + f"(r₀)  is positive we have oscillations about the circular orbit of frequency m^-½ O_SI(r₀)^½ and so period 2pm^½ O_SI(r₀)^-½ , a proportion (-3-r₀¦'(r₀)¦(r₀)^-1)^-½ of the orbit period 2p(-r₀^-1m^-1¦(r₀))^-½, while if O_SI(r₀) is negative we have hyperbolic deviation.
    If ¦'(r₀) + 3r₀^-1¦(r₀) vanishes, we have harmonic deviation equation mx" » x²( ½¦"(r₀) - 6r₀^-2) = -P_SI(r₀)x² where
      P_SI(r₀) º r₀^-2(6+½r₀²f"'(r₀)) .
    This is analytically problematic since while x" = ax² has a ready particular solution x(t)=6a^-1(t+6a^-1x(0)^-1)^-2 this is not general enough to match a given xdt(0) .

    The normalised orbilital stability indicator of a circular orbit of radius r₀ under a central force ¦(r) £ 0  is the value
    O_SI(r₀) º (3+r₀¦(r₀)^-1¦'(r₀))^½ , where ^½ denotes that for negative values we take the negative square root of the absolute value. In eliminating the magnitude of ¦ by multiplying O_SI(r) by |r^-1¦(r₀)|^½, we aquire an infinite O_SI(r) when ¦(r)=0. The sign of ¦(r) has also been eliminated so O_SI(r) of a repuslive central force is the O_SI of the attactive force -¦(r). For nonzero ¦(r), O_SI(r) is the number of orbits about 0 per peturbation oscillation.
    O_SI(r) provides a measure of deviation from shell condition p(t)² = p(0)² under acceleration p"(t) = -f'(|p(t)|)p(t)^~ where |p(t)| º |p(t)²|^½, assuming p(0)¿p'(0) is small. Large O_SI suggests rapid oscillations across the shell with integer values suggesting petal forms. O_SI<1 suggesting slowly corrected slightly noncircular orbits.

    "Power law" attractive force f(r)=-ar^k has O_SI(r) = (3+k)^½ so stable circular orbits require k>-3 . Inverse square forces have unit O_SI(r), so the period of the deviation coincides with that of the orbit, stretching it into an ellipse for example. An integer O_SI(r) indicate a single repeated orbit while rational O_SI(r) pq^-1 implies that after q orbits each of which contain p deviation cycles the trajectory will repeat itself. An r^-1 orbit takes ½^½ orbits to complete an oscillation and, as this is irrational, the trajectory in theory never repeats. However, as the oscillatory nature of deviations is only correct to first order in small deviations, such multi-orbit analysis is profoundly suspect.
    High O_SI(r) indicate rapid oscillations about the circular orbit while high negative O_SI indicate rapid hyperbolic deviations, with the caviat that ¦(r) changing sign gives infinite O_SI when ¦(r)=0 corresponding to deviations from the straight line zero forces trajectory. Zero O_SI indicates paraboloid deviations from u with q correspending to paraboloid deviation from u₀ with q while O_SI j as for ¦(r)=-ar^j2-3 corresponds to oscillating j times faster than the orbit.

    To maximise O_SI(r) º r^-13f'(r) + f"(r) we consider O_SI'(r) =  -3r^-2f'(r) + 3r^-1f"(r) + f"'(r) .

Circular Orbits

    -f'(r)r^~ = -mw²r where m represents a positive inertial mass yields w = ±|m^-1r^-1f'(r)|^½ Þ wr = ±m^-½ |rf'(r)|^½ , and so circular orbits under central force ¦(r)r^~ = -f'(r)r^~ have

• period 2pw^-1 = 2p|m^-1r^-1f'(r)|^-½ = 2pm^½|rf'(r)^-1|^½   ;
• speed rw = ±|m^-1rf'(r)|^½     and so sublight orbits require |rf'(r)| < m ;
• classical kinetic energy ½mw²r² = ½rf'(r) ; and
• angular momentum mr²w = m^½r^3/2f'(r)^½ = |mr³f'(r)|^½ ;
• radial acceleration -rw² = m^-1f'(r)

    The potential f(r)=-lr^b with l>0 has f'(r)=br^-1f(r) and hence squared angular momentum mlbr^b+2 and kinetic energy |bf(r)|. For b<0 the toal orbit energy is thus (b+1)f(r) and so in a Coloumb potential with b=-1 all circular orbits have total energy zero, while if b=-2 all circular orbits have angular momentum (2ml)_sqdrtcnj and positve total energy -f(r).  

    The Coulomb Potential f(r)=-lr^-1 with f'(r)=lr^-2 thus has w = ±|m^-1lr^-3|^½ ; period 2p|m^½l^-½ r^3/2 ; speed  ±|m^-1lr^-1|^½ ; angular momentum ± m^½r^½ ; classical kinetic energy ½lr^-1 ; and radial acceleration -m^-1lr^-2.
    O_SI(r) = r^-13f'(r) + f"(r) = -lr^-3 while O_SI(r) = 1.

    More generally a small test particle launched from r₀e₁ with velocity r₀w₀e₂ + s₀e₁ will seek a circular orbit having mr₀²w₀ = m^-½r^3/2f'(r)^½ and f(r₀) + ½m(r₀²w₀²+s₀²) = f(r) + ½rf'(r) .

 Quantised Orbits
    There are two natural ways to particularise orbits under a central oscillatory potential f(r). The maxima of O_SI(r) provide the stablest orbits, while orbits at r satisfying |rf'(r)| = mv² provide the orbits having a given speed v. In particular, v=1 yeilds the lightspeed orbits with inertial mass m regarded as a coulping constant rather than a mass-energy. with a particular speed v orbits which we might   

Ring Potential
    The ring potential foRz(r) º 2R ò0p dq f((z2+r2+R2-2rR cosq)½) is the potential at r of a circle of radius R lieing in a plane containing 0 and r.
    Setting s º  (z2+r2+R2-2rR cos(q)) we have ds = 2rR sinq dq.
    cos(q) = (2rR)-1(s-z2-r2-R2) so for q Î [0,p] we have
    sin(q) = (2rR)-1 (4r2R2 - (s - z2-r2-R2)2)½ = (2rR)-1 (2s(z2+r2+R2) -s2 + 4r2R2 - (z2+r2+R2)2)½    = (2rR)-1 Q(s)½ where
     Q(s) º -(s2 - 2s(z2+r2+R2)   + (z2+ r2+R2)2   - 4r2R2 )
    = -s2 + 2s(z2+r2+R2) - z4 - 2z2(r2+R2) - (r2-R2)2
    =?= -(s-z2-(r-R)2)(s-z2-(r+R)2) is a quadratic in s with roots z2 + (r±R)2.
    Thus foR(r) = 2r-1 òz2+(r-R)2z2+(r+R)2 ds f(s½) ( sinq)-1 = 4R ò(r-R)2 (r+R)2 ds Q(s)-½ f(s½) .

    For z=0 , Q(s) = -(s² - 2s| (r²+R²) + (r²-R²)² ) ,

Stream Function

    Setting scalar stream function y_p º ò_C dV² V¹ - ò_C dV¹ V² for any path C from 0 to p yields V¹ = Ð_e2y_p ; V² = - Ð_e1y_p , ie. V_p = Ñ_p×(y_pe₃) = (Ñ_PÙ(y_pe₃))e₁₂₃^-1 .
    If y_p is any ananalytic scalar field over ² satisfying this then its analyticity provides incompressability condition Ð_e1V¹ + Ð_e2V² = 0, ie. Ñ_PV_p=0 and also Ñ_p²y_p = 0.
    Hence V_p = (Ñ_P(y_pe₃))e₁₂₃^-1 = -Ñ_Py_pe₁₂ .
    Also note that Ð_Vp y_p = (V_p¿Ñ_p)y_p = 0 so y_p is constant when following the flow.

Velocity Potential

    For a steady irrotational incompressible flow with Ñ_pÙV_p = 0 we can constuct a 0-potential f_p = ò_C d¹p¿V_p for any path C from 0 to p so that V_p=Ñ_pf_p , ie. Vⁱ = Ð_eif_p . If the flow is conserved we also have Ñ_p¿V_p=0 so Ñ_pV_p = Ñ_p²f_p = 0.

Complex Potential

    The Complex 0-potential f_p º f_p + iy_p satisfies Ñ_P²f_p = 0 and it is natural to here associate i with e₁₂₃. It is defined for P=x Î Â^N₀=² but by defining z=x¹+ix² where p=x¹e₁+x²e₂ we can regard f_p = f_z,t) as a t-dependant analytic function mapping C®C with
    df_z,t)/dz = df/dx¹ + idy/dx¹ = V¹ - iV²   and |d_cvphiz/dz|₊ = |V_p| .

    The complex potential thus fully embodies an incompressible irrotational 2D flow.

Morse Potential
    Typically used for r representing the seperation between two covalently bonded atoms, f(r) = f_¥(1-(-(½kf_¥^-1)^½(r-r₀))^↑)²     for dissassociation energy f_¥>0, equilibrium or zero force distance r₀, and bond force constant k>0.   

Yellow shows potential f(r); green is force f'(r); while red is the OSI (3+f'(r)-1rf"(r))½ with imiginary values focred negative real.     For rÎ[0,r0] we have f(r)£f0=f(0)= f¥(1-((½kf¥-1)½r0)↑)2 with f(r)³f¥ for r£ r0 - (2f¥k-1)½(2)↓.     f(r) ® f¥ from below as r®¥ and to ¥ as r®-¥ though we might typically impose r³0. We have a global minimum at r=r0 with f(r0)=0 , f'(r0)=0, and f"(r0)=k.

[ Proof :  Set m(r) = (-(½kf_¥^-1)^½(r-r₀))^↑ = 1-f_¥^-½f(r)^½ so that m(r₀)=1 and f(r)=f_¥(1-m(r))² with m^(k)(r) = (-1)^k(½kf_¥^-1)^½km(r)
    f'(r) = -2f_¥(1-m(r))m'(r) = (2k)^½f(r)^½ m(r) Þ f"(r) = 2f_¥(m'(r)² -(1-m(r))m"(r))   = km(r)(2m(r)-1)  .] Ѳf(r)=r^-1f'(r)+f"(r) = r^-1(2k)^½f(r)^½ m(r) + km(r)(2m(r)-1) = m(r)(r^-1(2k)^½f(r)^½ + k(2m(r)-1))

    The comnplex Morse potential f(r) = b²(-2(r-r₀))^↑ - b(1+2D)(-(r-r₀))^↑ = (b(-(r-r₀))^↑ - (½+D))² - (½+D)² gives real energy eigenvalues E_n=-(n-D)² for integer n<D. [Saaidi]

Tangential occlusion potential
    Constructing an inverse square particle-mediated field is unrealistic because the r^-1 factor representing an inverse square "dispersion" of a constant signal fails for low r . If the absorbing particle is not attracting the carriers but merely absorbing those that reach a small perimeter distance h , then we would expect the proportion of carriers absorbed to be in proportion to the area content of the spherical cap defined by the cone tangential to the h radius sphere centered on the absorber. A sphere of radius h at distance r from 0 subtends a cone of demiangle q= sin^-1(u) where uºhr^-1 , which occludes an area 2p(1- cos(q))d² = 2p(1-(1-u²)^½)d² of the 4pd² surface area of a sphere of radius d at 0, rather than an area proportionate to r^-2.

Attractive central force     ¦(r) = (1-(hr-1)2)½ - 1 = (1-u2)½ - 1 for r³h (ie. u £1) has     ¦'(r) = h2r-3(1-u2)-½ = h-1u3(1-u2)-½ = h2r-3(1+¦(r))-1 .     For r>>h we have ¦(r) » -½(hr-1)2 - 1/8(hr-1)4   - K3(hr-1)6 - K4(hr-1)8 - ...   where Kk º 1*3*5*7*...*(_2k-3) / (2*4*6*8*...*(2k)) = 22-2k(2k-3)! / ((k-2)! k! ) .      For large r (so small u=hr-1), the inverse square "Coulomb" force will be massively dominant.     We find that circular orbits are stable only for   u2< 3/4 (ie. r>(3-½)2h) and that the 3 + r¦'(r)¦(r)-1 indicater tends to 1 from below as r®¥ .

    For a screened force F(r) = (-lr)^↑ ¦(r) we have Z(F)(r) = F(r) + 3^-1rF'(r)   =   (-lr)^↑ ( Z(¦)(r) - 3^-1lr¦(r) ) . Now -3^-1lr¦(r) is positive so exponentially damping ¦ makes circular orbits unstable whenever
     -1 + (1-u²)^-½(1-(2/3)u²) - 3^-1lr((1-u²)^½-1) ³ 0 Û -1 + (1-u²)^-½(1-(2/3)u²) + 3^-1lr ³ 3^-1lr(1-u²)^½ Û -1 + 3^-1lr ³ (3^-1lr - 1 + (2/3)u²)(1-u²)^½

    Comparing this with indicater (1-3^-1)r^k for ¦(r)=r^k we see that for large r, circular orbits are only just stable; verging on the the instability associated with inverse cube orbits.
    The apsidal angle p(3+r¦'(r)¦(r)^-1)^-½   =   p(3 + d²r^-2(1-u²)^-½ (-1 + (1-u²)^½)^-1 )^-½   =   p(3 + u²( (1-u²)^½ (-1 + (1-u²)^½) )^-1 ) ^-½   =   p(3 + u²( 1-u²-(1-u²)^½ )^-1 ) ^-½
    When u²=½ this is p(3 + ½( ½ - (½)^½ )^-1 ) ^-½ =p(3 + (1-2^½)^-1 )^-½ » 1.30p .
    It is a rational multiple of p only when 3 + u²( 1-u²-(1-u²)^½ )^-1 is a squared rational number, which occurs when  1-u²-(1-u²)^½ is a squared rational, .

      We will here refer to V(r) = ò_r^¥ dr ¦(r)   =      r¦(r) + h sin^-1(hr^-1)   =      r((1-(hr^-1)²)^½-1) + h sin^-1(hr^-1)   =      h(u^-1((1-u²)^½-1) + sin^-1(u)) as the tangential occlusion potential.
[ Proof :   ò_r^¥ dr ¦(r)   =   ò_u⁰ (-r²h^-1 du) ¦(r)   =   h ò_u⁰ du -u^-2¦(r)   =      h ò₀^u du ((1-u²)^½ - 1)u^-2 =   h [ -(1-u²)^½u^-1 - sin^-1(u) + u^-1 ]_0,u   =     h(u^-1(1-(1-u²)^½) - sin^-1(u))   =      -r¦(r) - h sin^-1(hr^-1)  .]

    (¶/¶r)²V + 2r^-1¶V/¶r   =   -r^-1(1-u²)^-½¦(r)²
[ Proof :  ¶V/¶r = -¦(r) and (¶/¶r)²V + 2r^-1¶V/¶r   =     =   -¦'(r) + 2r^-1¦(r)   =   -h²r^-3(1-u²)^-½ + 2r^-1((1-u²)^½-1)   =   r^-1(1-u²)^-½(-u² + 2(1-u²) - 2(1-u²)^½)   =   r^-1(1-u²)^-½(2-u² - 2(1-u²)^½)   =   r^-1(1-u²)^-½(1 - (1-u²)^½)²   =   -r^-1(1-u²)^-½¦(r)²  .]

    Setting V(r) = (1-u²)^a gives ¶V/¶r = a(1-u²)^(a-1)2h²r^-3 so r² ¶V/¶r = 2h²a(1-u²)^(a-1)r^-1 whence
    Ѳ V(r) = r^-2 (¶/¶r)(r² ¶V/¶r)   =   2r^-2h²a ( (a-1)(1-u²)^(a-2)2h²r^-4 - (1-u²)^(a-1)r^-2)
      =    2r^-4h²a (1-u²)^(a-2)( (a-1)2u² - (1-u²) )   =    2r^-4h²a (1-u²)^(a-2)( (2a-1)u²  - 1 )

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Copyright (c) Ian C G Bell 1999
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 01 Apr 2007.