Zonal Toroidal Harmonic Expansion

of External Gravitational Fields for Ring-like Objects

aka

Gravitational Field of BaumkuchenToshio FUKUSHIMA (NAOJ)

Submitted to AJ in Revised Form

ResearchGate Fukushima Click

Baumkuchen is Not

Only For Party

External Grav. Field of Ring-like Object

! Integral Expression! Pro: General Purpose! Con: CPU Time

! Harmonic Expansion! Pro: Fast Computation ! Con: Lack of Proper Formulation! WARNING: Curse of Brillouin

( )''

'V

dρ

Φ =−∫x

x

x x

n

n

Φ = Φ∑

Curse of Brillouin Manifold

! Brillouin (1933)! Brillouin Scattering, B of WKB

! Brillouin Manifold= Minimum Circumscribing

Manifold! Convergence Limitof Harmonic Expansion

! Sphere/Spheroids/Toroid

Toroidal Coordinates ( ), ,u θ φ

2 1

cos

sin

cos

a ur

u

az

u

θθ

θ

−=

−

=−

cos

sin

x r

y r

φ

φ

=

=

Toroidal Coordinate Surfaces

! Morse & Feshbach (1953)

! Blue: Constant u! Red: Constant θ! Constant Ring

Radius meansDifferent Center of Toroids

New Theory of Toroidal Harmonic Expansion

! Solution of Laplace’s Eq. (Hobson 1931)

! NEW Finding: Term 0 = Ring Potential at r=a! Key: Fast Computation of Radial Functions

cosV u θΦ = −

( )( )( )

( )( )

1/ 2

1/ 2

1/ 2

n

n n

n

n R R n R

P u F uup u

P u u F u

+

−

−

≡ =

( )0

cosn n

n

GMV C p u n

aθ

∞

=

= ∑

Ring Potential! Classic Result: Ring Radius = a

! K(m): Complete Elliptic Integral of 1st Kind! Acceleration Vector & Fast Procedure

! Fukushima (2010, Cele. Mech. Dyn. Astron.)! Fukushima (2015a, J. Comp. Appl. Math.)

( )2K mGM

p π

Φ =

2 2r x y≡ +2

4arm

p≡

( )2 2p r a z≡ + +

It’s Show Time

Case of Thick Disk! r0=0.1! r1=0.5! r2=2.0! H0=0.003! γ=9/8! Surface:10-8

Density: Thick Disk! Axisymmetric Volume Density Profile

! Power-Law & Scale Height Functions

( ) ( )( ) ( )

2

0

1 1, exp

22D

DD

zr z P r

H rH rρ ρ

π

= −

( ) 0

0

D

rH r H

r

γ

=

1 2r r r≤ ≤

( ) 0 01r r

P rr r

= −

Effective Cross Section of Thick Disk

Potential: Thick Disk

Spectrum: Thick Disk

Error Curve: Thick Disk

CPU Time Comparison

Conclusion! New Theory of Expansion! Recursive Computation of Radial Func.! Seeds by Complete Elliptic Integrals! 3000-10000 Faster Evaluation! Convergence outside Brillouin Toroid! Divergence inside Brillouin Toroid! Accretion Disk => Piecewise Disk Model

Piecewise Disk Model

References! Bannikova et al., 2012, MNRAS, 424, 820! Fukushima, 2010, Cele. Mech. Dyn. Astron., 108, 339! Fukushima, 2011, Math. Comp., 80, 1725! Fukushima, 2015a, J. Comp. Appl. Math., 63, 17! Fukushima, 2015b, Astron. J., revised! Hobson, 1931, The Theory of Spherical and

Spheroidal Harmonics, Cambridge Univ. Press! Morse and Feshbach, 1953, Methods of Theoretical

Physics, McGraw-Hill

Best isYet to Come