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