g s - アルマ望遠鏡 · Core Galaxy NGC 4260 Hubble Space Telescope / Seconds of aas Oust 117...
Transcript of g s - アルマ望遠鏡 · Core Galaxy NGC 4260 Hubble Space Telescope / Seconds of aas Oust 117...
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Zonal Toroidal Harmonic Expansion
of External Gravitational Fields for Ring-like Objects
aka
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Gravitational Field of BaumkuchenToshio FUKUSHIMA (NAOJ)
Submitted to AJ in Revised Form
ResearchGate Fukushima Click
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Baumkuchen is Not
Only For Party
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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∑
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Curse of Brillouin Manifold
! Brillouin (1933)! Brillouin Scattering, B of WKB
! Brillouin Manifold= Minimum Circumscribing
Manifold! Convergence Limitof Harmonic Expansion
! Sphere/Spheroids/Toroid
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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
=
=
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Toroidal Coordinate Surfaces
! Morse & Feshbach (1953)
! Blue: Constant u! Red: Constant θ! Constant Ring
Radius meansDifferent Center of Toroids
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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
∞
=
= ∑
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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≡ + +
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It’s Show Time
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Case of Thick Disk! r0=0.1! r1=0.5! r2=2.0! H0=0.003! γ=9/8! Surface:10-8
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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
= −
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Effective Cross Section of Thick Disk
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Potential: Thick Disk
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Spectrum: Thick Disk
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Error Curve: Thick Disk
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CPU Time Comparison
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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
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Piecewise Disk Model
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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
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Best isYet to Come
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