1. No category

中性子星の固有振動?
accreting millisecond X-ray pulsar XTE J1751-305
(Strohmayer & Mahmoodifar 2014)
an example of millisecond X-ray pulsars
SAXJ1808.4-3658
ミリ秒X線パルサーの発見
(Wijnands & van der Klis 1998)
中性子星の自転周期より
短い周期のバースト振動
(Chakrabarty et al 2003)
中性子星の固有振動：
複雑な構造・状況を反映して様々な振動モードが存在
1. 表面流体層（fluid ocean）、固体殻（solid crust）、流体核（fluid core）
(a) 音波モード（p-mode）や内部重力波モード（g-mode）
(b) 固体殻に特有な音波モード（toroidal and spheroidal crust modes）
2. 超流動（固体殻や流体核）
：超流動特有の振動モード（２流体モデル）
3. 自転：振動スペクトルへの影響
(a) ロスビー波（r-mode）や慣性波（inertial mode）
4. 磁場：振動数スペクトルや脈動安定性への影響
(a) Alfv´en 波
5. 相対論的効果：重力波による振動の安定化や不安定化
Figure 6: Cross section of a neutron star (Shapiro & Teukolsky 1983)
摂動（振動）の表現
1. 自転はないし、磁場もない：変数分離可能
p′ (r, θ, φ, t) = p′ (r)Ylm (θ, φ)eiωt ,
etc
2. 自転があったり、磁場があったり：変数分離不可能
!
′
p (r, θ, φ, t) =
p′l (r)Ylm (θ, φ)eiωt , etc
l≥|m|
3. 変位ベクトル ξ⃗ の spherodial 成分と toroidal 成分
#
!"
m
m
ξ⃗spherodial (r, θ, φ, t) =
Sl (r) Yl ⃗er + Hl (r) ∇H Yl eiωt
l≥|m|
ξ⃗toroidal (r, θ, φ, t) =
!"
l′ ≥|m|
#
−Tl′ (r) ⃗er × ∇H Yl′ eiωt
m
ここで、∇H = ⃗eθ ∂/∂θ + ⃗eφ sin−1 θ∂/∂φ である。自転があったり、磁
場があったりすると ξ⃗spheroidal と ξ⃗toroidal が混ざる
4. モードの分類に使われる整数（量子数？）
$
l;
degree
m; azimuthal wavenumber, |m| ≤ l
n; radial order
振動数 σ はしたがって、
σlmn
ただし、自転や磁場が無ければ振動数 σ は m に依存しない
振動数の規格化
σ0 =
!
GM
,
3
R
f0 =
σ0
,
2π
P0 =
1
f0
1. 太陽の場合（M = M⊙ 、R = 7 × 1010 cm）
f0 ≃ 0.1 mHz,
P0 ≃ 2.78 h
2. 中性子星の場合（M = 1.4M⊙ 、R = 106 cm）
f0 ≃ 2170 Hz,
P0 ≃ 0.461 ms
例えば、最速パルサーの自転周期は
Pfastest
pulsar
=
2π
Ωfastest
pulsar
≃ 1.4ms ∼ 3 × σ0
Ωfastest pulsar
1
∼
σ0
3
色々な振動モード
1. p-modes と g-modes
σg ∼
< σ0 ∼
< σp
(a) p-modes:
σp ∼ n
!"
dr
c
ここで、c は音速。n は正の整数。
#−1
(b) g-modes:
1
σg ∼
n
"
dr
N
r
ここで、N は Brunt-V¨
ais¨al¨a 振動数。
2. 自転に起因する振動モード：r-modes や inertial modes
σrot ∝ Ω ∼
< σ0
ここで、N は Brunt-V¨
ais¨al¨a 振動数。
2. 自転に起因する振動モード：r-modes や inertial modes
σrot ∝ Ω ∼
< σ0
例えば、r-modes は toroidal mode で、Ω → 0 の極限で
σ → σr−mode
$
ξ⃗ → ξ⃗toroidal
%
2
(l′ + 2) (l′ − 1)
≡ ′ ′
− 1 mΩ = −
mΩ
′
′
l (l + 1)
l (l + 1)
ω = σ + mΩ → ωr−mode
2mΩ
≡ ′ ′
l (l + 1)
3. 中性子星の個体殻（solid crust）を伝播する音波
√
σcrust ∝ n µ0
ここで、µ0 は shear modulus。
4. Alfv´en 波
σB ∝ nB
中性子星固有振動の計算例
（自転も無ければ、磁場も無い場合）
toroidal crust mode
(McDermotto etal 1988)
中性子星の固有振動？
•
•
残念ながら、観測的にはあまり例がない（見つかっていない）
QPOs in Giant flares in Soft Gamma-ray Repeaters (SGR)
•
•
•
•
SGR1806-20
SGR 1900+14
burst oscillations: QPOs in X-ray bursts
a non-radial oscillation mode in an accreting millisecond pulsar
? (Strohmayer & Mahmoodifar 2014)
２００４年１２月２７日の巨大γ線フレアー
SGR1806-20
Israel et al 2005
~92.5HzのQPOの発見
Israel et al (2005)
スマトラ沖地震のような巨大地震か？ 地震学？
The first giant flare was recorded on 1979 March 5 from SGR 0526−66
(Mazets et al. 1979). The source is well localized in the LMC, and so the
isotropic energy of the flare was 5×1044 ergs − some ten thousand times
larger than a typical thermonuclear flash.
Giant flare of SGR1900+14
Figure 80: The giant flare from SGR 1900+14 as observed with the gammaray detector aboard Ulysses (20-150 keV). Note the strong 5.16s pulsations
clearly visible during the decay. (Hurley et al 1999; Woods & Thompson
2006)
The second giant flare was not recorded until almost 20 years later, on
found to exist in the SGR 1900+14.
Figure 82: Key properties of QPOs in giant flares. ∗ : the frequency is the
centroid frequency from a Lorentzian fit; the quoted width is the associated
full witdh at half maximum. All amplitudes are the values computed from
rotational phase dependent power spectra. The duration is given with respect to the main flare at time zero. The nomical energy band in which the
QPOs are seen is < 100keV except where noted. † : Appears to drift upwards
in frequency and amplitude over time. ‡ : Seen at early times in 100−200keV
data at high amplitude, then at later times with lower amplitude/coherence
and different rotational phase. (Watts 2011)
11.2.3
oscillations of magnetized neutron stars
QPOs and burst oscillations with frequencies ∼ 500Hz, which are discovered rather recently from LMXBs by RXTE (Rossi X-ray Timing Explorer)
satellite. kHz OPQs, the name of which is after the typical high frequencies
∼1 kHz, from LMXBs usually appear as a pair of QPO frequencies in the
power spectrum of X-ray flux, and the frequencies are variable, depending
on X-ray intensity, but interestingly enough the separation between the pair
of QPOs is rather constant. Burst oscillations from LMXBs, on the other
hand, are detected in type I X-ray bursts, and the QPO frequencies are
usually slightly increasing towards the tail of the bursts.
kHz QPOs and burst oscillations:
NSs in binary systems (LMXBs)
kHz QPO
burst oscillations
Figure 42: Average power spectra computed from 1996 February 15 at
11:50:22 to February 16 at 10:13:01 UTC showing the evolution of the two
kiloherts QPOs from 4U1728-34. The source count rate was increasing from
figure bottom to top. Also note the QPO between 20 and 40 Hz and the
broadband noise component between 0.1 and 10Hz that decreases in strength
as the source intensity increases. (Strohmayer et al 1996)
These high frequency QPOs are observed in LMXBs, particularly, low
Figure 43: Light curve of the burst that occured at 10:00:45 UTC on 1996
magnetic field neutron star systems, which are subdivided into Z sources,
4U1728-34
105
Feburary 16. The main panel shows the total PCA counts in 31.25 ms bins.
The inset panel shows a portion of the power spectrum computed from 32s
of 122µs data. Each bin is 0.25Hz wide and represents the average of eight
original power spectral bins. (Strohmayer et al 1996)
atoll sources, and, probably, weak LMXBs. Z sources are most luminous,
Burst Oscillations (1)
4U 1728-34
Figure 48: Light curve of the burst that occured at 10:00:45 UTC on 1996
Feburary 16. The main panel shows the total PCA counts in 31.25 ms bins.
The inset panel shows a portion of the power spectrum computed from 32s
of 122µs data. Each bin is 0.25Hz wide and represents the average of eight
original power spectral bins. (Strohmayer & Markwardt 1999)
localized X-ray hot spot which expands in ∼ 1s to engulf the neutron star.
Burst Oscillations (2)
SAXJ1808.4-3658
ミリ秒X線パルサーの発見
(Wijnands & van der Klis 1998)
中性子星の自転周期より
短い周期のバースト振動
(Chakrabarty et al 2003)
Models for Burst Oscillations
1.Hot Spot Model
hot spot + rapid rotation
large amplitudes and strong harmonics
small frequency drift
2. Wave Model
slow wave + rapid rotation
small amplitudes and weak harmonics
large frequency drift
A non-radial oscillation mode
in an accreting millisecond pulsar ?
(Strohmayer & Mahmoodifar 2014)
1. XTE J1751-305: accreting millisecond pulsar
νspin = 435Hz
2. detection of coherent oscillation of the frequency
3. a
Figure 28: Examples of a typical HMXB (top) and LMXB (bot
neutron star in the HMXB is fed by a strong high-velocity s
νoscillation = 0.5727597
× ν beginning
= 249.3326Hz
and/or by spin
atmospheric Roche-lobe overflow. The neut
an LMXB is surrounded by an accretion disk which is fed by Roch
(RLO). There is also observational evidence for HMXBs a
non-radial oscillation mode?flow
harbouring black holes.(Tauris & van den Heuvel 2006)
(a) rotationally modified g-modes
r-modes
in theclassified
surface
X-rayor
binaries
are generally
intolayers
two groups called
(b) r-modes in the fluid
X-ray binaries (HMXBs) and low mass X-ray binaries (LMXBs)
corepanion stars in high mass X-ray binaries are massive O, B type s
mass larger than ∼ 10M⊙ , while those in low mass X-ray binar
mass dwarfs having mass less than ∼ M⊙ . Mass accretion mo
72
(Strohmayer & Mahmoodifar 2014)
(Strohmayer & Mahmoodifar 2014)
モデル計算
• 前提
• 中性子星表面への質量降着でできるホットスポットが非動径振動による周期的な摂動を受ける
• ホットスポットは中性子星表面に固定され、無限遠から見ると星の自転にともなう運動をする
• ホットスポットが受ける摂動の振動を無限遠で観測すると、中性子星の共回転系でみた振動数が観
測されると考える（Numata & Lee）
• ここでは二つのモデルをためす：脈動不安定なモードの振動数が観測を説明できるか
• 中性子星表面の降着物質の薄い層を伝播するr-mode
• 中性子星の流体核を伝播する振動モード
モデル１：中性子星表面のr-mode
中性子性表面の質量降着層：定常な質量降着を仮定している
伴星からの降着物質はほとんどヘリウム
10
9
0.7
10
8
10
7
10
6
0.1
T
10
-2
10
0
10
2
10
4
y
10
6
10
8
10
10
モデル１：中性子星表面のr-mode
6
10
2
2
Log N
10
2
0
2
0
10
10
10 l
2
Log N
Log N , Log L
10 l
4
2
2
Log N , Log L
2
4
6
-2
Log L
10 l
-4
-6 -2
10
2
10
0
10
2
-2
Log L
10 l
-4
10
4
y
10
6
10
8
10
10
-6 -2
10
2
10
0
10
2
10
4
y
10
6
10
8
10
10
モデル１：中性子星表面のr-mode
核（ヘリウム）燃焼による不安定性: ε-mechanism
振動数
m=1 & 2
モデル１：中性子星表面のr-mode
m=2
2
U. Lee
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("#$%!
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,-./
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Figure 1. Frequency ratio κ ≡ ω/Ω and the growth timescale τ in second as functions of the spin frequency νspin = Ω/2π for the
l′ = |m| = 2 r-modes propagating in the surface fluid ocean, where steady burnings of hydrogen and helium are assumed to take place
˙ = 0.7M
˙ Edd . The solid, dashed, dotted, and dash-dotted lines are respectively for the cases of
in the ocean for the mass accretion rate M
the hydrogen abundance X = 0, 0.01, 0.02, and 0.03 at the top of the ocean where Z = 0.02.
!"#&
!"#$
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モデル１：中性子星表面のr-mode
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,-./
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((!
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Figure 1. Frequency ratio κ ≡ ω/Ω and the growth timescale τ in second as functions of the spin frequency νspin = Ω/2π for the
l′ = |m| = 2 r-modes propagating in the surface fluid ocean, where steady burnings of hydrogen and helium are assumed to take place
˙ = 0.7M
˙ Edd . The solid, dashed, dotted, and dash-dotted lines are respectively for the cases of
in the ocean for the mass accretion rate M
the hydrogen abundance X = 0, 0.01, 0.02, and 0.03 at the top of the ocean where Z = 0.02.
m=2
!"#&
*"#$%'
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)"!$%&
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"
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''!
'+!
,-561
˙ = 0.1M
˙ Edd .
Figure 2. Same as Figure 1 but for the mass accretion rate M
j = 1, · · · , jmax . Note that in this paper no general relativistic effects are considered for the shell calculation and
discussed by Lee (2004), the frequency ω of the r-mode propagating in the surface thin shell is saturated for rapid rota-
モデル２：中性子星液体核の振動モード
m=2
crust mode
r-mode
detected oscillation
Yoshida & Lee 2001
モデル２：中性子星液体核の振動モード
不安定性：excitation and stabilization
excitation by gravitational wave emission
stabilization by dissipative processes
stability of a mode
growth timescale
モデル２：中性子星液体核の振動モード
4
U. Lee
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!'()*+,
!
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Figure 3. Frequency ratio κ ≡ ω/Ω and the growth timescale τ in second as functions of the spin frequency νspin = Ω/2π for the r-mode
of l′ = m = 2 and inertial modes of m = 2 in the core and toroidal crust modes of m = 2 in the crust, where the red and black dots
in the left panel indicate unstable and stable modes, respectively, and only unstable modes are plotted for the growth timescale in the
right panel.
!"%!
!"$#
!"
%
Eigenfunction of the toroidal crust mode
with k=0.4999 at v=435Hz
40
amplitudes
30
20
10
0.0
-10
0
0.2
0.4
0.6
r/R
0.8
1
!"##
モデル２：中性子星液体核の振動モード
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Figure 3. Frequency ratio κ ≡ ω/Ω and the growth timescale τ in second as functions of the spin frequency νspin = Ω/2π for the r-mode
of l′ = m = 2 and inertial modes of m = 2 in the core and toroidal crust modes of m = 2 in the crust, where the red and black dots
in the left panel indicate unstable and stable modes, respectively, and only unstable modes are plotted for the growth timescale in the
right panel.
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Figure 4. Same as Figure 3 but for we use the artificially enhanced shear modulus µ = 5 × µ0 for the crust.
The mode along the red curve, after the crossing, running
almost parallel to the black line of κ ≃ 0.65 has the shortest
growth timescale τ and may be regarded as the r-mode of
l′ = m = 2.
Carroll B.W., Hansen C.J., 1982, ApJ, 263, 352
De Cat P. 2007, CoAst, 150, 2007
Dziembowski W.A., Moskalik P., Pamyatnykh A.A., 1993,
265, 588
Eigenfunction of the toroidal crust mode
with k=0.557 at v=435Hz
40
amplitudes
30
20
10
0.0
-10
0
0.2
0.4
0.6
r/R
0.8
1
まとめ
1. detection of a coherent oscillation frequency in an accreting X-ray millisecond pulsar XTE J1751-305
νoscillation = 0.5727597 × νspin = 249.3326Hz
νspin = 435Hz
2. νspin = 435Hz において振動数比 κ = νoscillation /νspin = 0.5727 · · · を満た
すような（脈動）不安定モードが存在するか確かめるべく、モデル計
算を行った
(a) 中性子星表面の降着 helium 層を伝播する r-mode は、helium 燃焼
で励起され、降着物質に少量の水素を混ぜれば、その振動数が比
κ = 0.5727 · · · をみたすようすることができる
(b) 中性子星の個体殻を伝播領域とする振動モード（crustal toroidal
mode）は液体核を伝播する振動モード（r-mode）と結びつくこと
で（その不安定性は r-mode の不安定性よりも弱いが）励起され
る。個体殻の shear modulus を調整することで、その振動数が比
κ = 0.5727 · · · をみたすようすることができる
振動モードの分類
1. モードの分類に使われる整数（量子数？）
!
l;
degree
m; azimuthal wavenumber, |m| ≤ l
n; radial order
振動数 σ はしたがって、
σlmn
ただし、自転や磁場が無ければ振動数 σ は m に依存しない
2. spheroidal modes と toroidal modes
"
spheroidal modes; p-modes, g-modes, crustal modes
toroidal modes;
crustal modes
そして、自転や磁場があるとその二つが混ざる
3. 回転星における
(a) prograde modes と retrograde modes
p′ ∝ Ylm (θ, φ)eiωt ∝ exp i (mφ + ωt)
ω > 0 とすれば
"
m < 0, prograde modes
m > 0, retrograde modes
(b) frequencies in the co-rotating frame and in an inertial frame
ω = σ + mΩ
•
恒星の固有振動
•
音波モード（p-modes）や内部重力波モード（g-modes）
•
自転：振動数スペクトルや脈動安定性への影響
•
•
•
ロスビー波（r-modes）や 慣性波（inertial modes）
振動数スペクトルや脈動安定性は星の進化に伴う構造変化を反映する
中性子星の固有振動：複雑な構造・状況を反映して様々な振動モードが存在
•
表面流体層（fluid ocean）、個体殻（solid crust）、流体核（fluid core）
•
音波モードや内部重力波モード
•
個体殻に特有な音波
•
超流動（個体殻や流体核）：超流動特有の振動モード（２流体モデル）
•
自転： 振動数スペクトルへの影響、 ロスビー波や 慣性波
•
磁場：振動数スペクトルや脈動安定性への影響
•
•
Alfven波
相対論的効果：重力波によるexcitationやdampingなど
Figure 6: Cross section of a neutron star (Shapiro & Teukolsky 1983)