Effective potential and stable circular orbits in magnetized Schwarzschild spacetime
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摘要: 利用赤道平面上的径向有效势研究带电粒子在有外部磁场的史瓦西黑洞附近的圆轨道运动,发现针对某特定的角动量和较小的外部磁场,稳定圆轨道很难存在;但随着磁感应强度的增大,稳定圆轨道容易形成,并且磁感应强度的增大会导致稳定圆轨道半径逐渐减小. 稳定圆轨道半径随着角动量的增大而逐渐增大. 角动量和径向距离函数曲线与径向距离和有效势在讨论圆轨道性质上具有同等效果.Abstract: An effective potential in the equatorial plane was used to study the circular motion of charged particles near the Schwarzschild black hole immersed into an external magnetic field. It is found that no stable circular orbits exist for some angular momenta and small magnetic fields. However, the stable circular orbit easily occurs and has small radius when the magnetic induction increases. The radius of stable circular orbit increases with an increase of the angular momentum. The relation between the angular momentum and a radial distance can show the features of circular orbits, as the relation between the effective potential and the radial distance can.
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Key words:
- black hole /
- magnetic field /
- effective potential /
- stable circular orbits
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磁感应强度 r $ E^{2}\left(V_{\text {eff }}\right) $ $V_{\text {eff } }^{\prime \prime}$ 稳定性 B=0 3.618937 1.170126 −0.162335 不稳定 17.541063 0.946911 0.000294 稳定 B=0.01 3.577530 1.149839 −0.174817 不稳定 16.351999 0.912641 0.000519 稳定 B=0.1 3.313035 0.988926 −0.272909 不稳定 7.986825 0.772967 0.016335 稳定 磁感应强度 L=6.5 L=8 r $ E^{2}\left(V_{\text {eff }}\right) $ $V_{\text {eff } }^{\prime \prime}$ 稳定性 r $ E^{2}\left(V_{\text {eff }}\right) $ $V_{\text {eff } }^{\prime \prime}$ 稳定性 B=0 3.250000 1.923077 −0.640874 不稳定 3.155590 2.719855 −1.163588 不稳定 39.000000 0.975071 0.000030 稳定 60.844410 0.983849 0.000008 稳定 B=0.01 3.231197 1.898294 −0.664131 不稳定 3.141993 2.690759 −1.194549 不稳定 26.817870 0.936274 0.000273 稳定 32.780111 0.945019 0.000238 稳定 B=0.1 3.090400 1.692793 −0.860438 不稳定 3.035547 2.445489 −1.460505 不稳定 10.398772 0.816597 0.017095 稳定 11.870141 0.836892 0.017346 稳定 表 3 不同磁感应强度影响下最内稳定圆轨道半径(rISCO)、最内稳定圆轨道角动量(LISCO)及E2的值
Table 3. Values of rISCO, LISCO and value E2 under influence of different magnetic field intensities
参数 ${\mathit{r} }_{{\rm{ISCO}} }$ ${\mathit{L} }_{\rm{ISCO} }$ $ {\mathit{E}}^{2} $ ${ {L}_{{\rm{ISCO}}} }^{\prime \prime}$ B=−0.1 5.025714 4.626777 1.428876 −4.560000×10−8 B=0 6.000000 3.464102 0.888889 −3.608436×10−8 B=0.1 4.696670 3.359201 0.706673 −6.101820×10−7 表 4 角动量
$L=4.6$ 不同磁感应强度影响下$ { r} $ 及$ {E}^{2} $ 值Table 4. values of
$ r $ and$ {E}^{2} $ under influences of different magnetic induction with angular momentum$ L=4.6 $ 参数 r $ {\mathit{E}}^{2} $ B=0 3.618937 1.170126 17.541063 0.946911 B=0.01 3.577530 1.149839 16.351999 0.912641 B=0.1 3.313035 0.988926 7.986825 0.772967 表 5 不同磁感应强度下
$ r$ 、$ {E}^{2} $ 及对应$ L $ 值Table 5.
$ r $ ,$ {E}^{2} $ and corresponding$ L $ under different magnetic induction参数 r $ \mathit{L} $ $ {\mathit{L}}^{\prime \prime} $ 稳定性 B=0, $ {\mathit{E}}^{2}=0.946911 $ 4.295401 3.773997 0.151900 不稳定 17.541063 4.599999 −0.011102 稳定 B=0.01, $ {\mathit{E}}^{2}=0.942903 $ 4.295814 3.706091 0.152276 不稳定 16.351999 4.599998 −0.024210 稳定 B=0.1, $ {\mathit{E}}^{2}=0.772967 $ 3.705074 3.740838 0.389130 不稳定 7.986825 4.599998 −0.492759 稳定 -
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