超光速運動

解釋

一些相反的證據

1993年 Thomson 等人提出，類星體3C273的外部噴流是幾乎和觀測者的視線是共線的。沿著3C273內部噴流觀測到的超光速運動最高達到約9.6倍光速[3]

M87星系的噴流較內側部分已觀測到6倍光速的運動。如果要以運動方向和觀測者視線夾角很小的模型來解釋的話，噴流和視線方向夾角不能高於19°[4]。不過觀測證據顯示實際的夾角達到43°[5]。同組的科學家後來修改其結果，並宣稱他們的觀測結果支持噴流中的整體超光速運動[6]

數學推導

${\displaystyle AB\ =\ v\delta t}$
${\displaystyle AC\ =\ v\delta t\cos \theta }$
${\displaystyle BC\ =\ v\delta t\sin \theta }$
${\displaystyle t_{2}-t_{1}\ =\ \delta t}$
${\displaystyle t_{1}^{\prime }=t_{1}+{\frac {D_{L}+v\delta t\cos \theta }{c}}}$
${\displaystyle t_{2}^{\prime }=t_{2}+{\frac {D_{L}}{c}}}$
${\displaystyle \delta t^{\prime }=t_{2}^{\prime }-t_{1}^{\prime }=t_{2}-t_{1}-{\frac {v\delta t\cos \theta }{c}}=\delta t-{\frac {v\delta t\cos \theta }{c}}=\delta t(1-\beta \cos \theta )}$ , where ${\displaystyle \beta ={\frac {v}{c}}}$
${\displaystyle \delta t={\frac {\delta t^{\prime }}{1-\beta \cos \theta }}}$
${\displaystyle BC\ =\ D_{L}\sin \phi \approx \phi D_{L}=v\delta t\sin \theta \Rightarrow \phi D_{L}=v\sin \theta {\frac {\delta t^{\prime }}{1-\beta \cos \theta }}}$

${\displaystyle \beta _{T}={\frac {v_{T}}{c}}={\frac {\beta \sin \theta }{1-\beta \cos \theta }}}$
${\displaystyle {\frac {\partial \beta _{T}}{\partial \theta }}={\frac {\partial }{\partial \theta }}\left[{\frac {\beta \sin \theta }{1-\beta \cos \theta }}\right]={\frac {\beta \cos \theta }{1-\beta \cos \theta }}-{\frac {(\beta \sin \theta )^{2}}{(1-\beta \cos \theta )^{2}}}=0}$
${\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )^{2}=(1-\beta \cos \theta )(\beta \sin \theta )^{2}}$

註釋

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