Relativistic Aberration of Light
==========================================

.. _relativistic-aberration:

.. figure:: assets/special-relativity/relativistic-aberration.png
   :align: center

   The relativistic aberration of light.

In astrophyics, object moving with a significant fraction of the speed of light :math:`v_S` with angle :math:`\theta` shown in :numref:`relativistic-aberration`, is measure from the observer in a direction

.. math::
   \cos \theta_O = \frac{\cos\theta_O - v_S}{1 - v_S \cos \theta_S}.

If the object is moving towards us, we observe :math:`\theta_O=0`.

Meanwhile the apparent transverse velocity is measured to be

.. math::
   v_{\perp,O} = \frac{v}{1-\beta \cos (\pi - \phi +\theta)},
   :label: eqn-relativistic-apparent-velocity

where :math:`\pi - \phi +\theta` is the angle between line of sight and the velocity, measured in the object's frame. One of the astonishing fact about Eq. (:eq:`eqn-relativistic-apparent-velocity`) is that it's maximun value can be larger than 1, which means we could observe superluminal objects.