  In this work we study the radiative decay widths of vector quarkonia for the
process of $J/\psi(nS)\rightarrow\eta_{c}(nS)\gamma$ and
$\Upsilon(nS)\rightarrow\eta_{b}(nS)\gamma$ (for principal quantum numbers
$n=1, 2, 3$) in the framework of Bethe-Salpeter equation under the covariant
instantaneous ansatz using a $4\times 4$ form of BSE. The parameters of the
framework were determined by a fit to the mass spectrum of ground states of
pseudoscalar and vector quarkonia, such as; $\eta_{c}$, $\eta_{b}$, $J/\psi$
and $\Upsilon$. These input parameters so fixed were found to give good
agreements with data on mass spectra of ground and excited states of
pseudoscalar and vector quarkonia, leptonic decay constants of pseudoscalar and
vector quarkonia, two photon decays and two gluon decays of pseudoscalar
quarkonia in our recent paper. With these input parameters so fixed, the
radiative decay widths of ground (1S) and excited (2S, 3S) states of heavy
vector quarkonia ($J/\Psi$ and $\Upsilon$) are calculated and found to be in
reasonable agreement with data.
