  The background field is assumed to play prime role in the erupting structures
like prominences. In the flux rope models, the critical decay index ($n_c$) is
a measure of the rate at which background field intensity decreases with height
over the flux rope or erupting structure. In the real observations, the
critical height of the background field is unknown, so a typical value of
$n_{c}=1.5$ is adopted from the numerical studies. In this study, we determined
the $n_c$ of 10 prominence eruptions (PEs). The prominence height in 3D is
derived from two-perspective observations of \textit{Solar Dynamics
Observatory} and \textit{Solar TErrestrial RElations Observatory}. Synoptic
maps of photospheric radial magnetic field are used to construct the background
field in the corona. During the eruption, the height-time curve of the sample
events exhibits the slow and fast-rise phases and is fitted with the
linear-cum-exponential model. From this model, the onset height of fast-rise
motion is determined and is considered as the critical height for the onset of
the torus-instability because the erupting structure is allowed to expand
exponentially provided there is no strapping background field. Corresponding to
the critical height, the $n_c$ values of our sample events are varied to be in
the range of 0.8-1.3. Additionally, the kinematic analysis suggests that the
acceleration of PEs associated with flares are significantly enhanced compared
to flare-less PEs. We found that the flare magnetic reconnection is the
dominant contributor than the torus-instability to the acceleration process
during the fast-rise phase of flare-associated PEs in low corona
($<1.3R_{\odot}$).
