  An analytical expression for the energy of N\'eel skyrmions in ultra-thin
nanodots considering exchange, uniaxial anisotropy, Dzyaloshinskii-Moriya, and
dipolar contributions has been obtained. In particular, we have proposed for
the N\'eel skyrmion, a general ansatz for the component of the magnetization
perpendicular to the dot, given by $m_z(r) = [1-(r/R_s)^n]/[1 + (r/R_s)^n]$,
where $R_s$ is the radius of the skyrmion and $n$ is an integer and even
number. As proof of concept, we calculate the energy of a N\'eel skyrmion in an
ultra-thin Co/Pt dot, and we find that the dipolar contribution cannot be
neglected and that both Dzyaloshinskii-Moriya interaction and anisotropy play
an important role to stabilize the skyrmion. Additionally, we have obtained a
good agreement between our analytical calculations and previously published
micromagnetic simulations for $n = 10$. For this reliable value of $n$, we have
obtained that for a Dzyaloshinski Moriya constant $D = 5.5 \, (mJ/m^2)$, it is
possible to stabilize a N\'eel skyrmion for $K_u$ in the range, $0.4 \,
(MJ/m^3)< K_u <1.3 \, (MJ/m^3)$, whereas for $K_u = 0.8 \, (MJ/m^3)$, the
skyrmion stabilizes for $5.0 \, (mJ/m^2) < D <6.0 \, (mJ/m^2) $. Thus, this
analytical equation can be widely used to predict stability ranges for the
N\'eel skyrmion in spintronic devices.
