A paper of Heath-Brown gives an heuristic argument for the density of rational points on two cubic surfaces: $x^3+y^3+z^3=kw^3,k=2,3$, say, the number of rational points of height less than $N$ on these cubic surfaces is about $cN$, where $c$ has an interpretation from circle method.

Definition: Let $$\Sigma=\{P|P=(x,y,z,w)\in\mathbb{Z}^4,x^3+y^3+z^3=kw^3,gcd(x,y,z,w)=1\}$$

and $$\Sigma'=\{P'|P'\in\Sigma\},$$

where $P'$ lies on a plane rational curve over $\mathbb{Q}$ and $P'$ cannot be a non-singular point on plane rational curves over $\mathbb{Q}$.

Let $h(P)=\max\{|x|,|y|,|z|,|w|\},P=(x,y,z,w)$.

Question: A numerical computation seems to suggests that $\#\{P'|P'\in\Sigma',h(P')<N\}\sim c'N$ for these two surfaces, but I have no idea about an heuristic interpretation for the constant $c'$ ($c'\approx 0.8 c$ in both cases,but not exactly the same). Is there any heuristic interpretation for such constants?