Why gravitational constant $G$ does not have the factor of $4π$? We know that electrostatic constant $K=\frac{1}{4\pi\epsilon}$.
This $4\pi$ came from the surface area of the surface in which charge is enclosed.
Then, why don't gravitational constant has the factor of $4\pi$?
 A: You can define it as so. I'll name $4\pi G$ as $\mathcal{G}$.
In electrostatics:
$$F=\frac{1}{4\pi \epsilon_0} \frac{Qq}{r^2}$$
and so Gauss's law is:
$$\nabla \cdot \vec{E}=\frac{\rho}{\epsilon_0}$$
Notice how the $4\pi$ is gone from the denominator.
Equivalently, Newton's law of gravitation:
$$F=-G \frac{Mm}{r^2}$$
has the poisson's equation:
$$\nabla \cdot \vec{g} = -4\pi G\rho$$
where $\vec{g}$ is the acceleration field, similar to $\vec{E}$.
If you used $\mathcal{G}$, then the equations are:
$$F=-\frac{\mathcal{G}}{4\pi} \frac{Mm}{r^2}$$
and
$$\nabla \cdot \vec{g} = -\mathcal{G} \rho$$
It's just a convention.
A: The utility in writing the constant in Coulomb's law as $$\frac{1}{4 \pi \epsilon_0}$$ instead of $$\frac{1}{ \varepsilon_0}$$ is that this convention makes Gauss' law, equivalent to Coulomb's law, look simpler: $$\nabla \cdot \mathbf{E} = \rho/\epsilon_0.$$ With the opposite convention, we would have to insert the $4\pi$ in Gauss's law, and it would disappear in Coulomb's law. My guess is that the convention was chosen so that the $4\pi$ doesn't appear in the version which is used the most, which I'm pretty sure is Gauss's law.
In gravitation, however, the 'Coulomb version' $$F=G\frac{m_1m_2}{r^2}$$ seems to be used the most, so it would make sense to adopt the convention such that this equation appears without the $4\pi$.
A: The electrostatic constant in Gaussian units is simply one. It's a choice. It's a weird choice for applied physics because the factors of 4 show up in rectangular problems but disappear in spherical ones. Hence, for practical problems, we use SI units.
Gaussian units are handier in theoretical physics because spherical problems are more common. In Gaussian units you don't need a lot of 4's in formulae expressed in spherical coordinates. A similar convention is convenient for gravitational problems since they commonly use spherical coordinates.
