The stars are the main elements of galaxies; compared to them, the planets have mostly much smaller masses. However, the distribution of celestial objects by mass continuously changes from giant planets to stars of the smallest masses, called brown dwarfs.

Mass is an important distinguishing feature of planets from stars, but there is no obvious boundary of transition between a high-mass planet and a low-mass brown dwarf. It is generally taken to be about 13 times the mass of Jupiter, after which the conditions for the start of thermonuclear fusion are reached. Numerous studies have suggested this value as the official distinction between the two types of objects. However, many scholars believe that this limit does not have any solid physical basis. Some astronomers argue that a more significant distinction between planets and brown dwarfs is their mode of formation. Therefore, there is currently no clearly accepted transition mass between giant planets and brown dwarfs.

Here we propose a theoretical method for calculating the transition mass.

According to Newton’s law of universal gravitation, the absolute value of the force of gravitational interaction between two objects is where m₁ and m₂ are the masses of the objects, and r is the distance between the centers of their masses, and G = 6.674×10^–11 m³/(kg∙s²) is the gravitational constant.

From Newton’s law follows the formula for the gravitational potential Ф (we mean its modulus), which characterizes the gravitational field of an ordinary body: where m is the mass of a body, r is the distance from the body’s center of mass to the point at which the potential is determined. Note that the gravitational potential has the dimension of the square of the velocity, [Ф] = [υ²] = (m/s)².

It should be recalled that the International System of Units (SI) is a composite system that includes, in particular, the m-kg-s system of mechanical units (MKS system) and the m-kg-s-A system of electromagnetic units (MKCA system). The second system differs from the first primarily in that, along with the existing three base units (meter, kilogram, and second), it has a fourth base unit – ampere (A).

For example, in the MKSA system, the elementary electric charge e = 1.6×10⁻¹⁹ C, and the proportionality coefficient, included in Coulomb’s law, k = 9×10⁹ N∙m²/C².

In 2018, an article [1] was published that showed that the electromagnetic units of the MKSA system (the ampere, coulomb, ohm, volt, etc.) can be converted using the base units of the MKS system: m, kg, s. In the paper, it was shown that in the MKS system where c = 3×10⁸ m/s is the speed of light in vacuum, and F₁ = 10^–5 kg∙m/s² (or 1 g∙cm/s² – the unit of force in the СGS system).

Using these results, it was shown in paper [2] that the constant where υ_g = (G/k)^1/2 = 0.8617×10^–16 m/s is the elementary speed, i.e., the lowest speed of movement in nature.

Substituting the expression for G into equation (1), the new formula for the law of universal gravitation was obtained: or where g₁ = m₁υ_g and g₂ = m₂υ_g are the gravitational charges of the interacting bodies, and the coefficient k = 9×10²¹ m/kg.

So, we obtained an expression exactly analogous to Coulomb’s law (in the MKSA system): where F is the absolute value of the force of electrostatic interaction in a vacuum of two point electric charges q₁ и q₂, r is the distance between them, and the coefficient k = 9×10⁹ N∙m²/C².

From Coulomb’s law follows the electric potential φ that characterizes the electrostatic field of a point electric charge q and is defined in a vacuum by the formula where r is the distance from the charge to the point at which the potential is determined.

In the MKS system, the electric charge has the dimension of the momentum, [q] = kg∙m/s, and the coefficient k has the dimension [k] = m/kg; thus, the electric potential has the dimension of velocity, [φ] = [kq/r] = [υ] = m/s.

Hence, analogous to the electric potential, we can introduce a new formula for the gravitational potential: where g = mυ_g is the gravitational charge of a body, r is the distance from the body’s center of mass to the point at which the potential is determined, the coefficient 9×10²¹ m/kg, and the potential Ф_υ has the dimension of velocity, [Ф_υ] = [υ] = m/s.

It turns out that we can use this potential to calculate the transition mass from giant planets to brown dwarfs.

As previously mentioned, when the transition mass is reached, the conditions for the start of thermonuclear fusion are formed. An object with such a mass begins to emit the light (electromagnetic waves) that can travel a distance equal to the radius of the observed universe (R). This radius is determined by the formula R=c/H, where H is the Hubble constant. The exact value of this constant is not yet known; measurements give a value of H ≈ 70 (km/s)/Mpc (1Mpc = 3.0856×10²² m).

Let us assume that for an object with a transition mass m, at a distance equal to the radius of the observed universe (r=R) the potential Ф_υ is equal to the speed of light (Ф_υ=с). Hence, according to equation (10),

Using the value H = 70 (km/s)/Mpc = 2.26×10^–18 s^–1, we calculate: where M_J = 1.9×10²⁷ kg is the Jupiter mass.

So, we obtained a value twice the accepted intermediate mass. Therefore, we must finally accept that for an object with an intermediate mass m_i , at a distance r=R the potential Ф_υ=с/2:

Hence, the intermediate mass

Thus, to accurately calculate the transition mass, we need to know the exact value of the Hubble constant.

There is no universally acknowledged criterion for distinguishing brown dwarfs from giant planets. Numerous studies have suggested a definition based on an object’s mass, taking the ~13 Jupiter mass limit for the ignition of deuterium. Here we have presented a method for calculating the transition mass using a new formula for the gravitational potential, which has the dimension of velocity. This formula establishes a clear boundary for the transition between a giant planet and a brown dwarf: for an object with an intermediate mass, at a distance equal to the radius of the observed universe (r=R), the potential Ф_υ is equal to half the speed of light (Ф_υ=с/2).