Let’s start with some fundamental relationships among four crucial physical characteristics: Mass, Radius, Density, and Gravity.
Note that in the last row, each value is defined as a function of all of the other three, but in the top three rows, as long as any two of the other three values are known, the third can be calculated.
Also note that the fourth row of equations are only needed if all of the known values have been independently determined (or randomized); if you are only specifying, say, mass and radius, then use the top row to calculate gravity and density from those values.
In the case of radius, when R = 1, then mass, gravity, and density will all always be equal to each other.
By “relative units”, we mean “in terms of the ‘standard example’”; that is to say, Earth or the Moon, for instance. For example, using the equation:
The following table compares the measured density (ρ) of the other seven planets in the Solar System to that calculated by using only the mass and radius equation (ρ2) and that calculated using the mass/radius/gravity equation (ρ3). As can be seen, the calculated values are in close agreement with measured values (ρ2/ρ) and (ρ3/ρ), and in even closer agreement with one another (ρ3/ρ2).
For non-stellar bodies, these relationships will be our primary means of determining the physical characteristics of invented objects, but in all cases, at least two attributes will have to be decided upon before either of the other two can be calculated.
Thus, it will be important to decide up-front which attributes are most desired; for instance, are a particular mass and gravity the most important considerations, or are the mass and radius the more important attributes?
Taking a page from recent news, TRAPPIST-1b is reported to have a mass of 0.85 that of Earth, and a radius of 1.086±0.035 Earth's (we'll use the base value, here).
So, what would its surface gravity and density be?
Starting with gravity:
What about its density? Let's use all three of the other attributes to calculate it, so that the density is determined by the measured values for the mass and radius, and the calculated value for the surface gravity:
We get precisely the same value for the density if we use only the mass-over-radius-cubed equation:
We now have a fuller mental picture of the nature of the exoplanet TRAPPIST-1b: it is only very slightly larger than Earth in radius, but 85% as massive; with a surface gravity about 72% that of Earth, or 1.92 times that of Mars; and a density 66% that of Earth's, or 93% that of Mars.
The mass and density figures tell us the most about its possible composition; since it is the size of Earth, but 85% as massive and 66% as dense, it likely has a lower heavy metal content—perhaps indicating a smaller core—and a correspondingly higher content of rocky material.
Finer and more detailed observations of the TRAPPIST-1 system will perhaps give us more observational data with which to compare our calculated values, but if we wanted to base an in vented world on TRAPPIST-1b, we've made a good start.