Before getting into the nuts and bolts of designing stars and planets, it’s best if some basics are established—known relationships and processes to which we will be referring frequently as we delve deeper and deeper into this study.

Let’s start with some fundamental relationships among four crucial physical characteristics: mass, radius, density, and gravity.

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, whenR = 1,then mass, gravity, and density willbe equal to one another.always

These relationships hold for any

Note that these equations assume relative units. 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:

*solid*object large enough to achieve hydrostatic equilibrium (thus, planets, rather than not stars).Note that these equations assume relative units. 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:

… if all of the variables are expressed in terms of

*Earth**= 1*, then, for Mars,*M**= 0.107*terran, and*R**= 0.533*terran, and so:… the surface gravity of Mars is 0.377 terran, which is the value we find if we look up Mars in a reference work.

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).

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).

The following table makes the same comparisons, instead calculating the values for gravity based upon measured values for mass, radius, and density, and we find the same degree of agreement.

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, perhaps, are the mass and radius the more important attributes?

Taking a page from recent news <1>, TRAPPIST-1b is reported to have a mass of 0.850 terran, and a radius of 1.09 terran (we'll use the central value, here).

So, what would its surface gravity and density be?

Starting with gravity:

Taking a page from recent news <1>, TRAPPIST-1b is reported to have a mass of 0.850 terran, and a radius of 1.09 terran (we'll use the central value, here).

So, what would its surface gravity and density be?

Starting with gravity:

We arrive at a surface gravity for TRAPPIST-1b of 0.715 terran, or about 7.007 m/sec/sec.

As for 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:

As for 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 a density value for TRAPPIST-1b of 0.656 terran, or about 3.615 grams per cubic centimeter, which is slightly less than the measured value for Mars of 3.9335 terran.

We get precisely the same value for the density if we use only the mass-over-radius-cubed equation:

We get precisely the same value for the density if we use only the mass-over-radius-cubed equation:

... so we have good reliability of our calculated value for the surface gravity.

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, and only 85% as massive; with a surface gravity about 72% terran; and a density 66% terran.

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.

## Notes

- "Exoplanet: TRAPPIST - 1b," Tsunami Historical Series: Sumatra - 2004 Dataset | Science On a Sphere, , accessed December 23, 2018, https://sos.noaa.gov/datasets/exoplanet-trappist-1b/.