1 Introduction

The dimensions of planets began to become apparent in the early days of telescope astronomy, at least to the extent that Jupiter and Saturn appeared larger than Mars and Venus. Absolute dimensions—i.e., those in specific units, as opposed to by angular appearance or relative to some other standard–had to wait until the scale of the Solar system was deduced. The first roughly accurate estimate of the size of an AU was found by Christiaan Huygens in 1659, with Cassini and Richter obtaining a similar value in 1672. An estimate close to the modern value was found by Newcomb in 1895 (Goldstein 1985, Conférence internationale des étoiles fondamentales 1896).

In 1964, Dole (p. 29) attempted to relate density to radius for rocky terrestrial planets, using Earth, Mars, Luna, and Earth’s surface rocks as data points. The relation he found was:

$$\rho = \rho_{0} \exp \left( {\text{AR}} \right)$$
(1)

where ρ is the planet’s mean density in cgs units, ρ0 is the surface-rock density, taken as 2.770 g cm−3, A is a dimensionless constant equivalent to 0.6904, and R is the planet’s radius in terms of the Earth (i.e. R/R⊕). It is apparent that Mercury falls far from this curve, and the fit is good, but not perfect, for the other terrestrial planets.

When exosolar planets began to be discovered in 1992, a much larger database became possible. Weiss and Marcy (2014, see also Marcy et al. 2014) found, based on a sample of N = 65 bodies, that density tended to increase with radius up to about 1.5 Earth radii and thereafter declined, probably due to the planet, in formation, retaining volatiles from the preplanetary nebula. However, the MW14 sample mixed m sin i values in with absolute masses.

When estimates for the relations among planetary masses, radii, and/or densities become available, this may provide important clues to the internal makeup of such planets. It is already apparent that greater density in a terrestrial planet means few volatiles, and in the case of Mercury, a large metal core relative to the rest of the planet. Constraints on planet sizes in general, deduced from a large database of planet size estimates, could aid in modeling terrestrial planet interiors.

With this in mind, the author of this paper compiled a list of all worlds which met the following criteria:

  • Mass known to be between 0.01 and 10 times that of Earth. Exoplanets with only m sin i figures available were excluded from the sample.

  • Radius also known.

From these two data, many other figures can be derived from formulae based on Newton’s law for gravity, and spherical geometry; including a planet’s cross-sectional area, surface area, volume, density, surface gravity, circular velocity and escape velocity. The sample is listed in Table 1. It includes N = 89 bodies, nine of which are planets and satellites in the Solar system, and 80 of which are exoplanets. No dwarf planets such as Ceres or Pluto were included, as all fell under the minimum mass cut-off. The column Planet gives the planet’s name or exosolar planet designation. M/M⊕, R/R⊕, and F/F⊕ columns give the planet’s mass, radius, and solar or stellar insolation in terms of the Earth, while the columns +er and −er show positive and negative error bars—if only the positive figure is given, it is meant to apply in both directions. For flux, the error bars were always single-figure. All numerical columns are dimensionless.

Table 1 The catalog
Full size table

2 Some Characteristics of the Data

It should be noted that much of this data is very preliminary, and no doubt subject to observational and even systematic error. The apparent density of Kepler-131c, for example, would be three times that of osmium if the given parameters are correct, which seems unlikely unless exotic physics are involved.

As many significant figures are listed as were given in the original sources.

For exoplanets, the radiative flux density received by the planet relative to Earth was derived by the following steps:

  1. 1.

    The bolometric luminosity L of the central star was derived from its radius R and effective temperature Te via the relation

    $$L = 4\pi R^{2} \sigma T^{4}$$
    (2)

    where σ is the Stefan-Boltzmann constant.

  2. 2.

    The radiative flux density was calculated from the inverse square law:

    $$F/F \oplus = \frac{L/L \odot }{{a^{2} }}$$
    (3)

    where a is the semimajor axis in AU. Semimajor axes, and thus flux densities, were not available for all worlds listed.

  3. 3.

    Error propagation methods were used to find the error figures on the relative flux densities.

3 Methods

The data as graphed (Fig. 1) are sloped down at both ends, implying, if density is taken as a function of radius, an inflection point between the two ends. The planets in the sample were ordered by radius. Chow tests (see, e.g., Doran 1989, p. 146) were then performed with a ten percent trim to find the maximum Fisher’s F value, and thus the most likely location of the actual break (Fig. 2). This occurred (F2,85 = 22.74, p < 10−8) at a radius of 1.53 times Earth, corresponding to a density 1.88 times Earth and a mass 6.75 times Earth; the values for the exoplanet Kepler-80d.

Fig. 1
figure 1

Log density versus radius

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Fig. 2
figure 2

Chow test F values. The horizontal axis is N1, the size of group 1. By definition, N1 + N2 = N where N is the total sample size

Full size image

The “winning” regressions were:

$$\begin{aligned} & \log \left( {\frac{\rho }{{\rho \oplus }}} \right) = - 0.4083 + 0.3969\left( {\frac{R}{{R \oplus }}} \right) \\ & {\text{N}} = 44,\;{\text{R}}^{2} = 0.203,\;p < 0.00214 \\ \end{aligned}$$
(4)

for smaller, presumably rocky planets, and

$$\begin{aligned} & \log \left( {\frac{\rho }{\rho \oplus}} \right) = 0.3862 - 0.3071\left( {\frac{R}{R \oplus}} \right) \\ & {\text{N}}\, = \,45,\;{\text{R}}^{2} = 0.883,\;p < 9.29 \times 10^{ - 22} \\ \end{aligned}$$
(5)

for larger, presumably volatile-rich planets. Note the oppositely signed slopes and intercepts.

At R/R⊕ = 1.53, Eq. 4 gives ρ/ρ⊕ = 1.58 (corresponding to M/M⊕ = 5.66), while Eq. 5 gives ρ/ρ⊕ = 0.825 (and M/M⊕ = 2.95). Partly this is due to the sparseness of the sample, but at the same time, it would seem that composition matters.

Flux was not considered in this analysis, although the figures are available for most (not all) of the planets in the sample. Low flux undoubtedly accounts for the low densities of the Galilean satellites and Titan (i.e., they formed beyond the ice line), while very high flux may explain either the devolatilization of rocky planets or the extended gas envelopes of sub-Neptune planets. Speculation on this point is deferred until further analysis can be done.

4 Conclusions

The chart of data from this paper’s sample (Fig. 1) shows the same effect as that in the study of Marcy and Weiss (2014), although not subject to the same methodological problem (MW14 included m sin i values along with masses). The upper limit to the linear size of a rocky planet appears to be near 1.5 Earth radii, and the upper limit to its mass is in the vicinity of six Earth masses under normal circumstances. The few known larger “rocky” bodies, invariably subject to much higher stellar radiative flux than Earth, may perhaps be the cores of evaporated gas or ice giants.