# Can a planet exist with the same mass as the earth, but have a different diameter?

Can there be a planet with the same mass, and a similar composition, as the earth, but with a larger or smaller diameter, and how could that theoretically occur? I've tried to see if there is any research or sites that tackle this idea, but I, understandably, haven't found anything.

Yes, a planet can exist with the same mass as Earth, but have a different diameter - if it has a different density. The lower the density, the larger will be the diameter. In terms of an exact formula, we have:

$$d(M, \rho) := 2 \cdot \sqrt[3]{\frac{3M}{4\pi \rho}}$$

The trick is that density is specifically a function of composition: hence, the extent to which the diameter can vary depends essentially completely on what value you have in mind for when you ask for a composition as being "similar". Dramatically different densities, and hence diameters, will only be possible with a dramatically different composition - e.g. a heavy gas envelope will provide a significantly reduced density, as would a significant mass percentage of water (think of this hypothetical Earth-mass planet as being like a big version of certain moons of Jupiter and Saturn, such as Europa and Enceladus. In fact, some percentage of discovered extrasolar planets are believed to be of this type: though you should not think of it as having liquid ocean all the way down, because after about 100 km of depth or so on comparable-to-Earth gravity, water freezes due to compression.).

If by "similar" composition we mean that, say, the planet is still to have a solid, hospitable surface, and to have all the same materials as on Earth but just perhaps in somewhat different abundances, one "easy" way might be a reduction in the mass of iron. A lot of the Earth's density is supplied thanks to its relatively large central iron core: compare it with the Moon, which has little to no iron core - the density for Earth is 5.5 g/cm^3, while that for the Moon is about 3.3 g/cm^3, althuogh one should also be mindful of compression effects due to gravitational crushing of the interior. A small iron core would give one a density in the intermediate range between these two.

With such a smaller iron core, to give one Earth mass (about 6000 Yg), it must be exchanged for a greater quantity of less-dense rock, hence the planet's diameter will be larger. Conversely, we could imagine having more of an iron core, similar to Mercury, and then the diameter would be smaller (and the surface gravity larger) as we should have less rock to avoid going over the one Earth-mass prescription.

• +1 Useful mention of Earth's core vs the Moon, but perhaps overlooking an important point: under the Theia hypothesis, gravitational separation had already occurred before impact, with the result that the Moon mostly formed from the lighter ejecta from the Earth's mantle. If the impact had occurred before significant planetary differentiation, the densities of Earth and Moon would be closer. Interrupted gravitational separation could result in a smaller core and thicker mantle, which would affect planetary diameter. Oct 2, 2019 at 2:46

Yes, it basically comes down to average density. This average density can be used as a proxy for the composition of the planet, an example of mass-radius relations for low mass rocky planets can be seen in the attached picture from Gettel, S., Charbonneau, D., Dressing, C. D., Buchhave, L. A., Dumusque, X., Vanderburg, A., Bonomo, A. S., Malavolta, L., Pepe, F., Cameron, A. C. et al. (2016), ‘The kepler-454 system: a small, not-rocky inner planet, a jovian world, and a distant companion’, The Astrophysical Journal 816(2), 95.