"The Earth Similarity Index, ESI or "easy scale" is a measure of how physically similar a planetary-mass object is to Earth. It is a scale from zero to one, with Earth having a value of one."
To have an ESI, an object needs to have a mean radius, a density, an escape velocity, and a surface temperature. Gas giants and stars, therefore, don't really have a definable ESI, since they lack a 'surface' where you can take a temperature.
If something has all of those properties, then because of the way it's defined, it has an ESI which is strictly > 0 and <= 1 (it can only be zero if, for one of its properties, x - x0 / x + x0 == 1, i.e. x - x0 == x + x0 - and since x0 is strictly nonzero, that can't happen). So nothing has an ESI of 0. But you yourself, to pick an object at random, probably have an ESI very very close to zero.
Incidentally, the formulation of the ESI is a good generic formula for creating 'similarity' measurements for, for example, matchmaking algorithms or similarity searches.
The ESI defines a hyperspherical gradient across several different axes of earth similarity. And since each of those axes are defined by a ratio to an Earth value, zeroes would only be found for objects with the following properties:
- Infinite mean radius.
- Zero mean radius.
- Infinite density.
- Zero density.
- Infinite escape velocity.
- Zero escape velocity.
- Infinite surface temperature.
- Zero surface temperature.
As such, the only object I can think of that might qualify would be a theoretical zero-entropy singularity that might have existed existed prior to the Big Bang. Zero radius, infinite density, infinite escape velocity, zero surface temperature.
Given the nature of the calculation, it probably would have been better to define perfectly Earth-like as zero, and assign positive indices to bodies, with higher numbers indicating greater dissimilarity to Earth. It would be a distance calculation along multiple axes of similarity, but adjusted by scale factors to coerce the differences between comparison axes with different units to have an order of magnitude proportional to perceived importance to humans.
If you have a difference function for each axis that yields a number from 0-1 for comfortable, 1-10 for tolerable, 10-100 for survivable, and 100+ for lethal, then you can take the square root of the sums of the difference functions for the overall index.
Anything less than 1 would be a near clone of Earth. Anything less than 10 would definitely be habitable. Anything less than 100 might be explorable. Anything greater than that, and you might just seed it with Earth extremophiles and ignore it for a few million years.
Sure, you could do that. I guess there's just not really much need for ESI to be defined across the domain of "all things". A similarity index of zero just means "not at all similar to Earth", which, yes, applies to the Sun and Jupiter, but also to a blob of tomato ketchup, or moonlight, or schadenfreude. Most things have an ESI of zero, then.
Jupiter is a "planet" in our parlance, though. But I guess that's just an accident of a bunch of very different things having the same range of brightness in our sky.
If you look at the formulation for the ESI [1], you'll see that the components are mean radius, bulk density, escape velocity and surface temperature (measured in Kelvins).
Mathematically you could zero out one or all of the multiplicands (e.g. with a surface temperature of 0 degrees K) but constraining oneself to the realm of the possible, you'll always have a non-zero ESI.
Mars is at 0.697, for reference.
https://en.wikipedia.org/wiki/Earth_Similarity_Index