Planet Measurements
If you could only know one measurement/quantity of a planet, what would you want it to be? And why?
For me, I would want to know the temperature. It could give clues to whether the planet has any type of atmosphere (if you compared the temperatures across the different regions... atmospheres can distribute heat more evenly) and how far it is from its star (if the type of star is known). And if the heat is not uniformly distributed, then we might conclude that the planet is tidally locked, which also gives clues about its mass and distance from the star.
Temperature is also a way to determine habitability (to life that we are familiar with... though there are some extremophilic exceptions (talking about you, tardigrades and others!)), being one of the parameters of the Earth Similarity Index.
Density is also important (but that's two measurements in itself), to figure out the planet's composition, which helps with classification (i.e. hot super-Earth vs hot Jupiter or mini Neptune, etc).
Orbital Period, though this might be a known depending on the planet discovery method in the first place. But I like being able to understand the dynamics of a system and this would be a first step towards that and it's a useful piece of information for calculating and observing more. Should be able to calculate the orbital radius and then the mass from that. Which will give an idea if the planet can support life (in range of the 'habitable zone'), does it have an atmosphere or is it too close/ small like Mercury and a little about more info about an expected temperature (though nothing precise at this point).
Now does observing the star count as more than one measurement, as you said for the planet so I assume its allowed? If so and there's a transit with its star, the orbital period would be able to identify which transits are related to the planet in question, this is possibly how the planet was discovered in the first place. But using this information and observing the brightness of the star and the relative drop when a transit happens would allow for a planet radius calculation, consequently the density too.
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