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How much is it possible to measure from Earth and how accurately without sending probes to the planets? I am curious about how atmospheric boundary (altitude), atmospheric pressure, scale height were measured.

Also, when were the probes sent to do more precise atmosphere measures? Was it known where the atmospheric boundary is, or would the probe slowly lower it's orbit constantly detecting for presence of an atmosphere?

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The only alternative to sending probes is by measuring through remote sensing. Remote sensing is any measurement without actually "being there" (which includes what your eyes are doing!). For remote sensing of atmospheric properties, pretty much the only way¹ is by measuring electromagnetic radiation: either radiation emitted by the atmosphere, or radiation emitted by another source and then absorbed by the atmosphere. Detailed information can be obtained through spectroscopy.

Everything emits and absorbs electromagnetic radiation. Different molecules have very specific spectral signatures for their emission as a function of wavelength. This emission further depends on temperature, pressure, and some other factors.

Jupiter spectrum
Spectrum of Jupiter, from LASP

Then, we compare the observed spectrum with lab measurements for known species: H₂, H₂O, CH₄, CO₂, etc. We try to add up the different species contributions, until they match the observed spectrum. That is how we can determine the composition.

For more information, see this page at LASP.

Note that this is very commonly used not only for planetary sciences, but also in remote sensing of the Earth atmosphere, such as for weather and climate monitoring. Without atmospheric remote sensing, the daily weather forecast would not exist.


¹Theoretically, gravimetry can be used to get mass distributions, but this requires being very close to the object and then still, atmospheres are not massive enough. But we can measure Earth ocean currents!

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Wikipedia has a series of articles about the atmosphere of each of the major bodies in the Solar system; you can learn quite a bit about the history of investigation of each. For the moon, we observed no visual distortion of the surface with telescopes, so we expected no atmosphere, and confirmed it with probes (Ranger flyby in 1961, Surveyor landing in 1966).

Observation suggested Mercury had essentially no atmosphere, and Mariner 10 confirmed this (something like a hundred-trillionth of Earth's atmospheric pressure) in 1974.

The optically fuzzy edge of Venus hinted at a thick atmosphere as early as 1761; Mariner 2 came close to Venus in 1962 with a pair of radiometers, one measuring microwave radiation and the other infrared, and determined that the upper cloud cover was relatively cool while the surface was hot.

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Don't forget with Venus that there were also the Venera probes which provided direct measurements. –  Snowman Aug 13 at 4:48

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