[Lewis Thomson (OP)]

Donald would like some clarification on this area of science.

"When measuring massive objects like planets or stars, a few km or a few hundreds of km is trivial.  However, as a matter of definition, what is measured.  Certainly for the Earth, the diameter is defined by the solid and liquid portions. But what boundaries are measured for gas planets that have no specific boundary, or the sun that has an unusual atmosphere that is continuously expanding/'ejected' as solar wind. Should the Earth's atmosphere be considered as part of it's diameter?"

Discuss in the comments below...

[Eternal Student]

Hi.

However, as a matter of definition, what is measured.

    Well, for stars there's a reasonable answer we can give you.
    Other than the sun, most stars are so far away that you can't really expect to measure their diameter by any ordinary observation through a telescope.   They are just going to be dots with almost no diameter at all.
    Instead, we assume a star is roughly spherical  (ball shaped) and behaves like a black-body (this doesn't mean it's black in colour just that it's a type of thing that radiates heat in the most natural way possible).
    We know a lot about blackbody radiation.   We know that a blackbody should have a certain luminosity and we know precisely what will determine that Luminosity.   Luminosity appears as a "brightness", we specify the Luminosity quantitatively by measuring the total power output of all the electromagnetic (e-m) radiation from the star.   Now this means we need to try and collect all the e-m radiation that strikes us (our telescope) here on earth from the star.  That's actually not too difficult because although the e-m radiation could span everything from microwave through to gamma ray frequencies, most of a stars radiation is roughly in the visible light spectrum, so if you just collect all of that then you've done a fair job of measuring the total power it radiates.   (We can make a fine correction anyway because we know a lot about blackbody radiation spectrums and we're going to estimate the temperature of the star anyway - but these are fine details that we can overlook for the moment).
    Anyway, the Luminosity we observe depends on the distance between us and the star, so we do need to be able to estimate that distance.  There's several ways that can be done, one easy method that works well for close stars is to use seasonal parallax measurements as planet earth orbits around our sun.   I'm going to leave all discussion of how to measure astronomical distances for the moment because that would be enough to fill a forum thread all on its own.  We'll just accept that the distance to the star can be estimated.
     Now, we're in a position where we have the Luminosity of the star.   We can then just apply what we know about black-body radiation.   The Luminosity, L,  should obey this relationship, which is called the Stefan-Boltzmann law:   
    L = (Surface Area of the body)  x  (a constant) x (Temperature of the star)⁴

The "constant" is just a number that we can find in a book, it's the Stefan-Boltzmann constant, it's the same value for any black-body.   We will need to estimate the temperature of the star, T.    We can do this because we know a lot about the emission spectrum of a black body.  We can compare the amount of red coloured light we are receiving from the star against the amount of blue light we are receiving and we can then estimate the temperature of the star quite well.   Red stars are cooler,  blue stars are hotter.
    Now we've got everything we needed for the Stefan-Boltzmann law and re can just re-arrange that equation to obtain the surface area of the star.   We assume it's spherical, so that surface area tells us precisely what the radius of the star must have been.

OK,   I know that's a fair amount of words.   Here's a summary:
1.   We can't really see the size of a star directly, fine details like a progressively thinning atmosphere and some solar flares being emitted just aren't going to bother us.
2.   We just assume a star is spherical.
2.   We assume it radiates energy like a black-body.
3.   We re-arrange some equations to determine its radius just from measurements on things like Luminosity.  So all we really obtain is the radius of an idealised spherical black body that emits energy in the same way as the star.

Here's quite a nice video that covers most of what was said, if you'd prefer the information in video format:

This post is already too long, so I'm going to leave any discussion of the measuring the size of planets for now.

Best Wishes.

[Eternal Student]

Hi again.

    So, let's do the easier bit next.
For planets that are in our solar system, we can see them in a telescope and they aren't just dots.
We can measure the diameter of the planet directly by observation.   Therefore, the earliest definition of where a planets edge or boundary was located would have been just where it looks like empty space through your telescope as opposed to some other coloured substance.
    For example the moon has a thin atmosphere but you can't really see that through a telescope, it's completely transparent, so it would not have been included as part of the moon when the radius of the moon was estimated.

But what boundaries are measured for gas planets that have no specific boundary

    As previously indicated, some decision is made that the region around the gas giant now looks like ordinary empty space.   This means that as telescopes improved and as we started to use e-m wavelengths outside the visible spectrum, the precise boundary of what you would call a planet like "Jupiter" probably has changed a little.  It's not too important.  We are aware that Jupiter's upper gas layer just thins out progressively instead of abruptly ending.

 

Should the Earth's atmosphere be considered as part of it's diameter?"

    If you like.  Philosophically it's a perfectly fair decision.   However, for right or wrong that isn't what is generally done in Science.   When we talk about the diameter of the earth we are all assuming that the diameter of the rocky and liquid part of the planet was being considered and that this was also probably being averaged as if the surface of the earth was perfectly spherical.   It's not very important if our existing notion of the diameter of the earth is the best definition,  all that's important is that we do all use the definition we have consistently and understand the same thing by the use of the term.

How do we estimate the size of a planet that isn't in our solar system?
    We have found some planets that aren't in our solar system, which means that once again they would look like dots in our telescopes with no direct method to measure their diameter.   However, it's even worse than that,  since they are so far away and they aren't stars, they aren't even all that bright.  We don't really have much hope of seeing them at all.
    Many of the exo-planets   (planets that are in another solar system)  were found by noticing that a stars luminosity is reduced while a planet passes in front of it, effectively blocking off some of the light from that star.    We can then try to estimate the size of that planet but at this point we really are just estimating a lot of things.   We have Kepler's laws of planetary motion, so we can estimate how far the planet is from the star that it is orbiting by the speed at which it clears out of the way of the star, if that's all we can do then we have to assume the orbit was circular.   If the orbit time is fairly small, we can directly measure how long it takes for the drop in Luminosity to be noticed again (i.e. we can measure the full orbit time of the planet, at least then we can assume the orbit was elliptical and not perfectly circular)*.   Using the information we have about the distance from the star to the planet and* the amount of light that was blocked out when the planet passed in front of the star, we can estimate the size of the planet.   So all we're really doing is estimating the size of the opaque (light blocking) part of the exo-planet and not the gaseous atmosphere that it might have.

OK.  I hope that's some useful information about measuring the size of astronomical bodies like stars and planets.

Best Wishes.

LATE EDITING:  * I rushed this, sorry.    We don't even need the distance from the exo-planet to the star it orbits,  it's all so far away we can assume all rays from the star to our telescope are parallel.    However, the orbital distance of the exo-planet is estimated for other purposes and can provide a more complete picture and a better model.
   Here's a reference that seems reasonable:  https://www.sfu.ca/colloquium/PDC_Top/astrobiology/discovering-exoplanets/calculating-exoplanet-properties.html