the calculations will look like a bunch of numbers and mathematical symbols, of course.
Astronomers use spectroscopic analysis of the light from the Sun, and from distant stars, to learn about them. Different elements and compunds emit different wavelengths of light, both visible and non visible to human eyes, when they are at different temperatures. And different elements and compounds absorb different wavelengths of light at different compounds. Scientists found that out by studying the scetra of various substances at various temperatures in laboratories.
A famous 19th century scientists once said that it was totally impossible for man to ever learn the chemical composition of the stars. In 1860 Bunsen and Kirchfof deomonstated that spectroscopy could be used to detect elements and that the absorbtion lines in the Sun's spectrum were due to specific elements.
In the 1860s William and Margaret Huggins demonstrated that stars were made of the same substances as the Earth.
Meanwhile, astronomers finally managed to measure the parallaxes, and thus the distances, of three stars during the 1830s, and more followed in later decades. Advances made it possible to measure the distances to more and more stars more and more accurately.
Many stars are double stars, two stars revolving around each other. If the distance to the double star is known, the angular distance between the two stars can be converted into an absolute distance between them, and the time it takes them to complete a revolution can then be used to calculate their masses.
Astronomers began classifying stars by their spectra, and at the present time most stars have classifications in sequence from hotter to colloer (surfaces) of O, B, A. F, G, K, and M. Each letter class is also divided into 10 numerical classes from 0 to 9.
Stars of the same spectral type can have different luminosities depending on which stage of stellar life cycle they are in, and are classed with Roman numerals from Ia for bright supergiants down to VII for white dwarfs. If a star's luminosity class can be found from its spectrum, the distance can be calculated from its apparent brightness and from from the known luminosity of other stars with the same classification and known distance.
If astronomers know what spectral class a star has, they know its temperature. If they know its temperature, they know how much total light in all frequencies it emits from a unit of its surface area. If they know how much light Earth receives from that star, and know how distant it is, they can calculate how luminous it is. If they know how luminous a star is, and how much light is emitted from each unit of its surface area, they can calculate its surface area and thus its diameter and volume.
If astronomers know the mass and the volume of a star, they can calculate its average density.
The spectrum of a star can also be used to measure its motion toward or away from Earth by the Doppler effect. It can also be used to measure the rotation rate of star. Study of small motions of a star deduced from its spectrum can demonstrate the existance of unseen companions - faint stars and even large planets.
From 1989 to 1993 the Hipparcos satellite measured the positions and distances of hundreds of thousands of stars much more accurately than ever before. The Gaia space observatory has been measureing distances to stars with even greater accuracy since 2013 and has measured the distances of millions of stars, perhaps a billion in total.
So astronomers now have much more accurate measurements of the distances to stars of various spectral types, and thus of their luminosities, diameters, etc. to compare to other stars of the same spectral classification.
You can find a lot of online information about about the properties of stars of various classifications. For example, the Wikipedia article Stellar Classification has a section about each of the main spectral types.
For each spectral type, it has link to a main article about that spectral type. And each of those main articles about a spectral type has a table giving the mass, diameter, luminosity, etc. of the subdivisions of that spectral class.
Of course there is no hard and fast line between each subdivision of a spectral class, and their masses and luminosities are on a continuous spectrum, but those figures should be good approximations for the average star within each class subdivision.
These tables refer only to main sequence stars of those spectral classifications, luminosity class V, and not to the other luminosity classes of varius giants, sub giants, supergiants, etc. which have the same spectral class but different luminosity classification.
User177107 answered the question:
How would the characteristics of a habitable planet change with stars of different spectral types?
His answer has a table giving the characteristics of various classes of stars, including their mass, luminosity, etc. It includes the distance at which a planet would have to orbit to receive exactly as much heat and light from its star as Earth gets from the Sun, what I call the Earth Equivalent Distance (EED).
Much of the data comes from this source:
If all three of those sources agree on the mass or luminosity or other trait of a star of a specific spectral type such as A9V or K3V, you can probably assume they are correct.
I wonoder if you are considering the possibility of a habitable planet in another solar system.
If so, you should recognized that an enviroment habitable for some types of life can be ldeadly for other types of life.
For example, the biosphere of the planet Earth, where various lifeforms are found, is not confined to the surface of the Earth, but extends kilometers upwards into the atmosphere, kilometers deep under the surface of the ocean, and even kilometers deep inside solid rock.
But an unprotected and unprepared human teleported to a random space within Earth's biosphere would swiftly die if they materialized kilometers high in the sky or kilometers below the surface.
Earth is full of lifeforms which fourish where humans need protection to live.
When scientists discuss the possbility of habitable worlds they usually discuss the possibility of worlds habitable for liquid water using lifeforms in general. Worlds habitable for human beings, or for multicelled land animals or intelligent aliens with similar requirement as humans, are a much smaller subset of habitable worlds in general.
The circumstella rhabitable zone of a star is the range of distances at which water can be liquid on the surface a planet, and thus liquid water using life in general could exist on planets also suitable in other ways.
And the obvious way to find the circumstellar habitable zone of a star is compare its lumnosity to that of the Sun. Then take the inner and outer limits of the Sun's circumstellar habitable zone and adjust them for the ratio between the luminosities of the two stars.
Unfortunately there is not much agreement on the size of the Sun's circumstellar habitable zone.
The table here includes wide variations in the inner and outer limits of the Sun's circumstellar habitable zone.
One way to be certain a planet is within the circumstellar habitable zone of its star is to put it at the Earth Equivalent Distance, or EED, of the star, since Earth is certainly within the Sun's circumstellar habitable zone.
The only scientific discussion of planets habitable for humans in particular, and not liquid water using life in general, that I know of is Habitable Planets for Man, Stephen H. Dole, 1964.
Dole decided that only main sequence stars (luminosity clas V) could have planets habitable for humans (or lifeforms with similar reqirments) and only some of them, from spectral class F2 downward.
Your question mentioned Mirach, or Beta Andromedae as an example.
Beta Andromedae/Mirach is a class MIII star, Which makes it a red giant stars.
When stars reach the end of thier main sequenc epriod sof shining with slowingincreeasing luminosity, they swell up and become much more luminousm. That makes the planets formerly in their habitable zones much too hot for life. The increased luminosoity may move the habitable zone out to the orbits of formerly frozen planets and make them warm enough for life. Those newly warm enough palanets may a have steady temperatures for tens or hundreds of millions of years, a billion years in some cases. That should be enough for primitive lifeformse to develop but not nearly long enough for a planet to become habitable for humans or for advanced lifeforms with similar requirements.
After a comparatively short span as a red giant, a star will go though rather drastic changes, which dependon the mass it started with, and which might destroy some or all of its planets, before becoming a white dwarf, a neutron star, or a black hole.
Stars usually become white dwarfs, and all such stellar remnants are much less luminous than the original main sequence stars were. A white dwarf could be luminous enough to have planets within its tiny habitable zone, and keep them f wrm for billions ofyears. But those planets would have to be so close that they should have been destroyed as the star went through drascic changes to become a white dwarf. Unless possibly some process caused those hypothetical planets to migrate inward toward the star.
So Beta Andromedae/Mirach is a red giant. Possibly some of the planets or moons in its pesent circumstellar habitable zone might have primitive forms of life. But none of the the planets in its pesent circumstellar habitable zoneshould have been able to form an oxygen rich atmosphere suitable for humans or for life forms with similar requirements yet.
Of course it is psosible that one of the wworld currently in the present circumstellar habitable zone of Beta Andromedae/Mirach, by some one a million or one in a billion chance, has developed an oxygen rich atmosphere and become habitable for humans many times faster than usual.
And it is also possible that some advanced civilization might have terraformed a planet in the present circumstellar habitable zone of Beta Andromedae/Mirach, to make it habitable for a few million years.
The luminosity of Beta Andromedae/Mirach is about 1,995 tims that of the Sun. So the EED of Beta Andromedae/Mirach, should be 1 AU, the distance of earth from the Sun, multiplied by 44.665, the square root of 1,995. So any hypotetical planet in the EED of Beta Andromedae/Mirach should orbit at a distance of about 44.665 AU.
Since Beta Andromedae/Mirach has a radius of about 100 times the radius of the Sun, it would appear 100 divided by 44.665, or 2.2388895, times as wide as the Sun as seen from a planet orbiting at a distance of 44.665 AU.