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From reading this question on calculating distance to stars and from a bit of background reading on the standard candle theory I still don't see how we can confirm that a star we see at one distance and a star we see at another distance (redshifted and at a different apparent luminosity) are the same type.

Beyond a certain distance, parallax is not possible, so how can we confirm star X and star Y are the same type therefore the difference in redshift and luminosity means Y must be at distance Z. How do we know it isn't a different type of star at a different distance?

Or are we just saying they look similar, so probably are?

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We determine the spectral type (i.e. temperature) of a star using multicolour photometry, or (ideally) spectroscopy. By guesstimating the temperature, mass and radius of a star, we can say that two stars that have pretty similar observational properties probably are closely related to each other.

Cepheid variables, for example, display periodic pulsations that depend quite strongly on their intrinsic luminosity -- this is why they're reasonably good standard candles. Their characteristic variability makes them clearly identifiable as a Cepheid, and observations of their pulsation are backed up by data that place them in the same region of the HR diagram.

If it looks like a duck, swims like a duck, and quacks like a duck, then chances are it's a duck. If it's as hot as a Cepheid and pulsates like a Cepheid, then it probably is a Cepheid.

So to answer your final paragraph: yep. Though with no spatial resolution and often (in the absence of parallax measurements) only tenuous distance data, classifying stars is a rather messy business.

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I would say that the redshift in the spectrum plays a big role. The absorption lines are kind of like a fingerprint for the elements in the star. If those are shifted, we can calculate the redshift, and hence the distance. –  Arne Nov 17 '13 at 22:03
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This is not generally true for stars. At the distances where Hubble's constant is our most effective measurement of distance, individual stars are too dim to resolve - we just see whole galaxies. We can really only resolve stars like Cepheids within our Local Group of galaxies - and within our galactic neighbourhood, the other galaxies move independently from Hubble's constant. –  Moriarty Nov 17 '13 at 22:37
    
Interesting. So basically we are mostly stuck with parallax measurements in our own galaxy? –  Arne Nov 18 '13 at 6:50
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@Arne Parallax is generally most accurate, though all other measurements I can think of should work too (except using redshifts). It's just the fact that galaxies in a cluster tend to move about relative to each other that means we can't use Hubble's constant for nearby objects. The relative movement of these nearby objects (i.e., Andromeda is actually moving towards us) "overpowers" the comparatively small effects due to the expansion of space. –  Moriarty Nov 18 '13 at 10:48

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