# At what distance can I assume peculiar velocity is negligible compared to expansion velocity, to use Hubble's law?

I am working on a high school physics assignment and am trying to figure out a method to determine if the star is a giant or main sequence from its spectral and photometric data (from SDSS). I picked two stars of the same spectral class K1 a giant and a dwarf, and to compare their flux graphs I needed to fix the axis as the stars are not the same distance away. Here's the two graphs

Using hubble's law $$v=Hd$$, and that red shift is calculated by $$z=v/c$$ I attempted calculating the distance of the stars from their red shift then used the inverse square law to scale up the flux of the star that was further away based on its red shift. This happened to be the main sequence star as its red shift is about 8 times higher than the giant, but scaling up by what I calculated as 67.5 times (about 8 squared) the main sequence star would have much more flux than the giant based on the graph which didn't make sense to me. Thus, I am assuming that the expansion due to distance is not the major cause of red shift in this case but peculiar velocity, leading to this error. To overcome this I thought to pick stars so far away that peculiar velocity is negligible compared to expansion, but I don't know what range of red shifts to look for. It may be too late to change questions now but is this a correct way of determining whether a star is a giant or dwarf. Also is there any other ways. For example, I believe the width of absorption valleys should decrease as the star's size increases, due to decreasing pressure and rotational velocity, but looking at the two spectra I can't tell if there is a difference, especially when the vertical axes are not the same. What to look for?

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
May 31 at 12:04
• Hubble's constant is estimated to be roughly 70km/s per Mparsec. Peculiar velocities of stars in galaxies are on the order of hundreds of km/s and galaxies in clusters can have peculiar velocities of 1000km/s, so that requires distances of, at least, 100Mparsec. If your two stars are within the Milky Way, then Hubble flow does not matter to begin with, as both stars are gravitationally bound. I don't understand the rest of your question. Please clarify. May 31 at 12:26
• Thanks for your answer. The rest of my question is trying to differentiate between giants and main sequence stars based spectral features such as strength and width of absorption lines (because I couldn't do it using luminosity). But I don't know how strong or how wide these should be in dwarves compared to giants or what elements to look for. May 31 at 23:54

It depends how accurate you want your distances to be. Typically you might choose a limit that lies well beyond the local group (who's dynamics are not much influenced by the Hubble flow) and probably even beyond the Virgo cluster at 17 Mpc.

For instance at 17 Mpc, the Hubble flow velocity is about 1200 km/s but some of the peculiar velocities in a cluster like Virgo can be as high as $$\pm 1000$$ km/s.

Thus to be safe you need to be observing galaxies that are further away than this.

Note that Hubble's law does not apply to stars in our own Galaxy, such as those observed as part of the SDSS surveys.

Distinguishing dwarf and giants is best done by looking at the relative strengths and widths of some of the stronger metal absorption lines (sodium, magnesium, calcium). For stars at a given temperature the strong absorption lines will generally be weaker and narrower in giants because their surface gravities and pressures are lower, leading to less pressure broadening, lower opacities and weaker lines. The SDSS spectra are not good enough to compare the widths of the lines, but the differing strengths of those lines in dwarfs and giants should be apparent.

Here's the first example I came across when doing a quick search, from Majewski et al. (2000). The top spectrum is a giant and the bottom a dwarf with a similar temperature. Compare the strength of the Na and Mg lines. I have to say however, your spectra do not look like stars with markedly different surface gravities to me...

• Thanks for your answer. I had a suspicion that I wouldn't be able to get distance from SDSS, But how can you find the relative strength of an absorption line when you don't know the distance, because the distance affects the flux/luminosity of the stars we see Also where can I get information like which metal absorption lines should appear stronger in what type of stars. Thanks again May 31 at 23:50
• @HossamDahhan the strength of the lines means look at their depth relative to the continuum. It doesn't depend on distance at all. Which lines? The strong metal lines caused by sodium, magnesium and calcium. They are labelled in your plot. Jun 1 at 7:22

Quoted from Wikipedia - Hubble's law (version of April 2019):

Objects observed in deep space — extragalactic space, $$10$$ megaparsecs (Mpc) or more — are found to have a redshift, interpreted as a relative velocity away from Earth;

That means, that Hubble's law is not applicable for near galaxies (distance $$< 10$$ Mpc) and for stars within our own galaxy (the Milky way), because their peculiar velocity is not neglectable compared to the expansion velocity.