Download lect3 — 1 Measuring stars: What can be measured?

lect3 — 1
Measuring stars: What can be measured?
- Position: RA & DEC (astrometry).
- Distance: Difficult! See below.
- Velocity: Doppler shift of spectral features.
- Temperature: Estimate from λmax .
- Composition: From strength of absorption lines.
- Mass: Only if the star is in a binary pair; otherwise estimate from T .
Distance for nearby objects can be measured via parallax: Look at the angular change in the
apparent position of the object over a 6-month baseline (θ). The earth moves 2 AU, so then
tanθ = 1A.U./D ≈ θ
(using the small-angle approximation).
An angle of 1 arcsecond then corresponds to D ≡ 3.086 × 1018 cm. This is known as a parsec
(PARallax arcSECond).
Unfortunately, this only works for quite nearby objects. The upcoming ESA mission GAIA
(launching next year) will extend the reach of parallax greatly, but it will still only be for
things mostly within our Galaxy.
Most remaining distance indicators rely on finding the brightness of the object, and by some
other means knowing how intrinsically luminous it is. So let us introduce magnitudes.
Magnitudes: Astronomers often use magnitudes to express absolute or apparent flux. It is a
logarithmic brightness, defined as
F1
m2 − m1 = −2.5 log
F2
To set the physical scale, one must therefore define the flux (in e.g. Jy) of a 0-magnitude
star.
A historically popular system is Johnson or “Vega” magnitudes, where the bright star Vega
is defined to be 0 magnitude. A more recent system is called AB magnitudes, where 0
magnitude is always 3631 Jy, which translates to
mAB = −2.5 log F − 48.6
Note that magnitudes are negative log of the flux. This means that bright objects have small
magnitudes, while faint objects have large magnitudes. This is a historical artifact from when
astronomers classified stars in the night sky by a system of ”first brightest, second brightest,
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...”. It is unfortunately hard to rid astronomy of this system and just go to normal units
like Jy; many subfields of astronomy have already done so, but optical and near-infrared
astronomers in particular remain stubbornly tied to magnitudes.
Filters: Typically, one observes the flux in a particular filter. This is a piece of optics on the
telescope that only transmits a certain range in wavelengths. A common set of filters is
UVBRI, shown below.
One measures the magnitude in a particular filter. In Johnson magnitudes, the 0-mag flux in
the U band is 1880 Jy, while in B it is 4400 Jy; other filters have different values. In AB
magnitudes it is always 3631 Jy, which seems more sensible (even setting aside the fact that
Vega is now known to be a slightly variable star!). For optical bands, Johnson and AB
magnitudes are (by construction) similar. But for other bands e.g. infrared or ultraviolet,
they can be quite different. So when someone quotes you a magnitude, it pays to ask which
system!
Absolute magnitude: In another historical quirk, we define the absolute magnitude as the
magnitude a given star (or object) would have it were placeed at 10 pc. Effectively, since
we have a flux and a distance, an absolute magnitude can be directly translated into a luminosity. Just remember that larger absolute magnitude means a lower luminosity! Absolute
magnitudes are canonically written with Capital letters (MB ), whereas apparent magnitudes
are lower case (mB ).
Colors: You eye can tell color, but how does an astronomer quantify it? By the difference in
magnitudes between two bands!
So a “color” is often taken as e.g. “B − R”, which means
B − R = −2.5 log fB + 2.5 log fR
Something that is blue will have a high fB relative to fR , and hence its color will be low
(remember magnitudes have the opposite sign).
Note that colors are independent of distance, i.e. whether you are discussing apparent or absolute
magnitudes.
Traditionally, colors are always taken to be the bluer band minus the redder band. Hence a
smaller value of color typically means a bluer object. For instance, in the Johnson system,
Vega has (by definition) B − V = 0, and it is a massive blue star; the Sun has B − V = 0.66,
which makes it redder than Vega.
The virtue of this definition is that we can extend the idea of color to beyond the visible
spectrum, to include infrared bands (yJHK) and so on.
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Distance modulus: The ratio of the apparent flux to the absolute flux is a measure of distance.
In logarithms, ratios are just differences, so we can define a distance modulus as
m − M = −2.5 log
F
F10pc
!
If the distance D is in parsec, then
m − M = −2.5 log(102 /D2 ) = 5 log D − 5
(recalling that Observed flux = Luminosity /(4πD2 )).
Hence by measuring the apparent magnitude m, and somehow knowing the absolute magnitude
M (i.e. the true luminosity), one can infer the distance D (in pc).
H-R diagram distance: We discussed the H-R diagram last time as a relation between temperature and luminosity. In observational terms, it is actually a color-magnitude diagram;
color corresponds to temperature, while magnitude corresponds to luminosity.
For a main sequence star, M can be estimated based on the color. Then by measuring the
apparent magnitude, one can infer the distance. One has to be certain one is looking at a
main sequence star though, and not a giant. Again, this has only limited applicability since
MS stars are only individually visible well within our Galaxy.
Standard candles: This generalizes to the idea of a standard candle, which is an object
for which you can determine the intrinsic luminosity (or M ) without knowing its distance.
Standard candles are hugely important in astronomy, since obtaining the distance to most
objects in the night sky is always a challenge.
While main sequence stars are decent enough standard candles, it is difficult to see a typical
main sequence star very far away; they are just not that bright. Worse, the brightest (OB)
stars tend to be somewhat variable, and their color doesn’t translate into luminosity quite
so tightly as for something like the Sun.
Cepheids: Classically, this is the most reliable stellar standard candle. This type of giant star
is seen to have regular pulsations, and the pulsation period turns out to be quite tightly
tied to its intrinsic luminosity. In addition to being a giant, these typically have masses
∼ 5 − 20M⊙ , which means they are very bright indeed, and can be seen far away, even in
other galaxies!
The pulsations arise because of Helium in its outer envelope. As the star contracts, its outer
layer becomes more opaque, which catches more of the radiation, thereby puffing it out. Once
it gets too puffy, the helium becomes more neutral and the opacity drops, which causes it
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to contract via gravity. It is essentially like a harmonic oscillator whose restoring force, and
hence period, depends on its luminosity.
The period-luminosity relation is calibrated using very nearby Cepheids for which we have
highly reliable parallax distances. Then, for more distant Cepheids, this P-L relation is used
to obtain M and hence the distance. Much work has gone into (and continues to go into)
accurately calibrating the P-L relation in various bands.
RR Lyrae stars are another type of regularly pulsating giant, but they are typically older and
smaller (and hence fainter) than Cepheids, and also rarer. Nonetheless, their P-L relation
has less uncertainty, so they are often preferred when they can be seen.
Finally, an incredibly bright and incredibly useful (for cosmology) standard candle is a particular
type of supernova called a Type Ia. Indeed, these were used to definitively identify the
accelerating expansion of the Universe. We will discuss these later.
All these concepts like magnitudes, colors, distance modulus, etc were historically developed for
stars, but today they are also still in wide use for galaxies and extragalactic objects.