Download Chapter 11. Stellar Brightness, Magnitudes, the Distance

Chapter 11. Stellar Brightness, Magnitudes, the Distance
Modulus and the HR Diagram
Please read the Wikipedia articles linked on the ”Links” section of the Homepage, as
well. They will give you additional important insights into this chapter and the.
1.
The Magnitude Scale
The brightness of a star in physical units is its flux which has the units of Watts per
square meter in mks. However, astronomers almost never use flux to describe brightness of
objects, they use magnitude. Magnitudes have no units, as they are always on a relative
scale. We only give the magnitude of a star relative to another star (often a standard star).
If the flux we are referring to is the flux from a star measured at the Earth (f: technically,
just above the Earth’s atmosphere, so that the influence of the atmosphere in reducing the
brightness of a star is eliminated) then the associated magnitude is referred to by astronomers
as a star’s apparent magnitude (m). The relationship between magnitude and flux is:
m1 − m2 = −2.5log10
f 1
f2
Note that a difference in flux of a factor of 10 corresponds to a difference in magnitude
of 2.5 mag. Also note that brighter objects have smaller magnitudes. See the Wikipedia
article for some apparent magnitudes of common objects. The brightest star, Sirius, has an
apparent magnitude of about m = -1.4, while the faintest star visible with the naked eye on
the Wesleyan campus is about m = 4.5. Hubble Space Telescope can record objects as faint
as about m = 27.
2.
Stellar Distances by Parallax
An obvious factor in determining m for a star is its distance. The closer a star is to
the Earth, the brighter it will appear. For relatively nearby stars we can determine their
distance using the parallax method. Recall that one requires two vantage points to observe
a nearby object and a set of background objects against which to measure the apparent shift
in position as one shifts vantage points. In the case of the distance to Mars, the two vantage
points were in opposite hemispheres on Earth. In the case of stars, they are too far away
to show any substantial shift in position when viewed from any two locations on Earth –
we need a longer baseline. The longer baseline is provided by the orbit of the Earth around
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the Sun. If we view the same star field six months apart in time, then we view from two
different points in the Earth’s orbit separated by approximately the diameter of the orbit
(not exactly, of course, because the orbit is elliptical – but lets ignore that small effect here).
The diameter of the Earth’s orbit is, of course, approximately 2 AU.
As in the case of Mars, there is a shift angle that the star moves through. The observed
shift angle can be converted to the parallax angle (p) that a star would shift through as we
shift our vantage point by precisely 1 AU. Since p is always a very small angle (even for the
closest star it is only 0.7 arc-seconds) we can use the small angle approximation to relate
the distance to the star (d) to p. If p is in radians, then
p=
1AU
d(inAU)
Usually, we prefer to measure p in arc-seconds. Since there are 206,265 arc-seconds in one
radian, astronomers define a new and more useful unit for measuring stellar distances, namely
the parsec (pc) as 206,265 AU. With this definition, the parallax equation becomes:
p(in arcseconds) =
1
d(in pc)
Note that distance measurement by parallax only works for stars that are quite close
to the Sun. More distant stars than about 100 pc have parallax angles so small they just
cannot be measured even by modern techniques. Astronomers use the Kiloparsec (Kpc) and
megaparsec (Mpc) to measure even larger distances in the Universe. The size of the Galaxy
is about 20 Kpc, while the size of the Universe is about 4600 Mpc.
Obviously it is only a tiny fraction of all stars out there for which we can get direct
distances by parallax – only those stars within about 100 pc. To get distances to other
stars, we use the fact that there is a one-to-one correlation between a star’s luminosity and
its spectral type, as described in the last chapter. If we can get the distance to at least
one star of each spectral type, then we can calibrate its luminosity and assume that all
other stars in the Universe, regardless of their distance, would have the same luminosity
if they are the same spectral class. (Note that I am simplifying this a little – there could
be other factors besides temperature and radius affecting a spectral class, such as chemical
composition, rotation, etc. But, in general, the other factors do not vary much and/or have
little influence on a star’s spectral appearance in the relevant wavelength range.) To see how
all of this works in more detail, we need to consider a different kind of magnitude – the kind
that relates directly to the luminosity of the star and is independent of its distance from us.
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3.
Absolute Magnitudes
The luminosity (L) of a star, which has the units of Watts (in mks), is a measure of the
intrinsic brightness of a star – i.e. how bright it truly is, as opposed to how bright it appears
from Earth. Astronomers use the quantity Absolute Magnitude (M) to describe this. They
further have agreed on the convention that M is defined to be the apparent magnitude (m)
that a star would have if it were at a distance of 10 pc. I am not really certain why they
chose 10 pc as opposed to 1 pc or any other standard distance to define M, but that is what
they did (in the 1920’s I believe) and that is what we have lived with ever since. So – get
used to it! Absolute magnitudes of stars range from about -7 for the brightest supergiants
out there, to about +20 for the very faintest little red dwarfs that we find. The absolute
magnitude of the Sun is near +5. The concept can be extended to talk about the absolute
magnitude of whole galaxies and of quasars and other astronomical stuff.
4.
Distance Modulus
Clearly there must be a relationship between m, M and d and this can be derived from
the inverse square law if we neglect any dimming of a star that might occur due to interstellar
absorption (i.e. “fog”). Assuming that the only factor causing a star to dim as it moves
away from us is the geometric effect of spreading its light over an increasingly large sphere,
we can write that
L
f=
4πd2
where L is the luminosity of the star, d is its distance and f is its flux measured at Earth.
This is a statement of the so-called inverse square law for light, that the flux of an object
changes as the inverse square of its distance. Using the definition of M as m (at 10 pc) and
employing the magnitude equation, we may write:
m − M = −2.5log10
f
f (at 10 pc)
Now, using the inverse square law, we have
102
f
= 2
f (at 10 pc)
d
so,
m − M = 5log10 d − 5
where we have employed some well-known properties of logs, as given in class.
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The quantity m - M is known by astronomers as the distance modulus. Clearly, if we
know it, we know the distance to the star. In practice, we always can measure m for a
star, since this is just dependent on its flux at Earth (where we live). If we can measure
d for at least one star of a particular spectral type (e.g. G5V) then we can use the above
equation to get M for that star and, by extension, for all stars of its spectral type. Once we
have calibrated the spectral type - M relation for stars of all spectral types, we can get the
distance to any star using the same equation, just by measuring its apparent magnitude and
its spectral type.
5.
Correction for Interstellar Absorption (Extinction)
In reality, space is not quite perfectly clear of any “fog”. There are tiny grains that are
produced in stellar outflows from novae, and supernovae and some stars and these permeate
space creating a kind of fog that is more intense in some directions than others. These grains
are largely confined to the plane of disk galaxies and within our Milky Way galaxy tend to
block our view along the plane of the galaxy, but not perpendicular to the plane of the
galaxy. Hence, if we look outside the Milky Way we can see to great distances because there
is very little obscuration due to these dust grains in those directions. When we look along
the Milky Way our view is much more obstructed and limited. The absorption depends
heavily on wavelength. At optical wavelengths we cannot see even to the center of our own
galaxy (about 8 kpc away) because of this effect. In the infrared we can see the center
because the dust obscuration is much less. Longer wavelengths are much less affected than
shorter wavelengths, just like scattering in the Earth’s atmosphere (it is the bluer, shorter
wavelength light that gets scattered most by our atmosphere and by the interstellar dust
grains).
Correcting for interstellar absorption (or, extinction, is the better term, because it
encompasses both absorption of light and scattering of light, both of which diminish a star’s
apparent brightness) is difficult. Formally, we add an extinction term to the distance modulus
equation. This is usually represented as the letter “A” (for absorption) and defined to be
the amount of extinction affecting the star’s light, in magnitudes. With this definition, the
distance modulus formula becomes:
m − M = 5log10 d − 5 + A
In practice, determining what value A has for any particular star or galaxy can be challenging.
We will discuss this more a little later. Here I note that we generally use the fact that
extinction depends on wavelength to calibrate the total amount of extinction. A star suffering
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from a great deal of extinction will look redder – its longer wavelength radiation penetrates
to us more effectively than its shorter wavelength radiation. If we know the spectral type of
a star then we know its surface temperature and we know what its “intrinsic color” should
be. For example a star like the Sun should appear yellow, putting out most of its light
near 6000 angstroms. If we see a star which appears redder than it should given its spectral
type we can identify that as an effect of interstellar dust and use its degree of reddening to
calibrate the amount of extinction the star suffers. In practice, as I say, this can be tricky
and often knowledge of how much interstellar extinction is affecting any particular object’s
light is hard to determine.
6.
The Hertzsprung-Russell (H-R) Diagram
This is a plot of absolute magnitude (or Luminosity) of a star versus its spectral class (or
Temperature or Color) first employed by the astronomers whom it is named after. Examples
were shown in class and can be seen on the Wikipedia links. Arguably, this is the most
important diagram in astronomy, since it allows one to characterize every (normal) kind of
star we find in the Universe. The main features are described in the Wikipedia reading but
are quickly summarized here. There is a main sequence of stars that cuts diagonally across
the diagram from upper left (hot and bright) to lower right (cool and dim). [Note that in
typical astronomical fashion, the diagram does not follow usual conventions of science in
having smaller values to the left and larger values to the right – the temperature sequence
has hotter to the left and cooler to the right!]. The main sequence is where we find the
vast majority of stars, including the Sun. The luminosity class of main sequence stars is
class V, or “dwarfs”. We will see later that the reason this sequence is so well populated is
that it represents the locus of stars that are burning Hydrogen in their cores. This is the
longest-lived phase of stellar evolution. Once a star exhausts its Hydrogen it evolves through
other phases quite rapidly, hence we do not find many stars in these other phases.
The second-most populated region in the H-R diagram is the so-called red giants. These
are class III (giant) stars. Most giants are of K or M spectral class, indicating relatively cool
atmospheres. They are quite luminous, in spite of being cool, because their radii are truly
enormous – a hundred times larger than the Sun’s, or even more. Again, we will discuss the
nature of these stars more when we deal with stellar evolution theory, but we do know that
they represent stars that have exhausted hydrogen in their cores but still shine by hydrogen
burning in a shell around the (helium-rich) core. When a star leaves the main sequence this
is where it heads on the H-R diagram and this is the next most long-lived phase of stellar
evolution for a star, which is why it is fairly well populated.
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Other regions where we find stars on the H-R diagram (although not that many) are
among the supergiants, which are the more massive examples of stellar evolution, and among
the so-called “white dwarfs”. Note that a white dwarf is not a class V star. These are much
less luminous than the class V (dwarf) stars that populate the main sequence. The term
“white dwarf” is always used with both words together, indicating that we are talking about
something totally different than class V, dwarf, stars. White dwarfs are the burnt-out cores
of previous main sequence stars after they have gone through the red giant phase. They are
extremely hot only because they were once the core of the star and have not yet cooled off
substantially. We will discuss them in greater detail later. They are very small, with radii
approximately equal to the radius of the Earth!
Masses of stars vary systematically across the H-R diagram. Along the main sequence
(class V stars) the earlier the spectral class the more massive the star. [Note: early type
refers to hotter stars like O or B-type, while later type refers to cooler stars like K or M.
These terms were originally thought to relate to the ages of stars as well, or when they
formed, but we now know this is not true. Nonetheless, the terms “early type” and “late
type” continue to be widely used in astronomy today.] O-type stars have masses up to
50 solar masses or perhaps a little larger. M-type stars have masses down below 0.1 solar
masses. The smallest mass object that can start its nuclear fusion of Hydrogen is thought
to be about 0.07 solar masses, so this represents the smallest stars we see. These are of class
M. We do find cooler objects in space that are classified as L or T type objects but these
are not stars, per se, they are called “brown dwarfs”. They shine only by the heat generated
through gravitational collapse. They are extremely low luminosity and hard to find!
If we took a small volume of the galaxy around the Sun, say the nearest 20 pc or so,
and plotted every star in that volume on the H-R diagram, we would find that almost all
of them were low luminosity, low temperature stars of class K or M. M-type dwarf stars are
the most abundant stars in the Universe by far. There are roughly 100 million M stars for
every O star! Most of the mass in stars is in the form of these low luminosity, low mass
stars, typically having masses of about 0.1 solar mass. On the other hand, if we plotted the
brightest stars (by apparent magnitude) we would find that most of these stars are more
massive than the Sun. This tells us that while the mass of our galaxy is primarily in the
form of low mass stars, the light is primarily coming from high mass stars (O and B stars as
well as K and M giants and supergiants). This turns out to be true of galaxies in general.
The light of a galaxy is like the tip of an iceberg, showing us only the tiny fraction of stars
that are exceedingly luminous because they are quite massive. The real mass of a galaxy
resides in its M dwarfs, which even collectively do not produce that much light.
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7.
Standard Magnitudes and Colors
The definition of apparent magnitude (m) given so far has not included one important aspect – namely the wavelength range over which the flux (f) is measured. Since stars
have different colors they can have different values of f depending on what range is chosen.
Astronomers define a set of “standard” magnitudes that are measured over agreed upon,
conventional, wavelength ranges using a set of “standard” stars as the flux standards. The
most common standard system is known as the UBVRI system and has five separate wavelength regions over which flux from a star is measured. In practice this is done by using
colored glass filters in front of the instrument measuring the flux. In the UBVRI system,
the width of the band passes of those filters are relatively broad, with widths of about 1000
angstroms or more. The five bands are centered on five different wavelengths and named
according to the spectral region that they sample:
U is for Ultraviolet, and the bandpass is center is around 3600 angstroms
B is for Blue, and the bandpass is center is around 4400 angstroms
V is for Visual, and the bandpass is center is around 5500 angstroms
R is for Red, and the bandpass is center is around 6700 angstroms
I is for Infrared, and the bandpass is center is around 7900 angstroms
We refer to the apparent magnitude of a star in the V bandpass as mV or, more commonly, just its V magnitude. Similarly we can speak of the B magnitude of a star as its
apparent magnitude in the B bandpass.
In practice, these standard magnitudes are determined for stars by observing them with
telescopes, using the standard filter sets and comparing brightnesses of stars you wish to
measure with standard stars whose magnitudes are agreed upon by convention. Astronomers
have set up a whole lot of standard stars spread around the sky so that at least some of
them can be observed at any time from any location. In making these observations, we
must also correct for the effect of the Earth’s atmosphere. This is done by the astronomer
before reporting the standard magnitude. Standard magnitudes, therefore, are really the
brightness in the appropriate band pass that one would observe from just above the Earth’s
atmosphere.
I note in passing that it is sometimes desirable to talk about the flux from a star integrated over ALL wavelengths. This is called the “bolometric” magnitude of a star and
often designated as mbol . It can be very difficult to determine mbol , in part because many
wavelengths of light are totally blocked by the Earth’s atmosphere and can only be accessed
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through satellite or balloon measurements. In practice, astronomers have calibrated a “bolometric correction” for stars of a given spectral type that indicates how much light is falling
outside of the standard band (e.g. V) which can be measured more easily. We will not deal
with bolometric magnitudes any further in this class, but I do note that they are important
for determining the true luminosity of a star, since stars can emit much of their energy at
wavelengths other than just UBVRI.
Using the definition of the magnitude scale, we can write the color of a star as its
difference in magnitudes through two of the standard bands. By convention, we usually take
the bluer bandpass first and subtract a redder bandpass, producing colors such as B-V and
V-R and R-I. The definition of B-V is just
B − V = −2.5log10
fB
fV
where fB is the flux measured in the B bandpass and similarly for V and for the other colors.
Hence a color is a measure of the ratio of the fluxes at different wavelengths. The B-V of
bluer stars is more negative than the B-V of cooler stars. O stars have B-V = -0.3 or so,
while the Sun, a G star, has B-V = +0.6 and M stars have B-V = +2.0 or so. If interstellar
extinction effects are unimportant, then color simply depends on the surface temperature of
the star and so it can be used in place of spectral class as the x-axis on the H-R diagram.
When this is done, the resulting plot of, say, absolute magnitude in the V-band (designated
MV ) versus B - V is referred to as a Color-Magnitude Diagram. It is equivalent to an H-R
diagram.
Finally, we note that standard colors quantify color measurement and allow us to correct
for interstellar extinction in a fairly precise way. In particular, we can define the “color
excess” (e.g. E(B-V)) of a star as
E(B − V ) = (B − V ) − (B − V )0
where B-V is the measured color of a star and (B-V)0 is the color appropriate to the star’s
spectral type – i.e. the color it would have if it were unaffected by interstellar absorption.
We determine (B-V)0 for a star from its spectral type, having calibrated it with nearby stars
that we believe are unaffected by absorption. Then we can compute E(B-V) for any star
from its measured color. Studies throughout the galaxy have generally found that the grain
properties are quite similar in most (but not all) directions and so we can calculate the
amount of total extinction (AV ) in some standard band (usually V) as
AV = 3.1E(B − V )
This is the value of A that then gets used in the distance modulus equation to correct for
the “fog” of interstellar space.