The Family of Stars. Chapter 9. Guidepost.
Science is based on measurement, but measurement in astronomy is very difficult. Even with the powerful modern telescopes described in Chapter 6, it is impossible to measure directly simple parameters such as the diameter of a star. This chapter shows how we can use the simple observations that are possible, combined with the basic laws of physics, to discover the properties of stars.
With this chapter, we leave our sun behind and begin our study of the billions of stars that dot the sky. In a sense, the star is the basic building block of the universe. If we hope to understand what the universe is, what our sun is, what our Earth is, and what we are, we must understand the stars.
In this chapter we will find out what stars are like. In the chapters that follow, we will trace the life stories of the stars from their births to their deaths.
I. Measuring the Distances to Stars
A. The Surveyor's Method
B. The Astronomer's Method
C. Proper Motion
II. Intrinsic Brightness
A. Brightness and Distance
B. Absolute Visual Magnitude
C. Calculating Absolute Visual Magnitude
III. The Diameters of Stars
A. Luminosity, Radius, and Temperature
B. The H-R Diagram
C. Giants, Supergiants, and Dwarfs
D. Luminosity Classification
E. Spectroscopic Parallax
IV. The Masses of Stars
A. Binary Stars in General
B. Calculating the Masses of Binary Stars
C. Visual Binary Systems
D. Spectroscopic Binary Systems
E. Eclipsing Binary Systems
V. A Survey of the Stars
A. Mass, Luminosity, and Density
B. Surveying the Stars
We already know how to determine a star’s
In this chapter, we will learn how we can determine its
and how all the different types of stars make up the big family of stars.
d in parsec (pc) p in arc seconds
Star appears slightly shifted from different positions of the Earth on its orbit
1 pc = 3.26 LY
The farther away the star is (larger d), the smaller the parallax angle p.
Nearest star, α Centauri, has a parallax of p = 0.76 arc seconds
d = 1/p = 1.3 pc = 4.3 LY
With ground-based telescopes, we can measure parallax p ≥ 0.02 arc sec, which is d ≤ 50 pc
This method does not work for stars farther away than 50 pc.
In addition to the periodic back-and-forth motion related to the trigonometric parallax, nearby stars also show continuous motions across the sky.
These are related to the actual motion of the stars throughout the Milky Way, and are called proper motion.
The more distant a light source is, the fainter it appears.
(SLIDESHOW MODE ONLY)
The flux received from the light is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d)
Both stars may appear equally bright, although star A is intrinsically much brighter than star B.
App. Magn. mV = 0.41
For a magnitude difference of 0.41 – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28
App. Magn. mV = 0.14
Rigel is appears 1.28 times brighter than Betelgeuse,
But Rigel is 1.6 times further away than Betelgeuse
Thus, Rigel is actually (intrinsically) 1.28*(1.6)2 = 3.3 times brighter than Betelgeuse.
To characterize a star’s intrinsic brightness, define Absolute Magnitude (MV):
Absolute Magnitude = Magnitude that a star would have if it were at a distance of 10 parsecs (pc).
Back to our example of Betelgeuse and Rigel:
Difference in absolute magnitudes: 6.8 – 5.5 = 1.3
Luminosity ratio = (2.512)1.3 = 3.3
If we know a star’s absolute magnitude, we can infer its distance by comparing absolute and apparent magnitudes:
mV – MV = -5 + 5 log10(d)
distance in units of parsec
We already know: flux increases with surface temperature (~ T4); hotter stars are brighter.
But brightness also increases with size.
Star B will be brighter than star A.
Absolute brightness is proportional to radius squared (L ~ R2).
Quantitatively: L = 4 π R2 σ T4
Surface flux due to a blackbody spectrum
Surface area of the star
Polaris (F7 star) has just about the same spectral type (and thus surface temperature) as our sun (G2 star), but it is 10,000 times intrinsically brighter than our sun.
Thus, Polaris is 100 times larger than the sun.
This means its luminosity is 1002 = 10,000 times more than the sun.
Stars have different temperatures, different luminosities, and different sizes.
To bring some order into that zoo of different types of stars: organize them in a diagram of
“Hertzsprung-Russell (HR) Diagram”
O B A F G K M Spectral type
It’s useful to compare an HR Diagram to a similar graph of cars with different weights and horsepower.
Most stars are found along the Main Sequence
“Giants” (and supergiants) are same temperature, but much brighter than main sequence stars.
Giants must be much larger than m.s. stars
Stars spend most of their active life time on the main sequence (m.s.)
Dwarfs are same temperature, but fainter and smaller than m.s. stars
10,000 times the sun’s radius
100 times the sun’s radius
As large as the sun
100 times smaller than the sun
Ia Bright Supergiants
II Bright Giants
V Main-Sequence Stars
Pressure and density in the atmospheres of giants are lower than in main sequence stars, so:
• Absorption lines in spectra of giants and supergiants are narrower than in main sequence stars
• From the line widths, we can estimate the size and luminosity of a star.
• Distance estimate (spectroscopic “parallax”) is found using spectral type, luminosity class and apparent magnitude
More than 50 % of all stars in our Milky Way are not single stars, but belong to binaries:
Pairs or multiple systems of stars which orbit their common center of mass.
If we can measure and understand their orbital motion, we can estimate the stellarmasses.
center of mass = balance point of the system.
Both masses equal => center of mass is in the middle, rA = rB.
The more unequal the masses are, the more it shifts toward the more massive star.
(SLIDESHOW MODE ONLY)
RecallKepler’s 3rd Law:
Py2 = aAU3
Valid for the Solar system: star with 1 solar mass in the center.
We find almost the same law for binary stars with masses MA and MB different from 1 solar mass:
MA + MB =
(MA and MB in units of solar masses)
a) Binary system with period of P = 32 years and separation of a = 16 AU:
MA + MB = = 4 solar masses.
b) Any binary system with a combination of period P and separation a that obeys Kepler’s 3. Law must have a total mass of 1 solar mass.
The ideal case:
Both stars can be seen directly, and their separation and relative motion can be followed directly.
Usually, binary separation a can not be measured directly because the stars are too close to each other.
A limit on the separation and thus the masses can be inferred in the most common case: Spectroscopic Binaries
The approaching star produces blue shifted lines; the receding star produces red shifted lines in the spectrum.
Doppler shift Measurement of radial velocities
Estimate of separation a
Estimate of masses
Typical sequence of spectra from a spectroscopic binary system
Usually, inclination angle of binary systems is unknown uncertainty in mass estimates.
Here, we know that we are looking at the system edge-on!
Peculiar “double-dip” light curve
Example: VW Cephei
Algol in the constellation of Perseus
From the light curve of Algol, we can infer that the system contains two stars of very different surface temperature, orbiting in a slightly inclined plane.
The higher a star’s mass, the more luminous (brighter) it is:
Masses in units of solar masses
L ~ M3.5
High-mass stars have much shorter lives than low-mass stars:
tlife ~ M-2.5
Sun: ~ 10 billion yr.
10 Msun: ~ 30 million yr.
0.1 Msun: ~ 3 trillion yr.
Mmax ~ 50 - 100 solar masses
a) More massive clouds fragment into smaller pieces during star formation.
b) Very massive stars lose mass in strong stellar winds
Example: h Carinae: Binary system of a 60 Msun and 70 Msun star. Dramatic mass loss; major eruption in 1843 created double lobes.
Mmin = 0.08 Msun
At masses below 0.08 Msun, stellar progenitors do not get hot enough to ignite thermonuclear fusion.
Determine properties of all stars within a certain volume.
Fainter stars are hard to observe; we might be biased towards the more luminous stars.
Faint, red dwarfs (low mass) are the most common stars.
Bright, hot, blue main-sequence stars (high-mass) are very rare
Giants and supergiants are extremely rare.
stellar parallax (p)
absolute visual magnitude (Mv)
distance modulus (mv – Mv)
absolute bolometric magnitude
H–R (Hertzsprung–Russell) diagram
visual binary system
spectroscopic binary system
eclipsing binary system