5.1THE FAMILY OF STARS

Star Distances

Recall, parallax is the apparent shift in the position of an object due to a change in the location of the observer. Must use a very long baseline the diameter of Earth’s

orbit (2 AU), opposite sides of Sun.

Since stars are so distant, their parallaxes are very small angles, usually expressed in arc seconds (“).

Stellar parallax (p) is half the total shift of the stars; the shift as seen over a 1 AU baseline.

Astronomers have defined a special unit of distance, the parsec (pc) ,for use in distance calculations. Distance to an imaginary star with a parallax of 1 arc

second. Equal to 2.06 x 105 AU, or 3.26 light years (LY).

Stellar Parallax Formula

d = 1/p

d = distance, in parsecs

p = parallax, in arc seconds

Star Distances

The star nearest to the Sun, Proxima Centauri, has a parallax of only 0.747”, and the more distant stars have a parallax of even smaller.

How far away is Proxima Centauri, given its parallax from above?

With ground-based telescopes, we can measure p > 0.02”, which is equivalent to d < 50 pc.

1.34 pc ~ 4.36 LY

Proper Motion

In addition to the periodic back-and-forth motion related to trigonometric parallax, nearby stars show continuous motions across the sky.

This is related to the actual slow motion of the stars throughout the Milky Way galaxy and is known as proper motion.

Brightness and Distance

The total amount of light a star emits is known as its intrinsic brightness. Intrinsic means “belonging to the thing.” However, how bright an object looks depends

not only on how much light it emits but also on its distance from the observer.

Brightness and Distance

The flux (F) received from a light source is proportional to its intrinsic brightness, or Luminosity (L), and inversely proportional to the square of its distance (d).

F ~ L/d2STAR

A STAR B EARTH

Both stars may appear equally as bright from Earth, however Star A is intrinsically much brighter than Star B.

Brightness and Distance Recall the following

formulas: Difference In Magnitude Intensity Ratio

Difference In

MagnitudeIntensity Ratio

1 2.512

2 2.512*2.512 = (2.512)2 = 6.31

… …

5 (2.512)5 = 100

For a difference in magnitude of 0.41 – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28

BETELGEUSEApp. Mag. (mv) = 0.41

RIGELApp. Mag. (mv) = 0.14

Brightness and Distance The apparent brightness of a

light source is described by the Inverse Square Law. If you double the distance, its

brightness decreases by a factor of (2)2 or 4 times.

If you triple the distance, its brightness decreases by a factor of (3)2 or 9 times.

So, Rigel may appear 1.28 times brighter than Betelgeuse but is 1.6 times further also.

Therefore, Rigel is actually (intrinsically) 1.28 * (1.6)2 = 3.3 times brighter than Betelgeuse.

BETELGEUSEApp. Mag. (mv) = 0.41

RIGELApp. Mag. (mv) = 0.14

Brightness and Distance Recall, the apparent visual magnitude

(mv) is the brightness of a star as it appears to human observers on Earth.

If all the stars were the same distance from Earth, you could compare one with another and decide which was emitting more light and which less.

Astronomers have adopted 10 parsecs as the standard distance and refer to the apparent visual magnitude a star would have 10 pc away as its absolute visual magnitude (Mv).

Brightness and Distance

Back to our example of Betelgeuse and Rigel:

BETELGEUSEApp. Mag. (mv) = 0.41

RIGELApp. Mag. (mv) = 0.14

Betelgeuse Rigel

mV 0.41 0.14

MV -5.5 -6.8

d 152 pc 244 pc

Difference in absolute magnitude (Mv):

6.8 – 5.5 = 1.3 Intensity ratio = (2.512)1.3 = 3.3

Magnitude-Distance Formula If we know a star’s distance, we can

calculate its absolute magnitude by using its distance and apparent magnitude.

Magnitude-Distance Formula

mv – Mv = -5 + 5 log10(d)mV = apparent

magnitude

Mv = absolute magnitude

d = distance, in parsecs

Example: A star has a distance of 50 pc and an apparent magnitude of 4.5.What is the star’s absolute magnitude?

Mv = 1.0

Magnitude-Distance Formula Astronomers also use this formula to calculate

the distance to a star if the apparent and absolute magnitudes are known. Equation can be rewritten as follows:

Magnitude-Distance Formula

d = 10(mv – Mv +5) / 5

mV = apparent magnitude

Mv = absolute magnitude

d = distance, in parsecs

Example: A star has an apparent magnitude of 7 and an absolute magnitude of 2. What is the distance to it?

100 pc.

Distance Modulus

The magnitude difference, mv – Mv, is known as the distance modulus, a measure of how far away the star is.

The larger the distance modulus, the more distant the star.

Luminosity The luminosity of a star is the total

energy the star radiates in 1 second.

The absolute magnitude of a star is based only on visible radiation. Slight adjustments need to be made to

make up for invisible energy (UV hot stars; Infrared cool stars).

To find the luminosity of a star, you compare it with our Sun. If a star is 2.5 times more luminous than

our Sun, we would write it as 2.5 L.

To find the luminosity of a star in J/s, you would multiply by the luminosity of the Sun, which is 3.83 x 1026 J/s. Example: What is the luminosity, in J/s, of

Aldeberan if we know its a luminosity of 150 L?

5.75 x 1028 J/s

Luminosity, Radius, and Temperature Stars are spheres and recall the surface

area of a sphere is equivalent to 4πr2.

Luminosity Formula

L = 4πr2σT4 If you divide by the same

quantities for the Sun, you can cancel out the constants and get a simple formula for the luminosity of a star in terms of its radius and temperature

L = luminosity, in J/s

r = radius, in meters

σ = Stefan-Boltzmann constant

T = temperature, in K

L/L = (r/r)2(T/T

)4

Luminosity, Radius, and Temperature Example 1: Suppose a star is 10

times the Sun’s radius but only half as hot. How luminous would it be?

Example 2: Suppose a star is 40 times the luminosity of the Sun and twice as hot. How large is the radius of this star in comparison to the radius of the Sun?

6.25 times more

1.58 times larger

The H-R Diagram

The Hertzsprung-Russell (H-R) Diagram, names after its originators, Ejnar Hertzsprung and Henry Norris Russell, is a graph separating the effects of temperature and surface area on stellar luminosities and enables astronomers to sort stars according to their diameters.

Let’s take a look at a similar diagram to sort automobiles that may help you to understand how an H-R Diagram is set up.

The H-R Diagram

The main sequence is the diagonal band on the H-R Diagram running from top left to bottom right. Includes roughly 90% of all normal stars.

Stars just above the main sequence are called giant stars, and are roughly 10 – 100 times larger than the Sun.

There are even supergiant stars at the top of the diagram that are over 1000 times the Sun’s diameter.

The H-R Diagram

At the bottom of the H-R diagram lie the stars very low in luminosity because they are so small.

Near the bottom end of the main sequence, the red dwarfs are not only small but also cool, giving them low luminosities.

In contrast white dwarfs, lie in the lower left of the H-R diagram and have low luminosities but high temperatures.

Luminosity Classification A star’s spectrum contains clues as to whether it

is a main-sequence star, a giant, or a supergiant. The larger the star is, the less dense its atmosphere is

and that affects the widths of the spectral lines. Broader Balmer lines of Hydrogen main-sequence star Narrow Balmer lines of Hydrogen giant star Very narrow Balmer lines of Hydrogen supergiant star

Luminosity Classification

Size categories derived from spectra are called luminosity classes because the size of the star is the dominating factor in determining luminosity.

Roman Numeral

Star Type

Ia Bright Supergiant

Ib Supergiant

II Bright Giant

III Giant

IV Subgiant

V Main-Sequence Star (Dwarfs)

VI Subdwarfs

VII White Dwarfs

Spectroscopic Parallax

Stellar parallax can be measured for nearby stars, but most stars are too distant to have measurable parallaxes.

These distances can be estimated from a star’s spectral type, luminosity class, and apparent magnitude in a process called spectroscopic parallax. Doesn’t involve a measurement of

parallax shifts. Relies upon the location of the star

on the H-R Diagram. Results are only approximate

values.

For example, Betelgeuse is classified M2Ib.

Plot what you know on an H-R Diagram, similar to the one at the left, and you find it should have an absolute magnitude of about -6.0

Use the Magnitude Distance formula to calculate your distance in parsecs.

The Masses of Stars More than 50% of the

stars in our Milky Way galaxy are not single stars, but rather a pair of stars orbiting a common center of mass known as binary stars.

As binary stars revolve around each other, the line connecting them always passes through the center of mass, and the more massive star is always closer to the center of mass.

Binary Stars

According to Newton’s laws of motion and gravity, the total mass of two stars orbiting each other is related to the average distance between them and their orbital period.

Binary Star Mass Formula

Ma + Mb = a3/P2

M = mass, in solar masses

a = average distance between the two stars, in

AU’s

P = orbital period, in years

Binary Stars Example 1: If you observe a binary system

with a period of 32 years and an average separation of 16 AU, what is the total mass?

Example 2: Call the two stars in the previous example A and B. Suppose star A is 12 AU away from the center of mass, and star B is 4 AU away. What are the individual masses?

4 M (solar masses)

Star B – 3 M

Star A – 1 M

Binary Stars Three type of binary stars are especially

useful for determining stellar masses:

1. Visual Binary – two stars separately visible in a telescope.

2. Spectroscopic Binary – two stars are so close together they can only be separated by looking at a spectrum, formed by light from both stars.

3. Eclipsing Binary – when one of the two stars is in front of the other, blocking light from the eclipsed star.

Star Overview Along the main-sequence,

the more massive a star the more luminous it is.

High-mass stars have muchshorter lives than low-mass

stars:Sun ~ 10 billion yrs10 Mο ~ 30 million yrs

0.1 Mο ~ 3 trillion yrs

Mass-Luminosity Relation

L/L = (M/M)

3.5

L = luminosity (in terms of the Sun’s luminosity)

M = mass, in solar masses

Mass-Luminosity Relation