Chapter 14Surveying the Stars

Luminosity and Apparent Brightness Luminosity - the measurement of the rate at which energy is outputted from a star. This could also be called the power output of the sun.

Where:

(Stefan-Boltzmann constant)

= The surface area of the star with a radius

= the temperature of the star in Kelvin

However, when we look at a star, we see not its luminosity but rather its apparent brightness

Apparent brightness is a measure not of a star’s luminosity but of the energy flux (energy per unit area per unit time) produced by the star, as seen from Earth. It depends on our distance from the star.

Thought Question

These two stars have about the same luminosity -- which one appears brighter?

A. Alpha CentauriB. The Sun

Luminosity:

Amount of power a star radiates

(energy per second = Watts)

Apparent brightness:

Amount of starlight that reaches Earth

(energy per second per square meter)

Luminosity passing through each sphere is the same

Divide luminosity by the area of the sphere over which the light is being spread to get the apparent brightness at every point on that sphere.

The further from the star your go the larger the area over which the light is spread so the dimmer the star will appear to be.

The relationship between apparent brightness and luminosity depends on distance:

Thus we can determine a star’s luminosity if we can measure its distance and apparent brightness:

Luminosity = 4π (distance)2 x (Brightness)

Thought Question

How would the apparent brightness of Alpha Centauri change if it were three times farther away?

A. It would be only 1/3 as brightB. It would be only 1/6 as brightC. It would be only 1/9 as brightD. It would be three times brighter

Most luminous stars:

106 LSun

Least luminous stars:

10-4 LSun

(LSun is luminosity of Sun)

So how far are these stars?

Parallax

Apparent positions of nearest stars shift by about an arcsecond as Earth orbits Sun

Parallax angle depends on distance

Parallax and Distance

p = parallax angle

d (in parsecs) = 1

p (in arcseconds)

d (in light - years) = 3.26 1

p (in arcseconds)

Our nearest neighbours

The closest star to Earth (besides the Sun) is called Proxima Centauri. It has the largest known stellar parallax, 0.76'', which means that it is about 1/0.76 = 1.3 pc away—about 270,000 A.U., or 4.3 light-years.

Our next nearest neighbor to the Sun beyond the Alpha Centauri system is called Barnard’s Star. Its parallax is 0.55'', so it lies at a distance of 1.8 pc, or 6.0 light-years.

Stellar Motion we must be careful with parallax.

Barnard’s Star 22 years apart:

This stellar motion has two components. A star’s radial velocity—along the line of sight—can be measured using the Doppler effect. For many nearby stars, the transverse velocity—perpendicular to our line of sight—can also be determined by careful monitoring of the star’s position on the sky.

The annual movement of a star across the sky, as seen from Earth and corrected for parallax, is called proper motion. It describes the transverse component of a star’s velocity relative to the Sun. Like parallax, proper motion is measured in terms of angular displacement

Spectral lines from Alpha Centauri are blueshifted by a tiny amount—about 0.0067 percent—allowing astronomers to find the radial velocity from the Doppler Effect

Use basic trig to find the hypotenuse.

THE MAGNITUDE SCALE

The scale dates from the second century B.C., when the Greek astronomer Hipparchus ranked the naked-eye stars into six groups. The brightest stars were categorized as first magnitude.

The 1–6 magnitude range defined by Hipparchus spans about a factor of 100 in apparent brightness—a first-magnitude star was approximately 100 times brighter than a sixth-magnitude star.

The physiological characteristics of the human eye are such that each magnitude change of 1 corresponds to a factor of about 2.5 in apparent brightness. [ 2.512 to the fifth power is 100]

We now define a change of 5 in the magnitude of an object (going from magnitude 1 to magnitude 6, say, or from magnitude 7 to magnitude 2) to correspond to exactly a factor of 100 in apparent brightness.

Because we are really talking about apparent (rather than absolute) brightnesses, the numbers in Hipparchus’s ranking system are called apparent magnitudes.

The scale is no longer limited to whole numbers

Absolute magnitude

To compare intrinsic, or absolute, properties of stars, however, astronomers imagine looking at all stars from a standard distance of 10 pc (arbitrary choice).

Because the distance is fixed in this definition, absolute magnitude is a measure of a star’s absolute brightness, or luminosity.

Knowing distance allows us to compute its absolute magnitude. As discussed further in lab the numerical difference between a star’s absolute and apparent magnitudes is a measure of the distance to the star

The Magnitude Scale

1 2 1 2

1 2 1 2

1/5

1/5

m = apparent magnitude M = absolute magnitude

apparent brightness of Star 1(100 ) 2.51

apparent brightness of Star 2

luminosity of Star 1(100 ) 2.51

luminosity of Star 2

m m m m

M M M M

How do we measure stellar temperatures?

Blackbody Radiation1. Hotter objects emit more light per unit area at all frequencies.

(Brighter)2. Hotter objects emit photons with a higher average energy.

(Bluer)

Hottest stars:

50,000 K

Coolest stars:

3,000 K

(Sun’s surface is 5,800 K)

Absorption lines in star’s spectrum tell us ionization level which is highly dependent on temperature.

Pioneers of Stellar Classification

• Annie Jump Cannon and the “calculators” at Harvard laid the foundation of modern stellar classification

Early computers were not allowed to be Laptops

Harvard “computers” were ladies who did the tedious data analysis by hand.

1897: Antonia Maury spectral study led to HR diagrams.

1898: Annie Cannon’s spectral classes still used. She ranked stellar spectra based on the strength of the Hydrogen spectral lines.

1908: H. Leavitt’s variable star work .

Lines in a star’s spectrum correspond to a spectral type that reveals its temperatureThe original classification of spectral type was alphabetical and based on the strength of the hydrogen lines. Spectra with the strongest hydrogen lines were call A type stars.

(Hottest) O B A F G K M (Coolest)

Spectral Types in Temperature Order

• Only Boys Accepting Feminism Get Kissed Meaningfully

Each spectral type is subdivided into numbered subcategories (0-9)0 being the hottest and 9 the coolest.For example; if we break the F spectral type stars into its subcategories the order would look likeO B A (F0, F1, F2…, F9) G K M

The sun is a G2 spectral class star.

How do we measure stellar masses?

The orbit of a binary star system depends on strength of gravity

Types of Binary Star Systems

• Visual Binary• Eclipsing Binary• Spectroscopic Binary

About half of all stars are in binary systems

Visual Binary

We can directly observe the orbital motions of these stars

Eclipsing Binary

We can measure periodic eclipses

Spectroscopic Binary

We determine the orbit by measuring Doppler shifts

Alcor and Mizar are optical doubles not true binaries. They just look close. But Mizar A & B is a true visual binary made up of two spectroscopic binaries (4 stars total).

Estimating Stellar MassesRecall Kepler’s 3rd Law:

If we choose to measure the Period in years, the average orbital separation in

astronomical units (AU), and the masses in solar masses then the

constants

So

Example:A binary system is observed with a period of P = 32 years

and separation a of a = 16 AU:

= 4 solar masses

Need 2 out of 3 observables to measure mass:

1) Orbital Period (P)2) Orbital Separation (a or r = radius)3) Orbital Velocity (v)

For circular orbits, v = 2pr / pr

M

v

Most massive stars:

100 MSun

Least massive stars:

0.08 MSun

(MSun is the mass of the Sun)

Back to Stellar Masses.

• We can add mass to the data we have on stars thanks to the binaries.

• We find…• Supergiants, giants and white dwarfs reveal no

clear pattern.• But main sequence stars show a clear trend.

MAIN SEQUENCE STARS ONLY

(a) Size and mass relation (b) Luminosity and mass

What have we learned?• How do we measure stellar luminosities?

– If we measure a star’s apparent brightness and distance, we can compute its luminosity with the inverse square law for light

– Parallax tells us distances to the nearest stars• How do we measure stellar temperatures?

– A star’s color and spectral type both reflect its temperature

What have we learned?• How do we measure stellar masses?

– Newton’s version of Kepler’s third law tells us the total mass of a binary system, if we can measure the orbital period (p) and average orbital separation of the system (a)