Stellar Magnitudes and Distances

Ways of measuring a star’s brightness and distance.

What’s there to see in starlight?

• The light from a star may only look like a twinkling dot at first. Look more closely, however, and you’ll notice that different stars are different brightnesses, and even different colors.

http://antwrp.gsfc.nasa.gov/apod/ap060501.html

What’s in Star Light?

• With the right instruments, astronomers can tell a lot about a star, just from its light.

• “Like what?” you might ask.

Buried in that star light is…

• the direction and speed a star is moving

• its mass • its brightness or luminosity • its chemical composition • its size • its age • its temperature

Buried in that star light is…

• its distance from us • its stage of life • how it makes its energy • even whether it has companions or

not (planets or other stars orbiting it)!

• We’ll concentrate on brightness and distance for right now.

What’s Luminosity Again?

• Let’s start with Luminosity. How did we define it?

The amount of energy at ALL wavelengths given off by a star into space in each second. Since the units are energy / unit time (Watts), luminosity gives the star’s POWER.

Magnitudes, not Watts • Can the luminosity of a star really be

measured?

• Yes, with some difficulty. But the Watt is an inconveniently large unit for the tiny amount of energy that we receive from a star at the earth’s surface.

• Besides, our brains perceive a star’s apparent brightness. Most of us don’t think in terms of power output.

So What are Magnitudes?

• Even thousands of years ago, ancient peoples noticed that the stars weren’t all the same brightnesses.

• Hipparchus, a Greek philosopher, invented a system of magnitudes. He called the brightest stars in the sky (like Sirius), first class or first magnitude stars.

More Hipparchus

• The next brightest group of stars were 2nd class or magnitude 2 stars, and so forth, down to magnitude 6 stars, which were just barely visible to the naked eye.

• Hipparchus also estimated that the

brightest (mag. 1) stars were 100 times brighter than the faintest (mag. 6) stars.

http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Hipparchus.jpeg

What’s this mean to us?

• Astronomers today have inherited the magnitude system from the ancients.

• It takes some getting used to, because the scale appears to be backwards from the way we classify most things:

The brighter stars get smaller numbers. The fainter stars get larger numbers.

More Magnitudes!

• Because Hipparchus called a difference of 5 magnitudes (from 1 to 6) equal to a 100-fold change in brightness, each change of one magnitude = 1001/5 = 2.512 change in brightness.

• In other words, a Mag. 1 star is 2.512 times brighter than a Mag. 2 star, but 2.5122 (or 6.31 times) brighter than a

Mag. 3 star.

How do you do the math?

• Take the difference in magnitudes between two stars.

• Raise 2.512 to that power. • Example: How many times brighter is

Polaris (a 2nd magnitude star) than a barely-visible 6th magnitude star?

• 6 - 2 = 4. So 2.5124 = 39.8 times. Polaris is almost 40 times brighter than the faintest visible star!

Modern Magnitudes • Today, we’ve expanded the scale well

beyond the 1 to 6 range. • For example, the sun appears much

brighter than any other star in the sky. It has an (apparent) magnitude of -26.73.

• The full moon, at its brightest, has an (apparent) magnitude of -12.6 and Venus can be as bright as -4.4.

• On the other end, the Hubble Space Telescope can see objects of magnitude 30, way too faint for our eyes.

Pause for Practice

• How much brighter does the sun appear to our eyes than the faintest visible star?

6 - (-26.73) = +32.73 2.51232.73 = 12.4 trillion times (1.24 x 1013)

A word of caution: 2.51232.73 doesn’t mean to multiply the two numbers together! You need to use your powers key: 2.512 ^ 32.73 or 2.512 xy 32.73

(Apparent) Magnitude?

• A couple slides ago, I kept using the word “apparent” in front of magnitude.

• Apparent magnitude (m) is how bright a star appears from the earth’s surface.

• You know that not all the stars are at the same distance from the earth, so even if they were all exactly the same true brightness, they still wouldn’t all look equally bright.

Apparent Magnitude Questions

• If two stars have the same actual brightness (which we’ll call absolute magnitude later), but one star appears brighter at the earth’s surface, how do the distances of the two stars compare?

The brighter star must be closer to the earth.

Apparent Magnitude Questions

• If two stars appear to be equally bright from the earth’s surface, but you know that one of the stars is farther away, how do the actual brightnesses of the two stars compare?

The more distant star must actually be brighter.

The Brightness – Distance Connection

• If a given amount of light energy leaves a star, it passes through an imaginary sphere surrounding the star that is 1 AU from the star. 1 unit of light falls on every 1 unit of surface area of that imaginary sphere.

• As the light travels, it will pass through another imaginary sphere that is twice as far away.

The Brightness – Distance Connection

• Since the formula for the surface area of a sphere is A = 4πr2 the same amount of light must fall on an area that is 22 or 4 times larger. This makes the brightness of the light ¼ what it was at ½ the distance.

• We call this the Inverse Square Law for Light.

Pause for Practice

• When the light passes through an imaginary sphere that is 3 AU from the star, what will its apparent brightness be?

The light will have traveled 3 times the distance, so the same amount falls on 32 or 9 times as much area. This makes it 1/9th as bright.

Pause for Practice

• How about when the light travels 5 times as far? 1/52 = 1/25th as bright.

• 10 times as far? 1/102 = 1/100th as bright.

• 1000 times as far? 1/10002 =

1/1,000,000th as bright!

Pause for Practice

• What would be the apparent magnitude (m) of a star that appears 100 times brighter than a magnitude 3 star?

You know that a 100 times increase in brightness = 5 magnitudes. 5 magnitudes brighter than 3 is -2 (not 8!)

Interlude – Distance Units in Space

• Before we can define a star’s absolute magnitude, we have to define a couple of units that we’ll use shortly.

• You already know what a light-year (LY)

is: the distance that light can travel in 1 year’s time…about 6 x 1012 miles or

9.5 x 1015 meters.

Distance Units in Space

• Another unit of distance, even more commonly used, is the parsec (pc), which is a contraction for parallax arcsecond.

• 1 parsec = 3.26 light years.

• The nearest star, Proxima Centauri, is about 4.2 LY or 1.3 pc away.

http://chandra.harvard.edu/photo/2004/proxima/proxima_xray_scale.jpg

Absolute Magnitude (M) • A star’s apparent magnitude (m) is how

bright it appears at the earth’s surface.

• A star’s absolute magnitude (M) is how bright it appears from a standard reference distance of 10 pc or 32.6 LY.

• Since a star’s distance from the earth affects its apparent brightness, astronomers compare the brightnesses of stars on an absolute scale: absolute M.

Comparing m and M

• If we know how bright a star truly is at a set distance (M), and we know how bright the star appears at the earth (m)…

…then couldn’t we compare m and M to

determine the star’s distance from the earth!

Eureka! Hipparchus would be so proud!

Here’s the Equation

• Distance in parsecs = 10[ (m-M+5) / 5 ] • Everything inside the brackets is an exponent!

• Example: What is the distance to a star

like the sun (M = +4.6), if m = +12?

D = 10[ (12-4.6+5) / 5 ] = 10[12.4 / 5] = 102.48 = 302 pc = 985 LY

Concept Check!

• Rank these 3 stars from brightest to faintest…as they appear from the earth.

Star m M Sirius -1.44 -1.45 Polaris +1.97 -3.64 the Sun -26.7 +4.8

The sun, Sirius, Polaris.

Concept Check!

• Rank these 3 stars from brightest to faintest…as they actually are.

Star m M Sirius -1.44 -1.45 Polaris +1.97 -3.64 the Sun -26.7 +4.8

Polaris, Sirius, the Sun

Concept Check!

• If Polaris has m = +1.97 and M = -3.64, how far away is it?

Distance = 10[ (1.97-(-3.64)+5) / 5 ] = 10[ 10.61 / 5 ] = 102.12 = 132 pc or 432 LY

The Astronomical Chicken Another Way of Measuring Distance

The Astronomical Chicken

• Chickens have their eyes on different sides of their heads. They only see an object with 1 eye at a time. They don’t have binocular vision like we do, that is good for measuring distance.

• How then do they grab a grain of corn without slamming their heads into the dirt?

The Answer is Parallax

• A chicken bobs its head back and forth, viewing an object from different angles.

• By judging how big the angle is as it moves its head, the chicken determines how far away the object is.

• This method is called parallax.

Demonstrating Parallax

• Close your left eye, and put your index finger straight up at arm’s length. Now line your finger up with some vertical object on the other side of the room.

• Without moving your finger, quickly open your left eye, and close your right eye.

• It appears that your finger has moved! • Switch back and forth between eyes, and

your finger appears to jump back and forth.

Demonstrating Parallax

• Now, move your finger halfway in towards your eye. Repeat the experiment.

• Does your finger appear to jump a larger distance back and forth?

• You bet! The closer the object, the bigger the apparent jump.

Parallax and Stars

• As the earth orbits the sun, two positions on opposite sides of the orbit (6 months apart) act very much like when you rapidly switched eyes.

• A nearby star will appear to “jump” back and forth over a 6 month period, when viewed against the backdrop of very distant stars.

Credit: http://astrowww.astro.indiana.edu/~classweb/a105s0079/parallax.gif

Parallax – the Definition

• Heliocentric Stellar Parallax is defined as “The apparent movement of a nearby star against the background of distant stars, due to the observer’s change over a 6 month period.”

• The parallax angle is measured in seconds of arc (1/3600th of a degree).

• Seconds of arc = arcseconds.

Better definition of “parsec”

• Now that you know what parallax and arcseconds are, we can better define what a parsec is.

• A parsec is the distance between us and a star that would result in a parallax of exactly 1 arcsecond.

Here’s Another Equation!

• If you can measure the parallax of a star, it’s really easy to calculate the distance to that star:

distance in pc = 1/ parallax in arcseconds

or d = 1/p

Pause for Practice

• What is the distance in parsecs to a star with a parallax of 0.045 arcseconds?

1 / 0.045 = 22.2 parsecs. That’s all there is to it.

Pause for Practice

• What is the parallax of a star, if its distance is 100 LY? Be careful of the unit!

First, turn LY into pc: 100 LY 3.26 = 30.7 pc Then, 1 / 30.7 pc = 0.0326 arcseconds.

It’s Not Perfect • Using parallax to find distances has limits.

• Because we can only accurately measure a parallax angle to about 0.01 arcseconds, parallax is only accurate to about 100 pc or roughly 300 LY.

• Since the galaxy is 30,000 to 40,000 pc wide, parallax can only find the distances to stars that are in our immediate “neighborhood” in the galaxy.

The End … for now! Bwa Ha Ha!