Stellar Magnitudes and Distances Ways of measuring a star’s brightness and distance. Ohio University - Lancaster Campus slide 1 of 46 Spring 2009 PSC 100

This evening, we will investigate… the origin of the magnitude scale for measuring a star’s brightness. the difference between apparent and absolute magnitude for a star. two units of distance in space, the light year and the parsec. the concept of parallax, and how it relates to distances. Ohio University - Lancaster Campus slide 2 of 46 Spring 2009 PSC 100

What’s there to see in starlight? The Jewel Box Cluster in the Small Magellanic Cloud, 200,000 Light Years Distance Credit: NASA Ohio University - Lancaster Campus slide 3 of 46 Spring 2009 PSC 100

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 Ohio University - Lancaster Campus slide 4 of 46 Spring 2009 PSC 100

–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)! Intrinsic luminosity, distance, and size all contribute to how bright a star appears to us. Ohio University - Lancaster Campus slide 5 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 6 of 46 Spring 2009 PSC 100

Can the luminosity of a star really be measured? The Watt is an inconveniently large unit for the tiny amount of energy that we receive from a star at the earth’s surface. Our brains perceive a star’s apparent brightness. Ohio University - Lancaster Campus slide 7 of 46 Spring 2009 PSC 100

So What are Magnitudes? 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. Ohio University - Lancaster Campus slide 8 of 46 Spring 2009 PSC 100

The next brightest group of stars were 2 nd 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. Ohio University - Lancaster Campus slide 9 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 10 of 46 Spring 2009 PSC 100

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 = 100 1/5 = change in brightness. In other words, a Mag. 1 star is times brighter than a Mag. 2 star, but (or 6.31 times) brighter than a Mag. 3 star. Ohio University - Lancaster Campus slide 11 of 46 Spring 2009 PSC 100

Take the difference in magnitudes between two stars. Raise to that power. Example: How many times brighter is Polaris (a 2 nd magnitude star) than a barely-visible 6 th magnitude star? = 4. So = 39.8 times. Polaris is almost 40 times brighter than the faintest visible star! Ohio University - Lancaster Campus slide 12 of 46 Spring 2009 PSC 100

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 The full moon, at its brightest, has an (apparent) magnitude of and Venus can be as bright as On the other end, the Hubble Space Telescope can see objects of magnitude 30, way too faint for our eyes.

How much brighter does the sun appear to our eyes than the faintest visible star? 6 - (-26.73) = = 12.4 trillion times (1.24 x ) A word of caution: doesn’t mean to multiply the two numbers together! You need to use your powers key: ^ or x y Ohio University - Lancaster Campus slide 14 of 46 Spring 2009 PSC 100

(Apparent) 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. Ohio University - Lancaster Campus slide 15 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 16 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 17 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 18 of 46 Spring 2009 PSC 100

Since the formula for the surface area of a sphere is A = 4πr 2 the same amount of light must fall on an area that is 2 2 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. Ohio University - Lancaster Campus slide 19 of 46 Spring 2009 PSC 100

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 3 2 or 9 times as much area. This makes it 1/9 th as bright. Ohio University - Lancaster Campus slide 20 of 46 Spring 2009 PSC 100

How about when the light travels 5 times as far? 1/5 2 = 1/25 th as bright. 10 times as far? 1/10 2 = 1/100 th as bright times as far? 1/ = 1/1,000,000 th as bright! Ohio University - Lancaster Campus slide 22 of 46 Spring 2009 PSC 100

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!) Ohio University - Lancaster Campus slide 23 of 46 Spring 2009 PSC 100

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 miles or 9.5 x meters. Ohio University - Lancaster Campus slide 24 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 25 of 46 Spring 2009 PSC 100

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! Ohio University - Lancaster Campus slide 27 of 46 Spring 2009 PSC 100

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 [ ( ) / 5 ] = 10 [12.4 / 5] = = 302 pc = 985 LY

Concept Check! Rank these 3 stars from brightest to faintest…as they appear from the earth. StarmM Sirius Polaris the Sun The sun, Sirius, Polaris.

Rank these 3 stars from brightest to faintest…as they actually are. StarmM Sirius Polaris the Sun Polaris, Sirius, the Sun Ohio University - Lancaster Campus slide 30 of 46 Spring 2009 PSC 100

If Polaris has m = and M = -3.64, how far away is it? Distance = 10 [ (1.97-(-3.64)+5) / 5 ] = 10 [ / 5 ] = = 132 pc or 432 LY Ohio University - Lancaster Campus slide 31 of 46 Spring 2009 PSC 100

The Astronomical Chicken Another Way of Measuring Distance

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? Ohio University - Lancaster Campus slide 33 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 34 of 46 Spring 2009 PSC 100

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.

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. Ohio University - Lancaster Campus slide 36 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 38 of 46 Spring 2009 PSC 100

Go to this website and view the nearby red star move back and forth as the earth orbits the sun: Ohio University - Lancaster Campus slide 39 of 46 Spring 2009 PSC 100

Credit:

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 (1/3600 th of a degree). Seconds of arc = arcseconds. Ohio University - Lancaster Campus slide 41 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 42 of 46 Spring 2009 PSC 100

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 Ohio University - Lancaster Campus slide 43 of 46 Spring 2009 PSC 100

What is the distance in parsecs to a star with a parallax of arcseconds? 1 / = 22.2 parsecs. That’s all there is to it. Ohio University - Lancaster Campus slide 44 of 46 Spring 2009 PSC 100

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 = arcseconds. Ohio University - Lancaster Campus slide 45 of 46 Spring 2009 PSC 100

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. Ohio University - Lancaster Campus slide 46 of 46 Spring 2009 PSC 100