Transcript The Relationship Between a Star`s Brightness and its Distance

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Stellar Magnitudes and
Distances
Ways of measuring a star’s
brightness and distance.
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• 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.
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• What’s there to see in starlight?
The Jewel Box Cluster in the Small Magellanic Cloud,
200,000 Light Years Distance Credit: NASA
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• 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
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– 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.
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• 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.
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• 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.
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• 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.
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• 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
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• 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.
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• Because Hipparchus called a difference of
5 magnitudes (from 1 to 6) equal to a 100fold 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.
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• 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.
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• 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
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(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.
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• 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.
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• 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.
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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.
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• 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.
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• 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.
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• 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!
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• 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!)
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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.
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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.
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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!
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 ]
= 102.48
=
=
302 pc
10[12.4 / 5]
= 985 LY
Concept Check!
• Rank these 3 stars from brightest to
faintest…as they appear from the earth.
Star
Sirius
Polaris
the Sun
m
-1.44
+1.97
-26.7
The sun, Sirius, Polaris.
M
-1.45
-3.64
+4.8
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• Rank these 3 stars from brightest to
faintest…as they actually are.
Star
Sirius
Polaris
the Sun
m
-1.44
+1.97
-26.7
Polaris, Sirius, the Sun
M
-1.45
-3.64
+4.8
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• 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
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• 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?
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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.
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• 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.
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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.
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• Go to this website and view the nearby red
star move back and forth as the earth
orbits the sun:
http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Movies/parallax.html
Credit: http://astrowww.astro.indiana.edu/~classweb/a105s0079/parallax.gif
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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.
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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.
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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
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• 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.
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• 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.
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• 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.