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Climbing the Distance Ladder
Climbing the Distance Ladder
Key Concepts
1) Distances within the Solar System can
be measured using radar.
2) Distances of nearby stars can be
measured using parallax.
3) Greater distances can be measured
using standard candles.
Finding the distance to astronomical objects is difficult.
You can’t use the same technique for each object.
Instead, there exists an astronomical distance ladder.
Rung #1: Distances within the Solar System
can be measured using radar.
Bounce a radio signal from Venus.
Round trip travel time ÷ 2 = One-way travel time.
One-way travel time × c = distance to Venus.
c = speed of light
Once you’ve plotted the orbit of
Venus, you know where the Sun is:
at a focus of the elliptical orbit.
V
Rung #2: We find the distance to relatively
nearby stars by measuring their parallax.
E
average Earth-Sun distance =
1 astronomical unit (AU) =
149,597,870.69 kilometers
Remember: parallax is too small to be seen by the
naked eye (< 1 arcminute).
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For the closest stars, the parallax is large
enough to be measured with a telescope.
How to find distance by measuring parallax:
In 1838, Friedrich Bessel found
a parallax of 0.3 arcseconds for
a nearby star.
Parallax and distance are related by
a simple equation.
d
1
p
p = parallax of star (in arcseconds)
d = distance to star (in parsecs)
1 parsec = distance at which a star has a parallax of
1 arcsecond = 206,000 AU = 3.26 light-years.
Parallax Second = Parsec (pc)
• Fundamental distance unit in Astronomy
“A star with a parallax of 1 arcsecond has a
distance of 1 Parsec.”
• 1 parsec (pc) is equivalent to:
206,265 AU
3.26 Light Years
3.085x1013 km
What’s the nearest star?
(other than the Sun)
How far is the
journey from here to
Proxima Centauri?
Proxima Centauri, a not-too-bright
star in the constellation Centaurus
d
1
1

 1.30 parsecs
p 0.769
= 4.24 light-years = 267,000 AU
p = 0.769 arcseconds
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The Parallax Problem
Light Year (ly)
• Alternative unit of distance
Best parallax measurements were
provided by the Hipparcos satellite.
“1 Light Year is the distance traveled by light in one
year.”
• Relation to other units:
1 light year (ly) is equivalent to
0.31 pc
63,270 AU
1013 km
How “Bright” is an Object?
• We must define “Brightness” quantitatively.
• Two ways to quantify brightness:
• Intrinsic Luminosity:
– Total Energy Output.
• Apparent Brightness:
– How bright it appears to be as seen from a
distance.
Parallaxes < 1/1000 of an arcsecond were
too small to measure.
Distances > 1000 parsecs can’t be found from parallax.
Luminosity
• Luminosity is the total energy output from
an object per second
• Measured in Power Units:
– Energy/second emitted by the object (e.g.,
Watts)
• Independent of Distance
• Luminosity of a star is a measurement of
its total energy production.
Apparent Brightness
• Measures how bright an object appears
to be as seen by a distant observer.
– What we measure on earth (“observable”)
• Measured in Flux Units:
– Energy/second/area from the source.
• Depends on the Distance to the object
d=1
B=1
d=2
B=1/4
d=3
B=1/9
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• Surface area of a sphere = 4  d2
Flux-Luminosity Relationship:
the inverse square law.
Relates Apparent Brightness (Flux) and
Intrinsic Brightness (Luminosity) through the
Inverse Square Law of Brightness:
Flux =
We don’t directly measure a star’s luminosity.
We measure its flux (f): the wattage collected
per square meter of our telescope mirror.
What does this have to do with finding
distances to stars??
If you know the luminosity L,
and you measure the flux f,
you can compute the distance r:
Flux of sunlight at the Earth’s location =
1400 watts per square meter
f
Rung #3: We use the “World-Famous InverseSquare Law” applied to standard candles.
Every star has a luminosity (L): this the wattage of
the star (how much energy it emits per unit time).
40 watts
4×1026 watts
Luminosity
4 d 2
L
4  r2
r
L
4 f
An object whose luminosity you know
is called a “standard candle”.
Alas! Stars have a range of luminosities.
Betelgeuse, Rigel = high luminosity
Sun = medium luminosity
Proxima Centauri = low luminosity
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Spectra of 6 stars
For nearby stars, we can measure flux and
parallax, then compute distance from parallax.
We compute luminosity:
L  4  r2 f
very hot & luminous
Eureka! Stars with identical spectra have
identical luminosity!
Find a star with a spectrum identical to
the Sun’s (for instance).
Measure the star’s flux f.
Assume the star’s luminosity is the same as
the Sun’s (L = 4 × 1026 watts).
Compute the star’s distance:
r
L
4 f
cool & less luminous
Standard candle #2: Periodic
Variable Stars
• Stars whose brightness varies with a
characteristic, periodic pattern.
• Distance-Independent Property:
Period of their brightness variations.
• Importance:
Period-Luminosity Relation exists for certain
classes of periodic variable stars.
Measuring the Period gives the Luminosity.
Cepheid Variables
Brightness
RR Lyrae Variables
Brightness
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Period-Luminosity
Relationship
Cepheid Variables
• Pulsating Supergiant stars:
– Luminosity of ~ 103-4 Lsun
– Period range: 1 day to ~50 days.
• Period-Luminosity Relation:
– Longer Period = Higher Luminosity
– P = 3 days, L ~ 103 Lsun
– P = 30 days, L ~ 104 Lsun
Importance of RR Lyrae & Cepheid
variables:
RR Lyrae Variables
• Pulsating Stars:
• The period—luminosity relation allows
distance determination!
– Luminosity of ~50 Lsun
– Period Range: few hours to ~ 1 day.
– Relatives of Cepheid Variables
• Period-Luminosity Relation
– Less strong than for Cepheids
Example: Cepheid with a 10-day period
Luminosity (Lsun)
104
Cepheid
P-L Relation
L=5011 Lsun
103
102
P=10d
0.5
1
3
5
10
30 50
100
Period (days)
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