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The Distance Ladder
Topics
Trigonometric Parallax;
Spectroscopic Parallax;
Cepheid variables;
Standard Candles
Type Ia Supernovae
…and others
Motivation
Learn how we measure the size of the Universe.
See how other quantities depend upon the Universe’s size.
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The Distance Ladder
How does one determine the distances to
cosmic objects?
Distance is not directly measureable;
it must be inferred.
This is one of the greatest challenges in
observational astronomy.
The philosophy is to measure the distances to close objects using the
most reliable methods, then use those results to calibrate following
methods to further objects: repeat, repeat, repeat.
This construction is called the Distance Ladder. It becomes so rickety,
that ultimately the distances of very distant objects may be uncertain
by a factor of two or more.
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A Factor of Two
Ex: if something is 2× further away than
we thought….
Its luminosity is 4× greater than we thought.
Its size is 2× greater than we thought.
Its volume is 8× greater than we thought.
Its mass (ρ×V) is 8× greater than we thought.
If it is an energy source using fuel (mass) at a certain rate
(luminosity), our estimate for its lifespan (mass/luminosity) would be
off by 8×/4× = 2×.
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Radar Ranging: Solar System
Radar signals are sent to the moon, or to inner planets.
The time delay for the return determines the distance.
Common frequencies include 8495 MHz, 8510 MHz
(λ~3.5 cm).
The trip to the Moon and back takes 2.56 sec.
An astronomical unit is therefore determined by this
method to be 149,597,871 km (500 sec; 8 min 20 sec).
Radar ranging cannot be reliably used for the sun. (The
sun is soft!)
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Parallax: Nearest Stars
Objects appear to shift back and forth, depending upon
your viewing distance.
Closer objects shift back and forth more.
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Trigonometric Parallax: Nearest Stars
This effect can be used to measure distances.
As the Earth moves in its orbit, stars shift back and forth
in the sky.
EXAMPLE
Two viewers, separated by 1 astronomical unit, are
observing an object 3.26 LY away. Their measures of the
object’s direction differ by 1 arc second.
By definition, the distance to a star is:
D=3.26 LY/(parallax angle) = 1 parsec/ (parallax angle).
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Trigonometric Parallax: Nearest Stars
Pre-1989, only a hundred or stars had accurate parallax
measures.
ESA’s Hipparcos satellite (1989-1993) measured
parallax angles with very high precision (0.002 arc sec)
for over 100,000 stars (to about 1600 light-years).
ESA’s Gaia mission (2013) will be able to measure
parallax angles of 2×10-5 arc seconds (5×104 pc), for 109
stars! It is currently in a calibration stage.
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Spectroscopic Parallax: Stars
Consider the Hertzsprung-Russell Diagram. Different
stars have different luminosities.
Even within a spectral class, there are different
luminosity classes: dwarfs, subgiants, giants, supergiants.
Luminosity class can be determined by details of spectra
(line widths).
Therefore, by determining a star’s spectral type and
luminosity class, we can determine its actual brightness.
…Then, we look at how bright the stars look in the
sky….
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Spectroscopic Parallax: Stars
Example: Consider the three stars below, which all have the
same apparent brightness.
2
1
3
Suppose Star 1 is a main sequence O star.
Suppose Star 2 is a main sequence G star.
Suppose Star 3 is a main sequence M star.
What is the order, from closest to farthest?
If Star 1 is 106 times more luminous than Star 2, how many
times further away is it?
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Cluster Fitting: Distant Stars
Star clusters are exceptional laboratories for studying
stars because all the stars in a cluster share the same
characteristics:
–
–
–
–
Age
Composition
Reddening/dimming from interstellar dust
Distance
An HR diagram made of only the stars in a cluster is
called a cluster diagram.
The distance of the cluster can be inferred by the
direction the cluster data are shifted on the cluster
diagram.
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So what do we have so far?
Radar ranging → astronomical unit measurement.
Trigonometric parallax → an understanding of stellar characteristics.
Spectroscopic parallax → information on other types of stars.
Cluster fitting → Distances to clusters…
…and since we know the distances to clusters, we know the distances
to all the objects in them.
This lets us search clusters for yet other objects that might be good
measures for distance.
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Pressure-Gravity Balance
Stars are in a state of balance, where the (inward) force of gravity is
balanced by the (outward) force produced by pressure differences—
the deeper into a star you go, the higher the pressure.
If a star collapsed inwards upon itself (for fictional reasons), it
would convert gravitational collapse energy into heat energy. The
thermal pressure would increase, and the star would re-expand to its
original condition.
By the same
mechanism, if a star
expanded, it would
cool, and recollapse to
its original
configuration.
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Temperature-sensitive opacity
For stars of a specific combination of temperatures and pressures, an interesting
effect can occur. As the star collapses and gets hotter, some of the energy goes into
ionizing helium atoms (each helium atom loses an electron).
He+ →He++ + e-
This ionized helium is more opaque to radiation, so as the star enters the reexpansion stage, it gets pushed by radiation, and expands too far.
Similarly, when it finally contracts, it contracts too far.
There is a specific “instability strip” on the HR diagram where you can find these
stars.
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Cepheid Periodicity
Much like the deep pitch (low frequency) of a large bell, and the
high pitch (high frequency) of a small bell, Cepheid variables ring
with different frequencies.
Big, bright Cepheids take longer to oscillate than small, fainter ones.
By measuring a Cepheid’s period, one can determine its luminosity.
By comparing the luminosity to the Cepheid’s brightness in the sky,
one can determine its distance.
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Calibrating Cepheids
The period-luminosity relationship must be calibrated by
Cepheids of a known distance. It was first done by Leavitt,
using stars in the LMC and SMC.
In modern times, techniques such as trigonometric parallax
from Hipparcos are used to refine the relationship.
Happily, Cepheids are inherently quite luminous (up to 3×104L), so
we can see them at tremendous distances.
Cepheids have been found in:
•
•
•
•
•
•
•
The Milky Way Galaxy;
Large Magellanic Cloud (50 Kpc);
Small Magellanic Cloud (60 Kpc);
Andromeda Galaxy (0.8 Mpc);
M100 (17 Mpc);
NGC 4603 (33 Mpc);
And other galaxies.
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Standard Candles
Based upon the methods described so far, the distances of
many star groups, clusters, star formation regions, galaxies,
and galaxy clusters have been determined.
As a result, all the objects in these groups/galaxies/clusters
have had their distances determined. This lets us examine
these objects, looking for “standard candles.”
A standard candle is an object that, to a reasonable degree
of reliability, has a predictable and consistent brightness, as
if it were factory-built, conforming to a set of
manufacturing standards.
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Proposed Standard Candles
Brightest HII region luminosities
Brightest HII region diameters
Novae
RR Lyrids/W Virginis stars
Brightest O star in a cluster/galaxy
Brightest K-M star in a cluster/galaxy
Brightest X-ray flashes on neutron stars
Brightest planetary nebula in a galaxy
Brightest ellipticals in a cluster
Third brightest galaxy in a cluster
Type Ia supernovae
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Type Ia Supernovae
Recall the scenario
A binary system in which the compact companion is a white dwarf
star near 1.44M. Mass flows from the expanding giant star, forming
an accretion disk. Frequent novae occur.
The white dwarf star approaches 99% the Chandrasekhar Limit.
Uncontrolled C and O burning sweep through the white dwarf,
releasing enough energy to unbind (blow up) the white dwarf. It is not
thought that much (or anything) will remain of it.
The giant may have its outer layers stripped off. In any event, it will
fly away in a straight-line trajectory, as a high velocity star.
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A useful tool
Type Ia Supernovae
Being highly repeatable, it is thought that Type Ia supernovae are very
consistent.
The maximum luminosity is thought to be about 1010L (MB=-20).
The brightness curve of a Type Ia can be identified (especially in the
Blue band), and the maximum brightness inferred.
Compared to the apparent brightness in the sky, the distance can be
determined.
ENORMOUS amounts of computational efforts have been made, to
model/calculate the details, so we can use this tool as accurately as
possible.
Warning: the exceptionally rare coalescence of white dwarf double
stars could also mimic the appearance of more conventional Type Ia
supernovae, without the reliable brightness.
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Concept
Tully Fisher Relation
Lgalaxy (brightness) should be proportional to Mgalaxy.
Lgalaxy should be proportional to its rotational speed (more
massive galaxies spin faster; Newton’s laws).
The rotation redshifts the spectrum on one side of a galaxy, and
blueshifts the spectrum on the other side.
Overall, this broadens the width of a spectral lines.
Therefore, the width of spectral lines is related to Lgalaxy.
Measure the galaxy’s linewidths (brightness), and
compare this to its measured brightness in the sky to
determine distance.
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This method ruled from around 1970-1990.
Faber Jackson Relation
The Tully Fisher relation can be adapted for elliptical
galaxies too.
While the stars in ellipticals do not orbit as stars in a
spiral galaxy’s disk, they buzz around the center like a
swarm of bees.
Concept
Brighter ellipticals are more massive.
More massive galaxies have more gravity.
More gravity means faster star motions.
Faster stars mean broadened spectral lines.
Measure the spectral linewidths, calculate the brightness,
and compare it to how bright the elliptical looks in the
sky.
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Hubble’s Law
Working at Mt. Wilson Observatory, Hubble
determined the distances to galaxies from a number
of methods (Cepheids and standard candles), and
from this he was able to infer that all distant galaxies
are receding from us, following Hubble’s Law.
v=Hd
H=72 km/s/Mpc
This law could be interpreted
using Einstein’s theory of
General Relativity.
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