Download Astronomical Distance Ladder

Astronomical Distance Ladder
Date:
08/03/01
To:
Minister Gedda, 32 Edington Park, Edinburgh
From:
Kevin McCabe
RE:
Astronomical Distances
In response to the request for information regarding interstellar distances I have composed this report
detailing current knowledge of where we are in the universe and the techniques used to determine this
belief. From some approximate calculations the size of the observable universe is roughly 10 billion light
years. According to Edwin Hubble the universe we live in is expanding and everything is expanding
uniformly. The larger the distance between galaxies the faster they are moving apart. This distance to a far
away galaxy (D) is determined by using the Hubble constant (Ho) and the recession velocity of a distant
galaxy (V) in the equation
D V
Ho
. The velocity is found using the red shift of light being received from
distant galaxies. However, there is some uncertainty in Ho that arise from the different ways that Ho is
determined. Some values of Ho (100km/s/Mpc) give an approximate age of the universe around 6.5 billion
years old while others (40km/s/Mpc) produce ages of 16.5 billion years. This inaccuracy arises from the fact
that Ho is found by using an astronomical distance ladder, where farther distances are calibrated using
known techniques which are more accurate, however, this makes the error cumulative.
The most accurate distances known are the found here in our own solar system. The distance to objects
such as the rocky planets and asteroids can be found by bouncing radar signals off of them and measuring
the time it takes for the signal to return. By using these distances and the relative movement of these
planets to the sun and the earth we can then determine the distance between the sun and the earth. The
easiest way is to wait until the moon is exactly at its first quarter, which means it is at a 90o angle between
the earth and moon. After measuring  and the earth-moon distance (EM) by radar, the earth-sun distance,
also the hypotenuse (ES) of the sun-moon-earth triangle can be determined by trigonometry.
ES  EM COS
This distance ES (149,597,892 km) is now known as 1 Astronomical Unit (A.U.).
Now a technique called trigonometric parallax can be used to calculate the distance to nearby stars. This
works by observing the apparent movement of nearby stars against the background stars as the earth orbits
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Astronomical Distance Ladder: Astronomical Distances
the sun. After measuring the angle  (arcseconds) that the star has moved in half a year a small angle
approximation can be use to determine the distance between the sun and the star. d=206265”/” A.U.
206265 A.U. is defined as one parsec (pc), this standard was chosen because d=1 pc when =1. The
space program Hipparcos has recently made observations of parallax of approximately 1/1000” which gives
a maximum parallax distance of a few hundred parsecs.
Beyond these distances a method called proper motion is used. This is done by observing the motion of
stars in a cluster that is moving away from the solar system. To an observer here on earth it would appear
that after an infinite time all the stars in the cluster would seem to vanish at one point an infinite distance
away called a convergence point. The angle between the cluster and the convergence point is equal to the
angle between the radial velocity and the true velocity of the cluster. The transverse velocity (vt) can then
be calculated using this angle () and the radial velocity (vr). vt  vr tan  The distance to the cluster is
calculated by measuring the angle the object moves in the transverse direction in one year (also called
proper motion angle []) and calculating the transverse velocity for one year and using a simple sin relation.
d cluster  d t sin 
This proper motion technique is capable of giving reliable distances up to around 800 parsecs.
With the distances to many stars in the galaxy calculated by parallax different ways to calculate distance
was needed to extend the astronomical distance ladder. It was also discovered that stars radiated like black
bodies. This means that for a star at a specific temperature there would be a peak emission of radiation at a
certain wavelength. It was later discovered that stars that were still burning hydrogen, as it’s main source of
fuel all lie near a curve on an absolute magnitude vs. temperature diagram. This curve is called the main
sequence of stars. It is then possible to use a standard candle that relates the apparent magnitude of the
star’s luminosity to the absolute magnitude of its luminosity. Since it is know that flux from a given object is
proportional to the inverse of distance to the object squared. This distance is found from the equation
D=100.2(m-M)+1 where m is the apparent magnitude and M is the absolute magnitude. This gives you a
distance in tens of parsecs. By this standard candle approach to main sequence stars distances of up to
10,000 parsecs.
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Astronomical Distance Ladder: Astronomical Distances
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After distances of a few hundred parsecs it is necessary to look at the constant intrinsic properties of certain
objects such as cepheid variable stars, RR Lyrae, novae, and supernovae. Distances to them are calculated
fairly simple using the standard candle approach. Both types of cepheid variable stars are pulsating stars
whose period and luminosity are dependent on the mass of the star. Thus at specific periods every cepheid
of one type has the same luminosity. The standard candles approach can be used for supernovae, novae,
and RR Lyrae stars as well. The luminosity for a nova is determined by measuring the time it takes for the
explosion to dim 6 magnitudes from its peak luminosity. The shorter the time for the decay the brighter the
novae is and the distance can be calculated. The luminosity of supernovae are found the same way as for
novae, however, a different stellar process causes the explosion. For novae the explosion is caused by a
white dwarf accreting matter from a binary star and when it reaches a certain mass it sheds an outer layer of
mass in an explosion to keep it from collapsing. A supernova explosion is caused by a star that has
accreted sufficient matter from a binary companion and the collapse of the core blows the entire star apart.
RR Lyrae stars are identified by their period much like cepheid variables, however, their period is much
shorter and the spectrum is different. So the same technique is applied to find their distances. For RR
Lyrae stars a distance of around 100 kiloparsecs is possible from this approach, while for Novae a distance
of around 1 Megaparsec is capable, cepheid variable distances of up to 6 Mpc are attainable, and finally
supernovae give distances of 100 Mpc rather reliably.
Another reliable method for determining extragalactic distances is by a relation discovered by Tully and
Fisher. It was shown that in disc galaxies the luminosity is related to the rotation velocity. For disc galaxies
the luminosity is proportional to the size of the galaxy in other words L=c1 x R2. However, it is known that at
a given distance R the velocity should be V2=GM/R, then solving for R gives R=GM/V2. But M=c2 x L,
substituting for M and R, L=c3 x L2/V4 which gives us L=(1/c3) x V4. The velocity can then be found by the
Doppler shift of certain wavelength for these galaxies, typically infrared wavelengths are used because they
are less prone to interstellar absorption. This technique is also very good when used to find the Ho because
it is capable of reaching distances of around 60 Mpc, which is a good distance to work out relative recession
velocities. “The diagram below shows two galaxies: a giant spiral and a dwarf spiral, but the small galaxy is
closer to the Earth so they both cover the same angle on the sky and have the same apparent brightness.”3
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Astronomical Distance Ladder: Astronomical Distances
Finally a technique known as gravitational lensing can be used to calculate the distance from the lensing
galaxy to the sun. This technique is only currently being used for quasars that are being lensed. This is
because quasars have a time dependant variance in the light they emit. If two quasars are found near each
other, and the signal received from both of them is the same except for a time phase change it could be
possible that the light from a single quasar is being bent around a very massive object, usually a galaxy. If
the time delay can be found it would be possible to tell the difference between the distances the two rays of
light that were emitted at the same time had to travel. These calculations give us a range of distances up to
5000 Mpc.
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Beyond the distances granted by gravitational lensing and supernovae only the Hubble relation could give a
relative idea of the true size of the universe. These red shift calculations are the main tool for finding
distances to the edge of the universe. Until technology can extend the capabilities of primary distance
indicators and increase the precision of the secondary distance indicators the Hubble constant will remain
very inaccurate. Therefore, the age and size of the universe cannot be confidently known. Without this
knowledge mankind will never know its place in the cosmos.
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References:
1. http://www.sciam.com/specialissues/0398cosmos/0398freedman.html
2. http://www.i5ive.com/article.cfm/astronomy/11999
3. http://www.astro.ucla.edu/~wright/distance.htm
4. http://www.roe.ac.uk/~jac/astronomy1bh/greatdebate1.pdf
5. http://phyun5.ucr.edu/~wudka/Physics7/Notes_www/node111.html
6. http://www.tac.dk/~lars_c/thesis/node16.html
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