iSkylab Manual – Stage 4
Dr. Uwe Trittmann, Otterbein University, Fall 2014
Due December 8, 2014
INTRODUCTION
An incremental Skylab or iSkylab is a term paper with an observational component. The paper
will be developing over the course of the semester as you add a new section every three weeks or
so. This allows for revisions, as the individual sections are graded and handed back. In the end
you will have explored one aspect of observational astronomy in detail, and documented your
results in a proper, scientific way.
I will grade and return your Stage 3 reports soon. You can and should revise them – and will
receive up to 25% of lost points. Incidentally, this is how good writing and science progress: start
by making a good faith effort, and then iteratively improve by listening to feedback and by
learning from mistakes.
STAGE 4 – Conclusion: Using the refined theory and additional observations to discover
the expansion of the universe
By observing the positions and appearances of the sun and planets in the sky, we were able to
build a consistent theory of planetary motion. From many, but isolated, data points (positions on
different dates), we induced a general, but abstract theory, that stipulates that the planets orbit the
sun in ellipses, moving faster when they are closer to the sun in a well-defined way (equal-area
law). We thereby explain the appearances in the sky by hypothesizing that the planets “really”
rotate around the sun, including the observer on Earth. Incidentally, the observer rotates both
around the sun, and about the earth’s axis. The first, orbital rotation explains the motion of the
sun, the moon and the planets with respect to the stars, the latter rotation explains the daily rising
and setting of stars, sun, moon and planets, i.e. their motion with respect to the observer’s
horizon. Note that we have no way of seeing the earth and the planets move around the sun in
their orbits. I pause here to let this sink in. All we can see is the apparent motion of stars, sun,
moon and planets in the sky, i.e. how their stipulated actual motion appears to us as projected
onto the sky seen from Earth. We cannot see behind the curtain of appearances (if you are
philosophically inclined, this is reminiscent of the Allegory of the Cave in Plato’s Republic); we
can only theorize about what is actually going on. Indeed, there is another possible explanation
of the rising stars, the seasons and the retrograde motion of the planets: the geocentric model of
Ptolemy. It was, however, falsified by additional observations made possible by the advent of the
telescope. Ptolemy’s model makes predictions about the phases of Venus that are wrong, i.e. not
observed. We thus had to tentatively embrace the other, heliocentric model of Copernicus as
correct due to the lack of an alternative. Copernicus’ model turned out to be wrong, too (planets
do not move on circles, there are no epicycles), and was improved by Kepler, as described above.
Note that you could argue that Kepler’s theory of the solar system discarded Copernicus’, since it
is very different: the ellipses-for-circles switch in fact replaced the old notion of orbs (solid
spheres) with the modern concept of an orbit. Later, Newton modified Kepler’s theory by finding
an explanation for the elliptical orbits. They are the result of the combination of the sun’s
gravitational force, and the planet trying to move in a straight line due to its inertia. Again, you
could argue that Newton toppled Kepler’s theory, because de facto the planets do not move on
ellipses, since planets also exert gravitational forces on other planets, and perturb, i.e. alter, their
orbits. The story of our refined description of the solar system goes on with Einstein, who
supersedes Newton by claiming that, in fact, there is no gravitational force, but only motion in a
space-time that is curved in the presence of a large mass like the Sun’s. We have no way of
observing curved space-time, in the same way that we had no way of observing the orbits of
planets, but we can predict where a planet should be in the sky at a given time according to
Einstein versus according to Newton – and then observe its position at that time. It turns out that
you need to use a pretty big telescope to see that Einstein’s prediction is better by a ridiculously
small amount (shift of the perihelion of the planet Mercury is a few tens of arcseconds larger per
century). So Newton was wrong, but his theory is such a good approximation, i.e. such a good
description of Nature, that we frequently use it today, because it is good enough for most
practical purposes. This takes us to the present day. We already know that Einstein’s theory of
General Relativity is not the last word, since it does not conform to the rules of quantum
mechanics, but we don’t know (yet?) what to replace it with. Stay tuned.
As this expansion of our understanding of the universe has taken place, it has allowed us to find
out about an actual expansion of the universe by carefully measuring the distances to very
remote objects. Being able to explore this cosmic expansion is the crowning of our efforts this
semester. Let us review how the interconnected distance measurements give rise to a cosmic
distance ladder that will reach from the Earth all the way to the most distant objects that we can
observe from earth.
The understanding of planetary motion described above is the basis of the first rungs on the
cosmic distance ladder. We computed the length of the Astronomical Unit (distance Earth-Sun)
by using Keplers’s laws combined with a measurement of the speed of light in the lab. This tells
us that the Earth’s orbital positions half a year apart constitute a baseline of 300 million km for
the triangulation of the distance of nearby stars with the help of the stellar parallax. Knowing the
distances to several stars allows us to understand how stars produce their energy, and how their
different properties (luminosity, spectral class, mass, temperature, size) are related. In particular,
if we can somehow determine the luminosity L of a star, say by its spectral class, and we see how
bright (B) the star appears in the night sky, we can determine its distance d, since B is
proportional1 to L/d2. This is the basis of the spectroscopic parallax that takes us out to larger
distances if we focus on very bright stars. We get even farther, i.e. climb the next rung of the
distance ladder, if we use the parallax to measure the distance to peculiar stars that change their
brightness by changing their energy output (luminosity) in a regular pattern. Their luminosity is
proportional to the pulsation period which we can measure easily by observing the night sky.
Since these stars are extremely bright, they can be seen in other galaxies, and thus give away the
distance to these galaxies. As the (tentative, barring future discoveries) final rung of the cosmic
distance ladder, it was then discovered, on the basis of the distances of several galaxies, that
there is a correlation between the distance of a galaxy and the amount of the Doppler red-shift in
its spectrum. Namely, the farther out the galaxy is, the greater the red-shift. The interpretation is
clear: red-shift means that the object is moving away from us, so all (distant) galaxies recede
from us. In other words, the universe is expanding.
Now let’s put these ideas to work and climb the distance ladder to measure how fast the universe
is expanding! (Hint: please refer to the Excel file linked on the course homepage to help you
with the math on the first two questions.)
1. Due to the orbital motion of the earth you observe the parallax of two Cepheids called
Otterbaran and Beinetnash. You observe during several nights that their apparent
brightness changes by comparing it to the brightness of other stars nearby in the sky. It
turns out that Otterbaran is brightest every 2.5 days, while Beinetnash is brightest every
12 days. You also observe that, in half a year, Otterbaran shifts 0.4” with respect to the
background stars, while Beinetnash shifts 0.02”. The maximal brightness of Otterbaran is
3•10-7 W/m2, and of Beinetnash 4•10-9 W/m2. Use these observations to construct a
period-luminosity relation for Cepheid variables. This is nothing but a straight line in a
xy-diagram where the luminosity is plotted on the vertical axis, and the period in days is
on the horizontal axis, see activity on Cosmic Yardsticks.
2. Now that you know how to determine the luminosity of a Cepheid by measuring the
period of its varying brightness in the night sky, you go and observe a Cepheid each in
two galaxies named Aries and Bootes. The Cepheid in Aries gets brightest every 10 days,
and the one in Bootes every 30 days. You observe the maximal brightness of the Cepheid
in Aries to be 5•10-20 W/m2, and of the one in Bootes to be 7•10-23 W/m2. Also, you let
the light of these galaxies fall through a powerful telescope onto a diffraction grating and
discern a lot of absorption lines. In particular, you measure that the prominent red
Hydrogen line, which you measured in your lab at 656.28 nm, is shifted. The one in the
Aries’ spectrum is at 680.43nm, and the one in Bootes’ spectrum is at 846.57nm.
a. Compute the distances and the recessional velocities (via the Doppler effect) of
the two galaxies.
To be specific, we have B=L/(4πd2), because the energy produced by the star each second (its power output or
luminosity) spreads out over the surface of a sphere which has the area 4πd2 when you are a distance d away from
the star.
1
b. Plot the velocities versus the distance and put a straight line through the two
points. Determine the slope of the straight line: this is the Hubble constant!
c. Note that the Hubble constant has dimensions of velocity/distance = 1/time.
Therefore its inverse is a time, namely the time it took the universe to expand to
its current size, i.e. it is the age of the universe. Calculate the age of your
universe and express it in billions of years. Compare it to the accepted value. Is
your result accurate, precise, both or neither? Explain.
3. Reconsidering the cosmic distance ladder, explain in prose (no equations!) what you did
in the first two questions, and why our ability to measure the expansion of the cosmos is
inexorably linked to measurements of distances on Earth.
4. Describe, in your own words, how the discovery of the expansion of the universe is
linked to the intellectual expansion of our understanding of the universe that has
happened since Ptolemy. You might want to consider (and read up on), e.g. why Ptolemy
could not have discovered or even fathomed an expanding universe, why Newton did not
know the distance even to the nearest stars, why Herschel had no way of getting the size
of the Milky Way right, why Shapley did think the Milky Way is the universe, etc.