Text given in blue is for Professors only.
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Divide students into groups of 3-4 members.
Each group should get a balloon, a student handout for the activity (separate file),
paper ruler (separate file). You should also provide 2 or 3 Sharpie pens for
sharing among the groups.
You may want to distribute the first and second parts of the student handout
separately, so that you can ensure all groups have finished the first part before the
class starts on the second part.
Put a set of axes on the board so that students can graph their results as they
finish; make the axes big enough so that all the data will fit, and have different
groups use different symbols/colors of chalk so you can tell the data sets apart.
The expectation is that all data points collected by the different groups will fall
into approximately the same linear relationship; there will be deviations from this
if some groups don’t blow up the balloons very far or if points are placed too
close to the neck or the far end of the balloon, which tend to inflate less uniformly
than the rest of the body.
Hubble’s Law: Part I
Take a balloon and make three marks and an ‘origin’ (i.e., a different type of mark) on it.
Be careful not to puncture the balloon. Place the marks at a range of distances from the
origin, with one fairly close and one as far away as possible. Before blowing up the
balloon, measure the distances from the origin to each of the three marks.
DISTANCES PRIOR TO INFLATION OF THE BALLOON
Point 1 to origin
Point 2 to origin
Point 3 to origin
(cm)
(cm)
(cm)
Now blow up the balloon (all the way!) and measure the distances again.
Point 1 to origin
(cm)
DISTANCES AFTER INFLATION
Point 2 to origin
(cm)
Point 3 to origin
(cm)
Assume that it took you 10 seconds to blow up the balloon. Based on the change in
distances (distance after inflation – distance prior to inflation) you measured above,
calculate the average speeds in cm per second at which the points on the balloon were
moving away from the origin during the expansion of the balloon. Additionally, for each
point calculate the ratio of distance from the origin after inflation to distance from the
origin prior to inflation.
Speed of Point 1 wrt origin
(cm/sec)
SPEED OF EXPANSION
Speed of Point 2 wrt origin
Speed of Point 3 wrt origin
(cm/sec)
(cm/sec)
DISTANCE AFTER INFLATION / DISTANCE PRIOR TO INFLATION
Point 1 distance ratio
Point 2 distance ratio
Point 3 distance ratio
Questions:
1. Are the ratios of distance after inflation to distance prior to inflation for the different
points similar? If so, what does this mean? If not, what does this mean?
They should all roughly be the same. This is because the ratio reflects the fact that the
balloon has grown (approximately uniformly) by some overall scale factor. If they aren’t
approximately the same, then there has been a mistake, or for some reason the balloon
didn’t expand very uniformly (such as not being inflated all the way).
2. Sketch a graph of speed versus final distance from the origin on a separate page. If
appropriate, add a trend-line. How would you expect this plot to look if your origin had
been located somewhere else on the balloon?
The graph should look like Hubble’s law.
If the origin was somewhere else, you ought to have about the same result. This is
because all the points get stretched about the same no matter which one you set as the
origin. In other words, all galaxies appear to move away from US simply because all
points move away from all other points.
Hubble’s Law: Part II
Another way to think about the observed velocities of galaxies in the Universe is that the
distance at which we observe an object today is simply determined by its recession
velocity times the age of the Universe (i.e., distance = velocity x time); that is, that faster
moving objects have gotten farther away from us in the available time. Thus, we can use
the plot below to estimate the age of the Universe.
Note: 1 Mpc = 3.1 x 1019 km. Data are from Freedman et al. 2001, The
Astrophysical Journal, Vol. 553, p. 55.
Questions:
1. In order to estimate the age of the Universe, you will need to extract data from the plot
above. Do you think it makes more sense to use the data points themselves, or the trend
lines? Why?
The trend line represent an averaging of the data points, so they are the better choices in
some sense, since otherwise you would have to estimate the age based on each data point
and average over them, basically duplicating the effort.
2. From the graph above, you can extract distance and velocity. Dividing one by the other
will yield a number with units of time (e.g., seconds). What major assumption do we
have to make in order to translate this number into a quick estimate of the age of the
Universe?
The assumption is regarding a constant velocity of expansion; additionally, we are
assuming that the time in the past when the positions of all the galaxies converge
corresponds to the beginning of the Universe.
3. Based on the plot, what is your estimate for the age of the Universe? If the data and
assumption from part 2 are all accurate, what do you think is the approximate error on
your estimate of the age?
Note that the graph uses units of Mpc, rather than Mly (mega light years). Mpc is a
conventional unit in astronomical practice and is about a factor of three larger than one
Mly.
Using 75 km/s/Mpc (the central trendline value), the fact that 1Mpc = 3.1x10^19 km, and
the fact that there are 3.16x10^7 seconds in a year, the answer is:
Age = 1 / (75 km/s/Mpc) x (3.1x10^19 km/Mpc) x (1 yr / 3.16x10^7 s) = 13.1x10^10 yr,
or 13.1 billion years.
The error will be related to the uncertainty in the best-fit slope (note that such fits to
Hubble’s Law data are nowadays always constrained to have zero intercept), also known
as the Hubble constant. The Hubble constant as determined from these data has an error
of +/- 10km/s/Mpc.
Bellow are some additional notes pertaining to the Balloon activity:
What's useful about the balloon as a model? How is it a good model?
- it produces an expansion law qualitatively similar to Hubble's Law, in that the velocity
of expansion (with respect to the origin) is proportional to distance (from the origin)
- it shows that no matter where you draw the origin (i.e., no matter your location), you
will always observe that all points are moving away from YOU
What are failures of the balloon model?
-the balloon is finite
-the surface of the balloon is only 2-dimensional
-the marks drawn on the balloon also expand (they shouldn’t, as galaxies are held by
gravity)
-the balloon may not expand uniformly (especially if not inflated fully)
Velocity proportional to distance:
Once the Hubble Law has been established (calibrated), we can use the measurements of
the recession velocities of galaxies (which are relatively easily obtained) to deduce their
approximate distances. This is valuable because measuring astronomical distances is a
very difficult problem, always requiring assumptions and independent verification where
possible.
Velocity proportional to distance, implications:
If you go to any point on the line (on the Hubble diagram) and ask, how long would it
have taken for this galaxy to get this far away if it traveled at the speed we now observe it
to have, you will always get the same answer: an amount of time equal to 1/slope of line.
That is, the data are roughly consistent with a picture in which all galaxies started at the
same location at some time roughly 13.7 Gyr ago and have expanded outward at a
constant speed ever since. Thus we can estimate the age of the Universe as the time since
the beginning of the expansion.
If we used Hubble's original data from 1929 we would get an age estimate about 7 times
lower, or about 2 Gyr. It turns out that Hubble's distance estimates were systematically
off due to calibration problems with his distance-finding method.
The scatter of points around the line could reflect several factors:
- errors in measuring the distance to each galaxy
- errors in measuring the Doppler-shift radial velocity of each galaxy
- the fact that galaxies are also affected by one another's gravity; that is, some of the
scatter is real, in that galaxies may have "peculiar" velocities related to being drawn
toward local strong concentrations of matter (e.g, galaxy clusters) - expansion velocity is
not a perfect indirect measure (proxy) for distance.
Expansion and redshift of the electromagnetic radiation (Cosmic Microwave Background
Radiation):
Another way in which the balloons are accurate is that wavelengths of light expand
(redshift) in proportion to the expansion. One can think of this as a Doppler shift
(galaxies moving away from us experience a redshift).
On a powerpoint that is posted there is a slide showing several "pie diagrams" with 2dimensional views of the distribution of galaxies in space. In each of these diagrams we
are located at the center. The distribution of galaxies is highly non-random; they cluster
together, and there are certain regions of space (called voids) in which galaxies are rare.