Laboratory #6
Fall 2017
Orbits in the Center of the Milky Way
1. This lab will be completed in class and you will need to work with a lab partner.
2. Working with your lab partner(s), go to blackboard and complete the pre lab assessment for Lab #6.
a) First complete the assessment using your account.
b) Once you are finished, log out, let your lab partner log in, and repeat the assessment so that each
person has completed it.
3. Once each of you has completed the assessment, begin the lab.
4. In this lab we will determine the orbits of stars near the center of the galaxy.
a) Stars orbit the center of the galaxy, which is called Sgr A*, in ellipses.
b) Using large telescopes we can see those stars moving around those ellipses.
c) Unfortunately, those ellipses are not always in the plane perpendicular to our line of sight; the
ellipses are distorted by the angle at which we see them. After the distortion the stars still appear to
be orbiting in an ellipse, just a different ellipse.
d) In this lab we will use data measuring the apparent ellipse in order to find the true ellipse in which
each star is orbiting.
5. Scientists share data in what are called “Peer Reviewed” publications. That means that in order to
publish their results another scientist (a “peer”) who is not involved in the work must look at the work
and determine if it is suitable for publication. In this lab we will use peer reviewed data published by
the Astrophysical Journal.
a) Go to the website http://iopscience.iop.org/article/10.3847/0004-637X/830/1/17(this link will only
work from on campus).
b) Scroll down to figure 5 and save this figure.
c) Next go to table 3 and download the table.
d) Next go to table 4 and download the table.
e) Finally go to table 7 and download that table.
f) Draw a slip of paper from your TA. You will be either working of the star SO-2 or SO-38. The slip
will tell you which star to use.
g) Table 3 contains the positions of the star SO-38; Table 7 contains the positions of the star SO-2. Use
the data from the table that corresponds to your star.
h) Make a plot of the position of the star (Delta RA and Delta Dec) as it orbits. You may do this with a
pencil and paper or in a program such as excel, numbers, or gnuplot. Your TA will have graph paper
and a straight edge to use if you would like.
i) Find the largest circle which fits inside the data.
j) Find the smallest circle, centered at the same spot as the last circle, which surrounds the data.
k) The diameter of the smaller circle which “contacts” the data on each side is the semi-minor axis of
the apparent ellipse of the orbit. The diameter of the larger circle which contacts the data on each
side is the semi major axis of the apparent ellipse.
l) Draw the ellipse.
m) There are many way to find the true ellipse from this apparent ellipse. We will use the procedure
below (on the third page of this document).
n) Using that procedure find the eccentricity e, and semimajor axis a of the TRUE orbit.
o) Compare your results, and the answers to the questions below, to those of your classmates and the
ones found in table 4.
6. Prepare a lab report using the data, the plots you made, the calculation you performed, and the answers
to the questions below.
a) You may hand write and scan it or you may use a word processor (e.g. Office or Powerpoint).
b) Your report must be in a single PDF format document.
c) You should do this lab with a partner and talk with them about the write up but this write up must be
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
your own work in your own words.
d) You can (and should) compare your results that other people took but for your report you must use
the figures you and your lab partner made.
e) You must each submit your own report.
f) The lab report should use complete sentences to describe what you did, and what you saw.
g) Your instructor will give you additional instructions for the correct format and form of your report.
h) When you have completed your lab report upload it to Blackboard using the link in the Lab 6 folder.
7. Do the post lab quiz on Blackboard. You may, but don’t have to, work with your lab partner on the quiz.
8. You must upload your lab report and complete your post lab quiz within 4 days of your assigned lab
time. 10 points will be deducted for each day either your lab report or your quiz is late.
Questions
1. What is the period of the orbit of your star?
2. What is the mass of the object your star is orbiting? Show your work.
3. What is the Schwartzchild radius of the object your star is orbiting? Show your work.
4. How far is your star from the object at perihelion?
5. How many Schwartzchild radii is that?
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
De-projection Procedure:
A little background: the eccentric circle
Given any ellipse, we can draw its eccentric circleby making the circle which
• has the same center as the ellipse
• has a radius equal to the semi-major axis of the ellipse
In other words, the eccentric circle circumscribes the given ellipse.
Now, if we start with an ellipse of e = 0.8and its eccentric circle,
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
and subject them both to an arbitrary tilt (in this case i = 60 degrees) around an arbitrary line of nodes
(in this case tilted by ? = 30 degreesrelative to the true major axis), we create two new projected
ellipses:
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
Note that the true foci of the orbit, shown as red dots, no longer appear along the principal axes of the
projected ellipse. Also note several properties of the projected eccentric circle:
• its principal axes no longer line up with those of the projected orbital ellipse
• but it does still share its center with the projected orbital ellipse
• it touches the projected orbital ellipse at two points; the line connecting these points is parallel
to the line of nodes
We call the projected version of the eccentric circle the auxiliary ellipseor eccentric ellipsewhich
goes together with the projected orbital ellipse. We are going to use this auxiliary ellipse extensively in
our work below.
Step 1: sketch the apparent ellipse of the orbit
The first step is to plot the measurements of right ascension and declination (labeled Delta RA and
Delta dec in the tables). They should define an ellipse.
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
Step 2: find projection of semi-major axis, and true eccentricity
Now, in the true orbit, we can draw a straight line connecting the center of the true ellipse, the location
of the primary star (at one focus), and the perihelion of the secondary. This line lies along the major
axis of the true orbit.
Yes, it would be more accurate to use the term “periastron”, but I’d rather stick to familiar terms right now.
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
The ratio of (center-to-focus) to (center-to-perihelion) is simply
(center-to-focus) ae
——————- = —- = e
(center-to-perihelion) a
Now, these three points (center, focus, perihelion) remain colinear in the projected ellipse, and retain
their relative positions. That means that we can draw this line on the projected ellipse:
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
The ratio of the lengths CS to CA will again yield the eccentricity of the true ellipse, e.
That was easy! But the next bits involve more work…
Step 3a: calculate the constant k
We are going to draw the projection of the eccentric circle, that is, the auxiliary ellipse. It will take
several steps.
The first thing to do is to calculate a constant value, k, based on the eccentricity eof the true orbit.
Step 3b: draw the projection of the minor axis of the orbit
Next, we draw the projection of the minor axis of the true orbit. Despite being projected, the major and
minor axes of the true orbit remain conjugate. That means we can draw the minor axis like so:
• pick any chord on the observed ellipse which is parallel to the projected major axis
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
• bisect the chord
• extend the line running from the middle of the chord through the center of the observed ellipse
so that it intersects the observed ellipse
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
This line, marked M-M in the figure above, is the projection of the minor axis of the true orbit.
Step 3c: project outwards to define the auxiliary ellipse
We’re ready to build the auxiliary ellipse graphically. It will take a bit of repetitive actions, but each one
is pretty simple. We can mark a single point on the auxiliary ellipse by doing this:
• pick any point X on the projected ellipse
• draw a line parallel to the true minor axis from X to the projection of the true major axis
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
• measure the length of this line; call it d
• extend this line outside the ellipse so that its length is kd, where kis the constant you
determined back in step 3a
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
If we repeat this procedure at different locations around the projected ellipse, we can build up a set of
points which define the auxiliary ellipse.
We can now connect the dots to draw the auxiliary ellipse.
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
Note that the auxiliary ellipse
• shares the same center as the projected orbit
• touches the projected orbit on the projected major axis
• may have principal axes rotated relative to the projected orbit’s principal axes
Step 4: find the true semimajor axis aof the orbit
Once you have drawn the auxiliary ellipse, measure its semimajor (a) and semiminor (ß) axes.
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
Now, recall that the auxiliary ellipse is a projection of the eccentric circle. The eccentric circle had the
same radius as the true orbit’s semimajor axis a. When we tilted it, we squished that circle into an
ellipse … EXCEPT along the axis of rotation. So the longest diameter of the auxiliary ellipse must still
be the same as the original radius of the eccentric circle; but that’s also the same as the semimajor axis
aof the true orbit. To make a long story short, the semimajor axis aof the auxiliary ellipse is the same
as the semimajor axis aof the true orbit!
This lab is adapted from http://spiff.rit.edu/classes/phys440/lectures/fix_tilt/fix_tilt.htmland is thus subject to the terms of the Creative Commons License.
