AS1001 ASTRONOMY & ASTROPHYSICS
LABORATORY PRACTICAL
PLANETARY MOTIONS
Introduction
All the planets are in orbit about the Sun, because the Sun is the most massive object in the Solar System - it contains about 99.85% of the total mass of the system. Kepler was the first to demonstrate that the apparent motions of the planets relative to the background stars in the sky could be explained by the planets having elliptical orbits with the Sun at one focus of each ellipse. Newton then showed that Kepler's empirical laws of planetary motion could be derived directly from his own laws of motion and his law of gravitation.
Just as the planets orbit the Sun because the Sun is the most massive object, so the natural satellites (or moons) of the planets orbit their parent planet, because the satellite-planet distance is much smaller than the planet-Sun distance. In this laboratory practical, you will analyse two sets of observations made to determine
a)	the motion of a satellite about a planet, and
b)	the motion of a planet about the Sun.
The types of observation are different, but the end in view is the same, namely, to determine the mass of the central body about which the orbital motion is taking place.
I. The orbits of Io and Callisto about Jupiter
The satellite Io is the innermost of the four Galilean satellites of Jupiter, whilst Callisto is the outermost. These moons were discovered by Galileo, in 1610, within the first few months of him building a telescope that had the necessary light grasp and resolving power. They can be seen clearly through the Meade telescopes at the Observatory as points of light moving in their orbits about Jupiter. The orbital periods of the two innermost satellites (Io and Europa) are short enough for significant movements across the sky to be seen in just one or two hours of observation with those telescopes - try the voluntary observing sessions at the Observatory!
This practical utilises a computer simulation of the orbital motions of the Galilean satellites about Jupiter, in conjunction with Kepler's third law, to determine the mass of Jupiter in units of the Sun's mass.
Kepler's third law may be written in the form

where G is the gravitational constant, MJ is the mass of Jupiter, MS is the mass of the satellite (negligible by comparison with that of Jupiter), a is the semi-major axis of the satellite's orbit, and P is its orbital period. Since the orbits of these four satellites are all very nearly circular, the semi-major axis is the radius of the orbit.
In units of solar masses for MJ, astronomical units (AU) for a, and Earth years for P, the constants G and 4p2 cancel out, and the third law is written
,
where MQ signifies solar masses. Thus the determination of the orbital period and orbital radius for a satellite will yield the mass of Jupiter in units of the Sun's mass. Since separate observations (the orbital velocity of a planet about the Sun) will yield the mass of the Sun in kg, we can then convert the mass of Jupiter into standard SI units. 
Alternatively, direct velocity measurements via spectroscopy of the orbital motions of the Jovian satellites would provide the mass of Jupiter directly. However, these observational techniques were not available before the early part of this century, and it is instructive to use the relatively simple technique of measuring angular separations on the sky to provide accurate relative masses.
Procedure
The computer simulation of the motions of the four Galilean satellites about Jupiter presents a realistic view of what we see via a telescope. That is, Jupiter as a disc with markings (bands and zones), and the four satellites as tiny points of light arrayed about it, nearly all in the same plane as Jupiter's equator.
Kepler's third law tells you directly that the satellite, Io, nearest to Jupiter, will orbit much more quickly than Callisto, furthest away. (To be able to explain these motions was a major success of Newtonian gravitation.)
Ask a demonstrator to help you start the CLEA - Jupiter's Moons program on a PC in the Laboratory. Start your observations with the default date, i.e., today. Make the interval between observations equal to 1 day (24 hours). Identify the satellite Callisto by positioning the mouse pointer and clicking on the Left button. When you position the pointer accurately, the name of the satellite will be shown; if not, try again......

On the screen you will see the date and time of the observation in the lower left corner, and the projected (east or west) separation X of the satellite from Jupiter in the lower right corner, e.g., X = 10.2W. The units of X are Jovian diameters - that is, angular measures on the sky have been converted to linear measures for you by means of the small-angle formula, knowing the linear diameter of Jupiter. One Jovian diameter is 9:54 x 10-4 AU.

Make a table of date of observation vs. separation X for Callisto. The easiest date to record is the Julian date (JD) which is a continuous running number rather than a calendar date. For this practical, it is not necessary to record the entire number. Only record the latter part, e.g., JD2450190.1234 becomes 190.1234.
When you have made and recorded your measurement of X and JD, click Next and move to the next image (24 hours later). Repeat the process to obtain 22 observations.
Then re-do the process for the satellite Io, restarting the program at the default date, but this time changing the observation interval to 3 hours, rather than 24 hours. Again you will need 22 observations.
When all measurements are complete, plot graphs of X as a function of JD for Callisto and for Io, then draw a smooth curve through all the points for each satellite. They should be elegant sinusoids. Measure the periods P and semi-amplitudes R of the two sine/cosine curves, not forgetting to estimate the uncertainty in P and R.
For each satellite, calculate its orbital radius, and hence the mass of Jupiter. The two values of MJ should be in good agreement at about 10-3 MQ. Do your values agree within the errors?
II. The Orbit Of The Earth About The Sun
We live on a planet which is continually moving in its orbit about the Sun. Since we conduct all astronomical observations from this moving platform, or from a satellite in orbit about it, it should be reasonably obvious that we have to know precise values for the parameters which describe that orbit. All astronomical observations must be corrected for the motion of the Earth about the Sun, so we have to determine that motion by observing stars which do not participate in that motion. Of course, the stars themselves are not standing still in space, but are in motion themselves (just like the Sun) in the gravitational field of the entire Milky Way Galaxy. This practical illustrates this point, by asking you to determine the speed of the Earth in its orbit by means of a series of observations of the bright star Aldebaran (a Tau).
This practical uses the Doppler effect to determine the speed of the Earth in its orbit. The Doppler effect is the apparent change in wavelength of radiation due to the relative motion between the source of radiation and the observer along the line of sight. Hence we may determine the size of the Earth's orbit and the mass of the Sun from these results.

In the above diagram, the Earth moves in an orbit (very nearly circular) around the Sun. The orbital plane is called the ecliptic. The star is at a very much greater distance than the Sun-Earth distance, and has its own motion in space. At point A in its orbit, the Earth is moving directly away from the star; half a year later, it is at point Band is moving directly towards the star. At C, the Earth is moving at right angles to the direction to the star and hence has no component of orbital motion towards or away from the star.
Because of the Doppler effect, when the Earth is at A, lines in the stellar spectrum are shifted towards longer wavelengths (red-shifted) relative to their position when viewed from C. At B, the lines are shifted towards shorter wavelengths (blue-shifted) relative to that seen at C. The Doppler shift will be proportional to the line-of-sight, or radial velocity difference between the Earth and Aldebaran. Thus, spectra of the star obtained over many months will show a range of velocities. The series of spectra used in this practical covers the intervals January - March and October - December, that is, around point B, and near point A in the diagram.
There are three complications:
(i)	the star may not be in the ecliptic plane, so that only a proportion of the Earth's orbital motion is projected into the direction to the star - but the star Aldebaran is only 5º from the ecliptic, so the correction factor of 1/cos 5º = 1.004 may be ignored;
(ii)	(ii) the Earth also rotates on its axis once per day, which adds or subtracts further velocity in the direction to the star - but since the maximum rotation speed (at the equator) is 0.47 km s-1 it may safely be ignored in this practical;
(iii)	the Earth's orbit is slightly eccentric (e = 0:017) resulting in orbital speed variations, but these do not exceed 3.4%, so again, for the purposes of this practical we may ignore the correction.
The equation for converting wavelength changes due to the Doppler effect to velocities is simply
,
where l0 is the stationary or rest wavelength of a given line in a spectrum, Dl=lobs-l0 is the Doppler shift of the line observed to be at a wavelength of lobs, V is the relative radial velocity between the Earth and the star, and c is the speed of light, 3x105km s-1.
Procedure
Collect a set of spectra of the star Aldebaran, taken with the spectrograph on the Leslie Rose Telescope at the University Observatory, St Andrews. These spectra, recorded via a Reticon diode-array detector, have been flat-fielded and wavelength calibrated. Only 50 Å of the original 700 Å of spectrum are plotted at a scale suitable for this practical. The dates of observation are given in the following table, which also shows the other columns you will need.
Spectrum No.	Calendar Date	Day No.	Dl (mm)	Dl
(Å)	DV
(km s-1)	phase
f	sin(2pf)
sin(360f)	
1418	6 Jan	6						
1499	28 Jan	28						
2330	12 Feb	43						
2357	25 Feb	56						
2389	5 Mar	64						
0625	21 Mar	80						
3567	12 Oct	285						
1883	3 Nov	307						
1939	19 Nov	323						
2015	2 Dec	336						
2078	13 Dec	347						
This practical introduces you to a rudimentary graphical version of a very powerful technique for determining Doppler shifts in spectra, namely cross-correlation. You will need a light-box and a 10x eyepiece.
Use the spectrum numbered 2389 (5 March) as the template or reference spectrum, and determine the Doppler shifts of the other ten spectra relative to it. Place the template on top of the spectrum to be measured and simply shift the former along the wavelength axis until you obtain a best-match by eye of the positions of the absorption lines. Do not concentrate entirely upon one absorption line, but use the 5 or 6 obvious lines, and ignore the "wriggles" in the continuum. The shifts in the spectra are small - 3 mm at most - but they are measurable with care.
When you have a match, use the eyepiece to measure the offset in mm between the two spectra. Note carefully whether the spectrum you are measuring is red or blue shifted with respect to the template.
Record these values in a table like the one above, together with +/-  signs for redshifts/blueshifts of observed spectra relative to the template. Convert these shifts to wavelength units (Å), and hence determine velocities relative to the template spectrum by means of the Doppler formula. Use l0 = 4160 Å , and c = 3 x105 km s-1.
For purely circular motion of the Earth about the Sun, and a star at constant radial velocity, you would expect this velocity difference to follow a sine curve:
,
where K is the semi-amplitude of the curve (the orbital speed of the Earth in its circular orbit) and f is the orbital phase.

The template spectrum 2389 was obtained close to point B in the diagram. Assuming that it is exactly at point B, calculate f for each observation from

Hence plot DV vs. sin(2pf) or sin(360f), running from -1.0 to 1.0. Your data should lie on a straight line (although the scatter may be large, owing to measuring errors). Determine K, and its uncertainty, from your graph.
Assume that the derived velocity K is a good approximation to the mean speed of the Earth in its slightly eccentric orbit, and that the orbit may be approximated by a circle. Given that the orbital period P = 1 year = 3.156 x 107 s, determine the radius a of the Earth's orbit and its associated uncertainty.
Then use Kepler's third law to determine the mass of the Sun in kg (and its error!) on the reasonable assumption that the mass of the Earth is small compared with that of the Sun (in fact it is about a million times smaller - an acceptable approximation!). Kepler's third law may be written
.
The gravitational constant G = 6.67x1011 m3s-2kg-1.
Questions
1)	Most stars have measured radial velocities of less than 100 km s-1. Would you expect to detect any colour changes as a result of the Doppler effect?
2)	Because the Earth itself is moving, all our radial velocity measurements are relative. Radial velocities of stars are usually corrected to a reference point which removes the effects of the Earth's orbital and rotational motion. Where is this reference point?
3)	(For those who like a challenge!) The form of the DV = K sin(2pf) relation suggests that the intercept of a velocity{phase plot should be zero. Is it zero on your graph? What might be the physical reason for a non-zero velocity intercept?