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The Earth's
Orbital Velocity
Discussion
By measuring the Doppler shift in a stellar spectrum, astronomers can determine the
orbital velocity of the Earth, the radial velocities of stars, and the Earth's orbital radius
(the astronomical unit). In this exercise you will use actual astronomical observations
(spectra) of the star Arcturus (α Boötis) to determine these values. You will also become
familiar with the difficulties of making astronomical measurements and the uncertainties
in the results.
This lab will follow the procedures outlined in the Sky and Telescope reprint entitled
"The Earth's Orbital Velocity." A diagram of the Earth's position with respect to Arcturus
is given below. The two spectra of Arcturus were obtained when the Earth was located
at position "A" and position "B" in this diagram. We are looking at the Earth’s orbit from
above the ecliptic plane, making the orbit appear much more elliptical. Find Arcturus on
the SFA Star Chart to verify that it is also above the ecliptic.
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Astronomy 105 Laboratory Manual
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Earth’s Orbital Velocity
At time "A" the motion of the star relative to the observer (Earth) is
VA = Vstar + Vearth .
At time "B" the motion of the star relative to the observer (Earth) is
V B = Vstar − Vearth
Solving these equations, the Earth's velocity is given by
1
Vearth =
2
(VA − VB )
0.86
,
(1)
and the star's velocity is given by
Vstar =
1
2
⋅ (VA + VB ) .
(2)
Vearth was divided by 0.86 to correct for the fact Arcturus is not located in the plane of
the ecliptic.
By measuring the Doppler shift in the spectra of the star Arcturus you can easily
calculate VA and VB using the Doppler equation. The Doppler equation is given by
v=
where
c ⋅ ∆λ
λ
(3)
v = relative velocity of source and observer
c = speed of light = 300,000 km/sec
∆λ = apparent shift in wavelength
λ = wavelength of comparison line (at rest)
Complete the following tables by measuring (in centimeters) the Doppler shift for the
spectral lines indicated. Measure as accurately as possible. Remember, if the
displacement is a blue shift, then ∆ λ is negative and a red shift gives a positive
displacement. Once you have measured the shift of the line, apply the plate scale to
convert the measurement to Angstroms, then use equation (3) to compute the radial
velocity.
It will be easiest to start with reference line 5. Each of the reference lines is an actual
line (dark area) in the spectrum of Arcturus. The Doppler shifts of the lines are small
and the figure at left identifies them for reference line 5. You will want to connect the
reference lines top and bottom with a very thin line from a sharp pencil. The blue bar
labeled “a” in the figure is the shift for spectrum A and the green bar labeled “b” is the
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Astronomy 105 Laboratory Manual
shift for spectrum B. These are real spectra of Arcturus and sometimes the spectral
lines are not as nicely imaged as for reference line 5. But the shifts of all the lines in
spectrum A should be very similar and all the shifts for spectrum B should be similar.
We have you measure several lines in order to reduce errors.
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Earth’s Orbital Velocity
Name:____________________________ Lab Section Online Date: _____________
P.S. = Plate Scale =
(Distance Between 1 and 7 in cm )
=


 4307.91 - 4260.48 A 


o
(



cm )
o

A 

P.S. =__________________cm/Ǻ
SPECTRUM A - July 1, 1939
Ref. No.
λ
(Å)
1
4260.48
2
4271.16
5
4294.13
7
4307.91
Shift (cm)
(Measured)
(±) ∆λ
Shift/P.S.
VA (km/sec)
Equation 3
Average VA = __________
SPECTRUM B - January 19, 1940
Ref. No.
λ
(Å)
1
4260.48
2
4271.16
5
4294.13
7
4307.91
Shift (cm)
(Measured)
(±) ∆λ
Shift/P.S.
VB (km/sec)
Equation 3
Average VB = __________
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Astronomy 105 Laboratory Manual
Questions
Show all calculations and use the correct units.
1. Calculate the orbital velocity of the Earth (Vearth) in km/sec (equation 1).
VEarth = _________________ km/sec
2. Calculate the radial velocity of Arcturus (Vstar) in km/sec (equation 2). Is the star
receding or approaching the earth?
VStar = ___________________ km/sec
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Earth’s Orbital Velocity
3. Since we know the velocity of the earth and the period of its orbit (one year), we
can calculate the radius (astronomical unit) of the orbit. Find the radius of the
earth's orbit in kilometers and miles (1 km = .62 mi).
Vearth =
distance circumference of orbit 2π R
=
=
time
period
P
where P = 1 year = 31,600,000 seconds and π = 3.1416 .
R = ________________ km
R = ________________ miles
4. Assuming its present right ascension, what would the declination of Arcturus
have to be in order for these spectra to reflect maximum velocity of revolution for
earth? (Hint: Find Arcturus on the SFA Star Chart.)
5. Does the apparent color of Arcturus change as a result of its radial velocity?
Explain.
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