Departm Physics In this ex the spect light, Arc Arcturus figure. W

Department of Physics, Universitty of Coloraado at Denveer
Physics 1052 (Generral Astronom
my)
Experim
ment #4 — Orbital
O
Velo
ocity of the E
Earth Usingg Stellar Speectra
xercise you will
w find the orbital veloccity of the E
Earth by meaasuring the D
Doppler shiftt in
In this ex
the spectrum of the star Arcturus. Among thee very brighhtest of stars,, shining witth a soft orannge
light, Arccturus lights northern sprring skies. While
W
orbitinng the Sun, thhe Earth mooves towardss the
Arcturus half of its orbital period
d and it movees away from
m the star reest of the perriod as shownn in
figure. When
W
the Earrth is moving
g towards an
nd away from
m the star, thhe observed llight is
respectiv
vely blue-shifted and red
d-shifted .
(a) Earth mooving away
Arrcturus
Sun
(b)) Earth moviing towards
The phottocopied figu
ure (taken from SKY AN
ND TELESC
COPE) below
w displays tw
wo spectrogrrams
of the staar Arcturus taaken six mon
nths apart. The
T top and tthe bottom sspectrogramss are of a
referencee light sourcee at rest with
h respect to the
t telescopee. Corresponnding to eachh of the referrence
lines is a dark line in the star spectrum, displaced slightlyy to the rightt (red-shiftedd) in a, and tto
the left (b
blue-shifted)) in b.
Select lin
ne 1 and meaasure the displacement of
o it in a andd b to tength of millimeteers. One wayy to
do it is to
o use a straig
ght edge to connect
c
matcching referennce lines as sshown in thee figure on thhe
lower rig
ght corner on
n the next paage. The shifft are shown by filled recctagles (red iin a and bluee in
b) in the figure.Meassure the shiftts with a scalle estimatingg to tength a tength of m
millimeter. A
magnifyiing glass would be handy
y. Try to be as
a accurate aas possible. T
The measureements give the
shift Δ in
i mm for th
he line. Draw
w a table sim
milar to one sshown below
w and note doown the
measurem
ments. Repeat the processs for lines 1,
1 3, 5, and 7 and enter innto the tablee.
line
original
o
no. waavelength
0 angstrom
1
426
60.48
shift Δ
Δ
spectrum spectrum
s
a mm
b mm
shifft Δ
spectrum
spectrum
m
a angstrom b angstrom
m
vr 

c
0
spectra a
spectra b
Department of Physics, Universitty of Coloraado at Denveer
Physics 1052 (Generral Astronom
my)
Fig: Sp
pectrum of Arcturus,
A
wheen the Earthh is at (a) andd (b) positioons.
The meassurments you obtained are
a in millim
meters in papeer scale. Theese are not inn the angstroom
scale in which
w
the thee reference wavelengths
w
are measureed. You needd to convert your
measurem
ments to the angstrom sccales. By thee way, 1 angsstrom = 10-110 m. To do thhis, measuree the
distance between linees 1 and 7 in
n millimeterss with the saame accuracyy mentionedd in the previious
paragraph
h. Divide thee difference in actual waavelengths oof line 1 (42660.48 angstrooms) and linne 7
(4307.91 angstroms) by the distaance between
n lines 1 andd 7.
i.e., Conv
version Facttor (CF) 
4
- 42260.48 
 4307.91

angsstroms /milllimeter
It serves as the conveersion factorr to obtain th
he shift in waavelengths inn
angstrom
ms. Multiply the numberss you have in
n the third coolumn and ffourth
coulumn in mm of th
he above tablle by the CF
F and write ddown the resuults in
fifth and sixth colum
mn of the tablle respectiveely. The new nubmers aree the
shifts in angstrom.
a
Now
N we know
w the shift in
n spectrum ddue to the rellative
motion between
b
sourrce of light (A
Arcturus ) an
nd the obserrver on the E
Earth.
Now you
u can use thee Doppler’s effect
e
Eq.(6.3) from youu text book too find
the relative velocity between
b
the Earth and th
he star.
vr 

c
0
m/s is the speeed of light and
a 0 is the wavelengthh of
where c = 300000 km
correspon
nding reference line. These values fo
or line 2, 3, 44, 5 and 6 arre
respectiv
vely, 4271.16
6, 4271.76, 4282.41,
4
429
94.13, and 42299.24.
a
b
Department of Physics, University of Colorado at Denver
Physics 1052 (General Astronomy)
You can use above formula for both red-shift and blue-shift. Find the average vr from the second
last column of the table. Let’s denote this number by vrr.
vrr = ……. km/s.
It represents the average relative velocity obtained from the red-shift in the spectrum a. This
velocity is related to speed of the star and the speed of the Earth by
vrr  vStar  vEarth .
Similarly, find the average vr from the last column of the table and denote it by
vrb = - ……. km/s.
Notice there is negative sign. We are adding this negative sign here for convenience. Actually it
should be with the numbers in the last column of the table because the shift Δ is negative for the
blue shift. Usually displacement in the right is taken as positive and in the left is taken as
negative.
vrb  vStar  vEarth .
Our goal is to find vEarth , the orbital speed of the Earth. We can obtain it by eliminating vstar
from the above two equations. Please show the mathematical steps involved in the elimination
process. I am writing the result below
vEarth 
vrr  vrb
.
2
Now plug in the values of vrr and vrb you found above and find vEarth . Remember to include –ve
sign in vrb.
This speed is obtained by assuming that the Arcturus is located at ecliptic plane (In actuality,
Arcturus lies at 31o latitude.). To get the true orbital value you need to divide above result by
0.86 because it is equal to cos31o. Compare your final result with the actual value of the orbital
velocity of the Earth found in book or internet. Don’t forget to discuss the possible sources of
error in the experiment while writing the conclusion section of your report.

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