Download Tutorial: Star Measurement Basics

Measurements of Star Properties
For a refresher on Trigonometry, please
consult the Tutorial on Measurement
Basics
Common Name
Scientific Name
Sun
Distance (light years)
Apparent Magnitude
Absolute Magnitude
Spectral Type
-
-26.72
4.8
G2V
Proxima Centauri
V645 Cen
4.2
11.05 (var.)
15.5
M5.5Vc
Rigil Kentaurus
Alpha Cen A
4.3
-0.01
4.4
G2V
Alpha Cen B
4.3
1.33
5.7
K1V
6.0
9.54
13.2
M3.8V
CN Leo
7.7
13.53 (var.)
16.7
M5.8Vc
BD +36 2147
8.2
7.50
10.5
M2.1Vc
Luyten 726-8A
UV Cet A
8.4
12.52 (var.)
15.5
M5.6Vc
Luyten 726-8B
UV Cet B
8.4
13.02 (var.)
16.0
M5.6Vc
Sirius A
Alpha CMa A
8.6
-1.46
1.4
A1Vm
Sirius B
Alpha CMa B
8.6
8.3
11.2
DA
Ross 154
9.4
10.45
13.1
M3.6Vc
Ross 248
10.4
12.29
14.8
M4.9Vc
10.8
3.73
6.1
K2Vc
10.9
11.10
13.5
M4.1V
61 Cyg A (V1803 Cyg)
11.1
5.2 (var.)
7.6
K3.5Vc
61 Cyg B
11.1
6.03
8.4
K4.7Vc
Epsilon Ind
11.2
4.68
7.0
K3Vc
BD +43 44 A
11.2
8.08
10.4
M1.3Vc
BD +43 44 B
11.2
11.06
13.4
M3.8Vc
11.2
12.18
14.5
Barnard's Star
Wolf 359
Epsilon Eri
Ross 128
Luyten 789-6
Procyon A
Alpha CMi A
11.4
0.38
2.6
F5IV-V
Procyon B
Alpha CMi B
11.4
10.7
13.0
DF
BD +59 1915 A
11.6
8.90
11.2
M3.0V
BD +59 1915 B
11.6
9.69
11.9
M3.5V
CoD -36 15693
11.7
7.35
9.6
M1.3Vc
Measurements of Star Properties
Show the steps required to develop the expression
d (pc) = 1 / θ (arc sec)
Begin with the right triangle
expression
Tan θ = R / d
And for small angles, θ (radians) = R /
d
Parallax Angle
R

d
Measurements of Star Properties
Show the steps required to develop the expression
d (pc) = 1 / θ (arc sec)
Using distance measurements in AU,
θ (radians) = R / d → d (AU) = R (AU) / θ (radians)
To make the units the same as the above expression, we need two
conversions:
Conversion of the distance to parsecs
d(AU) = d(pc) (? AU / ? pc)
Where the conversion factor will be left unstated at this point, and
Conversion of the angle in radians to arc sec
θ (radians) = θ (arc sec) (1 radian / 206265 arc sec)
Measurements of Star Properties
Show the steps required to develop the expression
d (pc) = 1 / θ (arc sec)
So
d (AU) = R (AU) / θ (radians)
d(pc) ( ? AU/ ? pc) = R (AU) / { θ (arc sec) (1 radian / 206265 arc sec) }
So
d(pc) = { R (AU) / θ (arc sec) } (206265 arc sec / radian) (? pc / ? AU)
If we DEFINE 1 pc = 206265 AU, then
d(pc) = { R (AU) / θ (arc sec) } (206265 arc sec / radian) (1 pc / 206265 AU)
d(pc) = R (AU) / θ (arc sec) (arc sec/radian) (pc/AU)
If the measurements use the orbit of the earth for the baseline, R = 1 AU
and
d(pc) = 1 AU / θ (arc sec) (arc sec - pc/radian)
Measurements of Star Properties
Show the steps required to develop the expression
d (pc) = 1 / θ (arc sec)
d(pc) = 1 / θ (arc sec) (arc sec - pc/radian)
For convenience, the units at the end are dropped, and
d(pc) = 1 / θ (arc sec)
Measurements of Star Properties
Barnard’s star is observed from the earth. Observations are made of the
location of the star from opposite extremes of the earth’s orbit around the
sun. What parallax would be observed for Barnard’s star?
The right triangle information is as shown below:
Parallax Angle
R

d
Measurements of Star Properties
Barnard’s star is observed from the earth. Observations are made of the
location of the star from opposite extremes of the earth’s orbit around the
sun. What parallax would be observed for Barnard’s star?
Using the small angle approximation described in class,
 (radians) =
R
d
We know from the previous table that the distance to Barnard’s star
is 6 light-years and the radius of the earth’s orbit (from class) about
the sun is 1 AU
Measurements of Star Properties
Barnard’s star is observed from the earth. Observations are made of the
location of the star from opposite extremes of the earth’s orbit around the
sun. What parallax would be observed for Barnard’s star?
Using Appendix 2 from the book,
1 light-year = 63,200 AU
Therefore, the distance to Barnard’s star is
d = 6 (63200 AU) = 379200 AU
As a result, the parallax (that is, the parallax angle) for Barnard’s star
is
θ = R/d = 1 / 379200 = 2.6 x 10-6 radians
Measurements of Star Properties
Barnard’s star is observed from the earth. Observations are made of the
location of the star from opposite extremes of the earth’s orbit around the
sun. What parallax would be observed for Barnard’s star?
Review:
It is common to provide the small angles in arc sec, so
θ (degrees)
= (2.6 x 10-6 radians ) (360 degrees / 2π radians)
= (149 x 10-6 degrees)
θ (arc sec)
= (149 x 10-6 degrees) (3600 arc sec / 1 degree)
= 0.54 arc sec
Measurements of Star Properties
Barnard’s star is observed from the earth. Observations are made of the
location of the star from opposite extremes of the earth’s orbit around the
sun. What parallax would be observed for Barnard’s star according to the
“shorthand” formula d (pc) = 1 / θ (arc sec)?
Using
θ(arc sec) = 1 / d (pc)
And
1 light-year = 63,200 AU
Finally
d(pc) = 6 (63200 AU ) ( 1 pc / 206265 AU) = 1.836 pc
θ(arc sec) = 1 / d (pc) = 1 / 1.836
= 0.54 arc sec
IT WORKS !!
Measurements of Star Properties
A star 15 pc from the sun has a proper motion of 0.1”/year. What is its
transverse speed? If spectral lines are red shifted by 0.001 %, what is the
magnitude of its true speed?
Refer to the following figure to see the situation
Measurement
made same time
during the year

w
d
 (radians) =
w
d
w = d x  (radians)
If the time interval between measurements is measured, then v = w/ t
Measurements of Star Properties
A star 15 pc from the sun has a proper motion of 0.1”/year. What is its
transverse speed? If spectral lines are red shifted by 0.001 %, what is the
magnitude of its true speed?
Using small angles θ << 1,
w=dθ
Where θ is measured in radians. Since 1 radian = 206265 arc sec, the proper
motion of this star is
{ (0.1 arc sec ) (1 rad / 206265 arc sec) } / year = 4 x 10-7 rad / year
And, since the proper speed is
W / year = d (θ / year) = ( 15 pc ) (4 x 10-7 rad / year) = 6 x 10-6 pc / year
Measurements of Star Properties
A star 15 pc from the sun has a proper motion of 0.1”/year. What is its
transverse speed? If spectral lines are red shifted by 0.0001 %, what is the
magnitude of its true speed?
The star moves 4 x 10-7 rad / year. This can be converted to a transverse
speed by determining the distance associated with the 4 x 10-7 rad traveled
by the star in one year. Using the triangle from the previous slides, the
distance traveled in one year is given by
w = d θ = (15 pc)(4 x 10-7 rad) = 6 x 10-6 pc
And, therefore the transverse speed is
d = w / year = 6 x 10-6 pc / year
Measurements of Star Properties
A star 15 pc from the sun has a proper motion of 0.1”/year. What is its
transverse speed? If spectral lines are red shifted by 0.001 %, what is the
magnitude of its true speed?
Now, pc / year is not a common unit on speed, so using the fact that (using
Appendix 2 in the book)
1 pc = 3.09 x 1016 m
And
1 year = 3.16 x 106 sec
vTransverse = ( 6 x 10-6 pc / year ) ( 3.09 x 1016 m / pc) ( 1 year / 3.16 x 106 sec)
= 5.9 x 104 m/sec
Measurements of Star Properties
Astrophysics and Cosmology
A star 15 pc from the sun has a proper motion of 0.1”/year. What is its
transverse speed? If spectral lines are red shifted by 0.001 %, what is the
magnitude of its true speed?
The radial part of the speed is determined by the Doppler shift:
Apparent Wavelength
True Wavelength
=
True Frequency
Apparent Frequency
=
Velocity of Source
1+
Wave Speed
Since the lines are red shifted, the apparent wavelength is 0.001 % longer
than the true wavelength, that is, λApparent = 1.001 λTrue , and therefore
1.05 = 1 + ( vsource ) / c
vsource = 0.001 c = 3 x 105 m/sec
NOTE: This is the RADIAL part of the true speed of the source.
Measurements of Star Properties
Astrophysics and Cosmology
A star 15 pc from the sun has a proper motion of 0.1”/year. What is its
transverse speed? If spectral lines are red shifted by 0.001 %, what is the
magnitude of its true speed?
Finally, the true speed of the star is determined from the Pythagorean
Theorem:
v
vt
Pythagorean Theorem:
v2 = vR2 + vt2
vR
Measurements of Star Properties
Astrophysics and Cosmology
A star 15 pc from the sun has a proper motion of 0.1”/year. What is its
transverse speed? If spectral lines are red shifted by 0.001 %, what is the
magnitude of its true speed?
Finally, the true speed of the star is determined from the Pythagorean
Theorem:
Therefore,
v2 = vR2 + vt2
= ( 3 x 105 m/sec)2 + ( 5.9 x 104 m/sec )2
= 9 x 1010 m2 / sec2 + 3.48 x 109 m2 / sec2
= 9.348 x 1010 m2 / sec2
v = 3.06 x 105 m/sec