Download H. Other Methods of Determining Stellar Distances

The
Memphis Astronomical Society
Presents
A SHORT COURSE
in
ASTRONOMY
CHAPTER 8
MEASURING DISTANCES
in the
MILKY WAY
Dr. William J. Busler
Astrophysical Chemistry 439
MEASURING DISTANCES
in the MILKY WAY
Overview:
• In this chapter, we will explore quantitatively some
of the concepts covered in a more narrative fashion
in the previous chapter (“The Early History of
Astronomy”).
A. Measuring the Distances in
the Sun-Earth-Moon system
• Recall from Chapter 7 that Aristarchus (310-230
BC) made the first attempt at measuring the scale
of the Sun-Earth-Moon system, using units based
on Earth’s diameter.
• He figured (correctly) that when the Moon is at its
First Quarter phase, it is not at right angles to the
Sun in the sky, unless the Sun were an infinite
distance away. (Refer to the diagram on the next
slide.)
A. Measuring the Distances in
the Sun-Earth-Moon system
• Aristarchus figured that when the Moon is at its
First Quarter phase, it is not at right angles to the
Sun in the sky, unless the Sun were an infinite
distance away. (Refer to the diagram below.)
Moon
1st Q.
Earth
Last Q.
Sun
A. Measuring the Distances in
the Sun-Earth-Moon system
• Aristarchus tried to measure the Sun-Earth-Moon
angle at First Quarter, but found it was impossible
to measure any difference from 90 using the
instruments available to him.
• He then surmised (correctly) that it should take
longer for the Moon to move from its First Quarter
position to its Last Quarter position than to move
from the Last Quarter back to First Quarter.
A. Measuring the Distances in
the Sun-Earth-Moon system
• After years of observations, which were made
difficult by inaccuracies in clocks and the
ellipticity of the Moon’s orbit, he detected an
average time difference of about an hour in the two
halves of the lunar month. (The actual difference
is 1 hour, 4 minutes, and 38 seconds.)
• Using his unreliable data, he calculated the various
distances and sizes in the table on the next slide:
A. Measuring the Distances in
the Sun-Earth-Moon system
Aristarchus’ various lunar distances and sizes:
Aristarchus
Modern
Moon’s Distance
10 Earth Diameters
30
Moon’s Diameter
0.33
0.272
Sun’s Distance
200
11,700
Sun’s Diameter
7
109
A. Measuring the Distances in
the Sun-Earth-Moon system
• Of course, we would now consider these results to
be seriously inaccurate.
• However, these were probably the first
astronomical measurements ever made.
• As such, they were quite an accomplishment for
the time of Aristarchus.
B. Measuring the Diameter
of the Earth
• Eratosthenes (276-195 BC) was a Greek
astronomer who made the first attempt to measure
the diameter of the Earth.
• He noticed that the Sun would shine directly down
a well on the first day of summer at Syene (modern
Aswân), while on the same day and time, the Sun
was 7.2 south of the zenith at Alexandria, about
500 miles north of Aswân.
• Eratosthenes believed that the Earth was spherical
and that the Sun was so far away that all of its rays
which strike the Earth are essentially parallel.
B. Measuring the Diameter
of the Earth
Alexandria
7.2°
Center of Earth
From
Sun
Syene
• Eratosthenes made the following calculation:
7.2 / 360 = 500 miles / x; x = 25,000 miles =
the circumference of the Earth. 25,000 miles / 
= 7960 miles = the diameter of the Earth.
(Accepted value = 7930 miles.)
C. Finding the Distance to the
Moon by Parallax
• This method was developed by Ptolemy, about 140
AD.
• Parallax is the phenomenon by which a nearby
object appears to shift its position against remote
background objects when the observer changes
position.
• (Try holding one finger about 12 inches in front of
your nose; look at it first with one eye, then the
other; notice the shift in your finger's apparent
position against a distant background.)
C. Finding the Distance to the
Moon by Parallax
• In this method, two observers simultaneously
observe the Moon against the starry background
from positions 6000 miles apart. The Moon’s
position against the stars differs by 1.4.
*
0.7
*
Earth
Moon
3000
miles
X
*
*
*
To calculate the Earth-Moon distance, X: sin (0.7)
= 3000 miles / X; 0.0122 = 3000 miles / X; X =
3000 miles / 0.0122 = 246,000 miles.
C. Finding the Distance to the
Moon by Parallax
*
0.7
*
Earth
Moon
3000
miles
X
*
*
*
• In actual practice, two simultaneous observations
are not necessary. The rotation of the Earth during
the night will carry a single observer from one
point to the other. During this time, however, the
Moon will have moved a short distance in its orbit.
This motion must be subtracted before the true
parallax can be determined.
D. The Scale Model of
the Solar System
• In his book De Revolutionibus Orbium Cœlestium,
published in 1543, Copernicus calculated and
tabulated the distances of the planets from the Sun
in terms of the Earth-Sun distance (AU).
• To do this, he used the time it took for each planet
to move from opposition (or conjunction) to
quadrature – right angles to the Sun.
• Since he knew the sidereal periods of the planet
and of the Earth, he could calculate the fraction of
a complete orbit which had been traversed, and
thus the angle with the Sun.
D. The Scale Model of
the Solar System
• From there, it was a matter of simple geometry to
calculate the planet’s distance from the Sun
compared to the Earth’s.
Modern
Planet
Copernicus
Mercury
0.38
0.387
Venus
0.72
0.723
Earth
1.00
1.00
Mars
1.52
1.52
Jupiter
5.22
5.20
Saturn
9.18
9.54
E. Using Kepler’s 3rd Law to
Calculate Sun-Planet Distances
• Recall from Chapter 7 that Kepler had used
Tycho’s data to formulate his three laws of
planetary motion.
• Once these mathematical laws had been obtained,
it was possible to make exact calculations which
were not dependent on any uncertainties in the
original data.
• For example, once a planet’s sidereal period (year)
is known, its exact distance from the Sun can be
calculated using the third (harmonic) law:
E. Using Kepler’s 3rd Law to
Calculate Sun-Planet Distances
• For Mars, the orbital period (P) is 1.88 years.
P2 = D3.
D =
1.882 = D3.
3
3.5344 = D3.
3.5344 = 1.52 AU.
• This remarkable equation allows the distance of
any planet to be calculated as accurately as its
sidereal period is known.
E. Using Kepler’s 3rd Law to
Calculate Sun-Planet Distances
• However, an error (of less than 1%) is introduced
for the most massive planets, Jupiter and Saturn,
because Kepler’s equation neglects the mass of the
planet orbiting the Sun.
• Newton made this correction years later, but for
small bodies orbiting the massive Sun, it is not
really necessary.
F. Newton’s More General Form
of Kepler’s 3rd Law
• In the Principia (1687), Newton explained and
derived Kepler’s laws from fundamental principles
of celestial mechanics, not observational data.
• The general (i.e., good for all cases) version of the
third (harmonic) law is as follows:
(mS + mP)  P2 = (DS + DP)3,
where mS and mP are the masses of the Sun and the
planet (in units of solar masses), and DS and DP are
the distances of the Sun and planet from their
common center of mass (in AU).
F. Newton’s More General Form
of Kepler’s 3rd Law
(mS + mP)  P2 = (DS + DP)3
• When the planet is much smaller than the Sun
(e.g., Mars), mP and DS may be dropped, and the
equation becomes identical to Kepler’s third law.
• For a very massive planet (e.g., Jupiter or Saturn),
this equation removes the discrepancies observed
between observational data and Kepler’s third law.
G. Measuring Distances to
Stars by the Parallax Method
• The parallax method used for the Moon won’t
work for stars; the baseline (the diameter of the
Earth) is too short.
• Instead, the baseline must be the diameter of the
Earth’s orbit.
• Even with a baseline of 186 million miles, the
parallax is very small – generally less than 1 arcsecond!
• Recall that many pioneer astronomers ruled out a
moving Earth because no stellar parallax could be
seen as a result of the Earth’s revolution.
G. Measuring Distances to
Stars by the Parallax Method
• There is a detailed example with an explanatory
drawing of the parallax method in Chapter 9, pages
3 and 4. It is strongly suggested that you study that
example at this time.
DISTANCE DETERMINATION
by PARALLAX
• Recall that as the Earth revolves around the Sun, it
changes its position in space by a distance of 2
astronomical units -- 186 million miles.
• This causes the nearer stars to exhibit parallax, i.e.,
to appear to move back and forth slightly against
the background of the more distant stars. (Ch. 8.)
• Through the use of trigonometry, parallax
measurements allow us to calculate the distances to
some of the “nearer” stars -- those within a few
hundred light-years of our Solar System.
The annual parallax of a star depends upon its
distance from the Solar System: The farther away
it is, the smaller the angle of parallax.
• Usually, the star is observed
from two points in the Earth’s
orbit 6 months apart.
• This provides a baseline of 2
astronomical units (186 million
miles) for triangulation.
• This drawing is not to scale!
The parallax angle for even the
nearest stars is less than 1
second of arc (1/3600 of one
degree).
• Suppose a star is at a distance such that
its parallax is exactly 1" either side of
its average position (i.e., 2" total) when
viewed from diametrically opposite
points in the Earth’s orbit.
• This distance is known as one parsec
(from parallax-second).
• We can easily calculate the distance of
the star, using trigonometry:
Opposite
•
Adjacent
•
•
•
•

opposite
tan  =
;
adjacent
7
9.3 10 miles
tan 00'1" =
x
7
9
.
3

10
miles
-4
tan 2.778 10  =
x
7
9
.
3

10
miles
-6
4.848 10 =
x
7
9.3 10 miles
13 miles
x =
=
1.92
10
4.848 106
= 19.2 trillion miles
= 3.261 light-years = 1 parsec.
G. Measuring Distances to
Stars by the Parallax Method
• An additional example: Sirius exhibits a total
annual parallax of 0.77; in other words, its true
parallax, based on the radius of the Earth’s orbit of
93 million miles, is 0.385. What is its distance in
light-years and in parsecs?
• Easy Solution: 1 / 0.385 = 2.60 parsecs;
2.60 psc  3.261 l-y / psc = 8.5 light-years.
G. Measuring Distances to
Stars by the Parallax Method
• Same problem; direct trigonometric solution:
• Sirius exhibits a total annual parallax of 0.77; in
other words, its true parallax, based on the radius
of the Earth’s orbit of 93 million miles, is 0.385.
What is its distance in light-years and in parsecs?
tan (0.385) = 9.3  107 miles / X.
0.385  1/3600 = 1.0694  10-4 .
X = 9.3  107 miles / tan (1.0694  10-4 )
= 9.3  107 miles / 1.8665  10-6
= 4.9825  1013 miles.
G. Measuring Distances to
Stars by the Parallax Method
4.9825  1013 miles  1 l-y / 5.87  1012 miles
= 8.5 light-years;
8.5 light-years  1 psc / 3.261 l-y = 2.60 parsecs.
G. Measuring Distances to
Stars by the Parallax Method
• Although Bessel (1838) used visual observations to
measure the parallax of 61 Cygni, it is preferable to
use photographic methods now that they are
available.
• It is possible to measure stellar distances up to
about 200 light-years with about 20% accuracy
using the parallax method.
• Computer measurement of photographic plates and
the Hubble Space Telescope have extended the
reach of the parallax method.
H. Other Methods of
Determining Stellar Distances
1. Radial Velocity / Proper Motion:
• Most stars have some intrinsic motion (space
velocity); they wander about through the Galaxy.
• If we assume that all stars have about the same
space velocity (risky!), then those with the most
apparent motion against the background of very
remote stars are probably the closest.
• This motion may be across the sky (proper
motion); it may be detected by comparing a series
of photographs taken over long periods of time.
H. Other Methods of
Determining Stellar Distances
1. Radial Velocity / Proper Motion:
• If the motion is along our line of sight (radial
velocity), it may be detected by looking for
Doppler shifts in the star’s spectral lines.
• The spectral lines will be shifted towards the blue
if the star is approaching; towards the red if it is
receding.
• (There will be further discussion of Doppler shifts
in Chapter 12.)
H. Other Methods of
Determining Stellar Distances
2. Moving star clusters:
• If a star cluster is moving through space, it may be
assumed that all of its stars have the same space
velocity and direction.
• However, if the paths of the stars appear to be
converging or diverging, this is a perspective
effect; the greater the degree of convergence or
divergence, the nearer the cluster.
• This method was useful for determining the
distances to the Hyades and the five middle stars of
the Big Dipper.
H. Other Methods of
Determining Stellar Distances
3. Spectral Class / Inverse Square Law:
• In Chapter 9, we will see that if a star’s spectral
class (essentially its color) is known, the
Hertzsprung-Russell diagram may be used to
provide a good estimate of its intrinsic luminosity.
• Comparing the star’s luminosity with its apparent
brightness leads to the determination of its
distance.
H. Other Methods of
Determining Stellar Distances
4. Binary Star Systems:
• (Refer to Newton’s version of Kepler’s third law,
in section F of this chapter.)
• We can measure the period of revolution (P) of the
system from its light curve (eclipsing binaries) or
from visual observations of the stars’ changing
positions and separation.
• For stars whose distance from us is already known,
we can easily obtain the true distance between
them, and then calculate the sum of the masses.
H. Other Methods of
Determining Stellar Distances
4. Binary Star Systems:
• If the distance is not known, we can estimate the
masses from the spectral classes (Chapter 9), and
then calculate the separation distance.
• By comparing this known separation with the
angular separation in the sky, we can compute the
distance of the system.
H. Other Methods of
Determining Stellar Distances
5. Interstellar absorption:
• There is a slight amount of gas in interstellar space
through which a star’s light must travel to reach us.
• The degree of absorption of a star’s light (as
evidenced by the intensity of the absorption lines
in its spectrum) is a function of its distance from
us.
H. Other Methods of
Determining Stellar Distances
6. Nova / Supernova light-shell front:
• After a nova or a supernova explosion, a “shell” of
light expands away from the residual star at the
speed of light.
• By measuring the angular expansion of the shell
over a period of time, and knowing that the true
velocity of expansion is the speed of light, the
object’s distance from us can easily be calculated.
H. Other Methods of
Determining Stellar Distances
7. Nova / Supernova Remnant Expansion:
• This is similar to the previous method, but uses the
much slower expansion of the physical material in
the remnant, rather than the light front.
• The true rate of expansion is nowhere near the
speed of light; it may be estimated from the
Doppler shift of material near the center of the
remnant.
MEASURING THE
SPACE VELOCITY
OF A STAR
b
a
A star moves through space from
point a to point b.
What is its true “space velocity”?
1
1. Measure its “proper motion”:
the angular speed across the sky
(arc-sec/year).

Solar System
Measuring the “space velocity” of a star
A star moves through space from
point a to point b.
b
a
What is its true “space velocity”?
1. Measure its “proper motion”:
the angular speed across the sky
(arc-sec/year).
2
1
2. Measure the distance to the star,
using the parallax method.

Solar System
Measuring the “space velocity” of a star
A star moves through space from point a to point b.
b
What is its true “space velocity”?
a
3
1. Measure its “proper motion”: the angular speed
across the sky (arc-sec/year).
2. Measure the distance to the star, using the parallax
method.
2
1

3. This allows us to calculate its “transverse
velocity”, the tangential component of its
true “space velocity”:
transverse velocity (3)
tan (or sin)  =
dis tan ce (2)
Solar System
Measuring the “space velocity” of a star
A star moves through space from point a to point b.
b
What is its true “space velocity”?
4
a
1. Measure its “proper motion”: the angular speed
across the sky (arc-sec/year).
2. Measure the distance to the star, using the parallax
method.
2
1
3. This allows us to calculate its “transverse
velocity”, the tangential component of its true
“space velocity”.
4. Using the Doppler shift, calculate the
“radial (line of sight) velocity” of the star:

v / c = D l / l0
Solar System
Measuring the “space velocity” of a star
A star moves through space from point a to point b.
b
4
What is its true “space velocity”?
5
a
1. Measure its “proper motion”: the angular speed
across the sky (arc-sec/year).
2. Measure the distance to the star, using the parallax
method.
2
1
3. This allows us to calculate its “transverse
velocity”, the tangential component of its true
“space velocity”.
4. Using the Doppler shift, calculate the “radial (line
of sight) velocity” of the star.

Solar System
5. Knowing the transverse and radial
velocities, use the Pythagorean theorem to
calculate the hypotenuse -- the “space
velocity” of the star.
Measuring the “space velocity” of a star
THE END