Lecture 11: Stars: The Distances and the Brightness

Chapter 17: 8th Ed. pages 433 - 440 or 3rd Ed. ???

Relating what we see to what happens with the star involves knowledge of distances

Observed by our instruments

Intrinsic Properties

Angles &alpha,&theta ; Brightness b, ... &hArr Size D ; Luminosity L , ...
Distances d
  • To establish distances to far-away objects we built a distance ladder : we find sizes and distances of local objects, use them to measure distances to next neighbouring ones, use those to go further and further

Back Next

Distance Ladder:
Earth size &rArr Moon size &rArr Distance to Moon &rArr Distance to the Sun

Eratosthenes (273 - ? BC) determined the size of the Earth.

The Ancient Astronomers Aristarchus and Eratosthenes used geometry to find the distances to the Sun and Moon.
  • Alexandria is 780 km due north of the ancient city Syene.
  • On a day when the Sun is at zenith (directly overhead) at noon at Syene, the Sun is 7o away from Zenith at Alexandria.
  • By circular geometry, the circumference of the Earth is
    C = 780 km x 360/7 = 39,000 km (modern value: 40,074 km !)
  • (The Earth is not exactly a sphere, so this is not exact.)
Figure 3-14

Back Next

Aristarchus (310 - 230 BC) determined the Size of the Moon.

  • During a lunar eclipse, the Moon travels through the Earth's shadow.
  • We can see that the Moon is smaller than the Earth's shadow so it must be smaller than the Earth.
  • Aristarchus judged that the Moon is about 1/3 the size of the shadow, so the Moon must be 1/3 the size of the Earth.
  • (Actual ratio is REarth = 3.7 RMoon.)

Back Next

Aristarchus determined the distance to the Moon.

  • The Moon subtends an angle &theta=0.5o
    (Half of the width of your little finger held out a arm's length.)
  • &theta = Angular diameter of the Moon (observable)
  • D = Diameter of Moon (Aristarchus found it!)
  • To find the distance to the Moon:
    • &theta in radians = 2 &pi x (&theta in degrees)/(360)
      = 2 &pi x 0.5/360 = 8.7 x 10-3 rad.
    • distance = d = D/(&theta in radians)
      = D/(8.7 x 10-3) = 3.5 x 103 km / (8.7 x 10-3)
      = 4.0 x 105 km.
    • (Actual distance is 3.8 x 105 km.)

Modern method for finding distance to Moon:

  • Astronauts left a mirror on the Moon.
  • We aim laser at the mirror and time how long it takes for the laser to bounce back to the Earth.
  • Laser light, travels at the speed of light.
  • 2 x distance = c x time

Back Next

Aristarchus' Method for finding Distance to the Sun

  • Once the distance to the Moon is known, use trigonometry to find the distance to the Sun.
  • When Moon is in the first quarter phase, the angle &theta between Sun and Moon (measured from Earth) is not exactly 90 degrees.
  • Measure angle, and then
  • dsun = distance to the Sun
  • dmoon = distance to the Moon
  • dsun = dmoon/cos(&theta)
Figure 3-15
  • Aristarchus measured the angle to be only 87o and found that the Sun was only 20 x further away from the Moon, but he had the right idea.
  • In 1630 Vendelinus used a telescope and measured an angle of 89.75o which would make the Sun 230 x further away than the Moon, which is a bit better than Aristarchus' measurement.
  • The actual angle is 89.853o
  • Therefore the Sun is 390 x further away from us than the Moon.

Using Moon as a step in the ladder is pretty inaccurate, because it is too close !

Back Next

Distance Ladder: Earth &rArr Venus &rArr Sun

Halley's Method For Finding the Distance to the Sun

  • In 1716 Edmond Halley thought about an alternative way to find the distance to the Sun.
  • One could easily measure the ratio of the distance between Venus and the Sun (VS) and the distance between the Earth and the Sun (ES) through trigonometry.
  • But we do not know VS (nor EV). We need another observation involving VS to exclude it.

    Back Next

Venus Transit

  • Approximately 2 times a century a straight line would connect the Earth, Venus and the Sun
  • When this occurred, we would see Venus move across the face of the Sun. This is called a Transit of Venus.
  • This happens rarely, because Venus orbit is slightly tilted.
  • Transits come in pairs, separated by 8 years
  • Most recent transit of Venus was June 8, 2004. Next transit of Venus is June 6, 2012.

  • Halley realized that two observers at different latitudes on the Earth could observe a transit of Venus along different tracks.
  • By comparing their measurements, they could find the physical distance between the tracks on the surface of Sun.
  • And by measuring angular distance between the tracks they can establish the distance to the Sun
  • The data was obtained from the transits of Venus in 1761 and 1769, including Captain Cook's famous voyage to Tahiti.

Primer: Physical distance between tracks knowing VS/ES=0.72 and size of Earth

  • On this diagram, d = distance between the observers on Earth and is a known quantity. We need to find D.
  • Since we have similar triangles,
    D/LV = d/(LE - LV) .
  • Thus (check the algebra)
    D = d x (LV/LE)/(1 - LV/LE)
    = d x 0.72/0.28 = 2.6 x d
  • So the distance D is now known !.

Back Next

Beyond the Sun: Earth-Sun &rArr neighbouring stars

Hipparchus' (200 - 100 BC) Parallax Method for Finding the Distances to Stars.

Figure 19-02 left panel Figure 19-02 right panel
Figure 19-2aFigure 19-b
  • Hipparchus' method is named the Method of Parallax
  • Suppose you know dsun = 1 AU.
  • If the Earth orbits the Sun, then our position in space changes and the apparent position of a nearby star will change over the course of 6 months.
  • Define p = parallax angle = 1/2 the change in angular position of the star.
  • parallax angle is measure in arc-seconds.
  • Hipparchus did not observe any change in position for any star, so he concluded that the Earth does not move!
  • Actual parallax angles are very tiny and difficult to observe.
  • First measurement of stellar parallax by Friedrich Bessel in 1835.
  • Recent satellite mission named Hipparcos measured the parallax angle for thousands of stars.

  • d = distance to the star
  • p = parallax angle in arcseconds
  • &theta = parallax angle in radians
  • &theta = p x 2 &pi / (3600 x 360) = p x 4.848 x 10-6 rad
  • Since the angles are small, the Parallax formula is
    d = AU / &theta = (1.496 x 1011 m) / (p x 4.848 x 10-6) = 3.086 x 1016m / p
  • Now define a new unit of distance called the parsec.
  • 1 parsec = 1 pc = 3.086 x 1016m = 3.26 light-years.
  • parsec is short for parallax-arc-second
  • 1 pc is the distance a star is away from us if its parallax angle is 1''.
  • Simple formula for distance: d = 1/p where
    • d = distance in units of parsec
    • p = parallax angle in arc-seconds
Back Next

Brightness of Light

  • "brightness" (or intensity) of light is what we measure by our instruments.
  • How bright an object appears to you depends on:
    • the luminosity of the object and
    • the distance between you and the object.
  • Light is emitted from a light bulb or star in all directions.
  • Imagine a spherical wavefront of light moving outwards like an expanding balloon.
  • As the wavefront expands, same energy emitted per second is spread out over a larger surface area.
Figure 17-4

Brightness b of the object at the distance d is the flux (energy per second per unit area) measured at this distance.
  • A sphere with radius d has surface area
    A = 4 &pi d2

  • The brightness of a source is luminosity divided by the surface area of the wavefront
    b = L/A=L/(4 &pi d2) ,

  • This is called the Inverse Square Law of Light. It's relative form is of most importance
  • The observed brightnesses of two objects with intrinsic luminosities L1 and L2 that are at distances d1 and d2 from the observer, are in the ratio

    b1/b2 = (L1/L2) (d2/d1)2


  • The Luminosity of a star is an intrinsic property of the star and is independent of who observes it.
  • The brightness of a star depends on how far away the observer is from the star.

Using inverse square law

  • If the distance to a star is known, an astronomer can determine the luminosity of the star
    L= 4 &pi d2 b
  • If the luminosity is know, the distance to the star can be determined
    d = (L/(4 &pi b))1/2


Next lecture: Classification of Stars
Read Chapter 17, 8th Ed. pages 442 - 456 or 3rd Ed. pages ???