• How do astronomers measure the distance to the stars?
Measuring Tape? Radar?
• Obviously, you cannot use a tape measure
• Bouncing radar off the surfaces of stars would not work because:
• (1) stars are glowing balls of hot gas and have no solid surface to
reflect the radar beam back
• (2) the radar signal would take years to just reach the nearest stars.
MEASURING DISTANCE TO THE STARS
• A favorite way to measure great distances is a technique used
for thousands of years:
• look at something from two different vantage points and
determine its distance using trigonometry.
• The object appears to shift positions compared to the far off
background when you look at it from two different vantage
points.
• The angular shift, called the parallax, is one angle of a triangle
and the distance between the two vantage points is one side of
the triangle.
• Basic trigonometric relations between the lengths of the sides
of a triangle and its angles are used to calculate the lengths of
all of the sides of the triangle.
PARALLAX
• The size of the parallax angle p is proportional to the size of
the baseline.
• If the parallax angle is too small to measure because the object
is so far away, then the surveyors have to increase their
distance from each other
• Ordinarily, you would use tangent
or sine, but if the angle is small enough,
you find a very simple relation between
the parallax angle p, baseline B,
and the distance d:
p = (206,265 × B)/d
PARALLAX CONTINUED
• Trigonometric parallax is used to measure the distances of the
nearby stars.
• The stars are so far away that observing a star from opposite
sides of the Earth would produce a parallax angle much, much
too small to detect.
• As large a baseline as possible must be used. The largest one
that can be easily used is the orbit of the Earth. In this case the
baseline = the distance between the Earth and the Sun---an
astronomical unit (AU) or 149.6 million kilometers!
• A picture of a nearby star is taken against the background of
stars from opposite sides of the Earth's orbit (six months
apart).
• The parallax angle p is one-half of the total angular shift.
EARTH’S ORBIT AS BASELINE
• The distances to the stars in astronomical units are huge, so a more
convenient unit of distance called a parsec is used (abbreviated with
``pc'').
• A parsec is the distance of a star that has a parallax of one arc second
using a baseline of 1 astronomical unit.
1 parsec = 206,265 astronomical units = 3.26 light years
• FYI: The nearest star is about 1.3 parsecs from the solar system.
• Using a parsec for the distance unit and an arc second for the angle,
our simple angle formula above becomes extremely simple for
measurements from Earth:
p = 1/d
THE PARSEC
• The angles involved are very small, typically less than 1 second or arc!
(Remember that 1 arc_second = 1/3600 of a degree).
• To determine the distance to a star we can approximate the equation given in the
previous section with the small angle approximation:
d=r/p
• where d is the distance to the star, p is the parallax angle expressed in radians
(see diagram), and r is the baseline, in this case 1 Astronomical Unit (A.U.) -- the
radius of the Earth's orbit. Since there are 206,265 arc-seconds per radian, the
formula can be re-written as:
d (in AU) = 206,265 / p
• with p measured in arc-seconds .
Or
• If we define the distance of one parsec as 206,265 AU, we get:
d (in parsecs) = 1 / p
• This is the distance unit astronomers use most frequently, and it is equivalent to
3.26 light-years.
ONE MORE TIME
• Parallax angles as small as 1/50 arc second can be
measured from the surface of the Earth.
• This means distances from the ground can be determined
for stars that are up to 50 parsecs away.
• If a star is further away than that, its parallax angle p is
too small to measure and you have to use more indirect
methods to determine its distance.
• Stars are about a parsec apart from each other on average,
so the method of trigonometric parallax works for just a
few thousand nearby stars.
PARALLAX ONLY GOOD FOR NEARBY
STARS
• When the direct method of trigonometric parallax does not work for a
star because it is too far away, an indirect method called the Inverse
Square Law of Light Brightness is used.
• This method uses the fact that a given star will grow dimmer in a
predictable way as the distance between you and the star increases.
• If you know how much energy the star emits, then you can derive
how far away it must be to appear as dim as it does.
• A star's apparent brightness (its flux) decreases with the square of the
distance.
• The flux is the amount of energy reaching each square centimeter of
a detector every second.
• Energy from any light source radiates out in a radial direction so
concentric spheres (centered on the light source) have the same
amount of energy pass through them every second.
• As light moves outward it spreads out to pass through each square
centimeter of those spheres.
DISTANCE-INVERSE SQUARE LAW
The same total amount of energy must pass through each sphere
surface.
• Since a sphere has a surface area of 4𝜋𝑟 2 :
• the flux on sphere-1 = (the flux on sphere #2) × [(sphere
#2's radius)/(sphere #1's radius)]2.