MEASURING DISTANCE TO THE STARS

• 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.

PARALLAX

• 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 CONTINUED

• 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

EARTH’S ORBIT AS BASELINE

• 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.

THE PARSEC

• 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

ONE MORE TIME

• 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.

PARALLAX ONLY GOOD FOR NEARBY STARS

• 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.

DISTANCE-INVERSE SQUARE LAW

• 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.

The same total amount of energy must pass through each sphere surface. • Since a sphere has a surface area of • the flux on sphere-1 = (the flux on sphere #2) × [(sphere

#2's radius)/(sphere #1's radius)]2.