The trigonometric parallax
If we watch a near object from different places, then the
object seems to change its position against the very far away background. e.g. (the
forefinger watches it, they once only hold this one the left and once only with
the right eye in front of the nose)!
We as well see that near stars change their position
particularly against the background of the very far away stars if we change our
position. Of course this change in the position has very smally and therefore to
be measured difficultly also for the greatest possible place change. The
greatest place change which we can execute is twice as the big half axis of the
orbit of the earth as big, i.e. we measure temporal distance of half a year in
this. This looks in an outline so:
We connect P with the sun's p.s A and B are the positions of the earth with a
halfyearly distance which is orthogonal to hp. So the distance to the sun is 1
ae in very good approximation (because of the low eccentricity of the orbit of
the earth.) The angle J sun earth from the star P would the mean distance see,
parallax angle or simply annual or trigonometric parallax is called, under which
one. He is smaller, the wider the star is remote. In the course of the year the
star from the earth seen describes an ellipse trajectory (with a not
ascertainable parallax) against the background of the very far away stars.
Is valid in the rightangled triangle APS:
Is necessary tan J = J if we measure J in the radian measure
for the appearing little angles. Valid, if we mark the distance of the star by
R, then then:
If the angle is half so big, then the distance is twice so large. The parallax 1
one introduces a new distance unit now, it is the distance for which " would be.
1 parallax second, short are then called 1 Parsec (1 PC) the distance.
By the transformation submits to the radian measure
and with that
Is 1 PC = 206265 aes and the distance can with that
being calculated with J in the angular measure (in seconds).
Example: PC is necessary J = 0.5 PC is so R = 2 J = 0.1 R =
10 ", is necessary ", is so.
The greatest measured parallax, that is that one of the
neighbor star of the sun, are Centauri, is only 0.772 ". The other parallaxes
still have much smaller and therefore to be measured difficultly. PC denies the
method to a distance of approx. 100 since the parallaxes are then so small that
they lie in the range of the measuring errors as of.
For stars further removed one is so no longer possible for a geometrical
distance regulation, one must think up other methods.
Exercises
The trigonometric parallax of the Pole Star is 0.050 ". Calculate its distance
in ae, PC, light years.
Trigonometric parallaxes can be measured up to a distance of approx. 300 light
years with sufficient precision. Which parallax angle is part of this distance?
