How far away in parsecs is 61 Cygni if Bessel measured a parallax
of 0.29"?
d = 1 / p
so if p = 0.29 arcsec, then d = 1 / 0.29 parsecs
∴d = 3.5 parsecs
What would the distance to 61 Cygni be in light years?
1 parsec = 3.26 light years
∴ 3.5 pc = 3.5 x 3.26 light years
so 61 Cygni is 11.4 light years distant.
Given a ground-based optical limit of 0.01", what is
the furthest star we can directly measure a distance to?
d = 1 / p
so if p = 0.01 arcsec, then d = 1 / 0.01 parsecs
∴ d = 100 parsecs
If the distance to the centre of our galaxy is 8.5 k pc, what
% distance of this can we measure directly using ground-based
optical trigonometric parallax?
8.5 k pc = 8 500 pc
Using ground-based trigonometric parallax from our answer to question 3
we can measure 100 pc.
so 100/8,500 x 100/1 = 1.2 % of the distance to galactic centre.
Betelgeuse (α Ori) has a measured parallax of 7.63 ±
1.64 mas (milliarcseconds). What is the range in distance
to Betelgeuse?
The smallest parallax to Betelgeuse is:
7.63 - 1.64 = 5.99 mas = 0.00599 arcsec.
This gives a distance of:
1/0.00599 = 167 pc
The highest value parallax to Betelgeuse is:
7.63 + 1.64 = 9.27 mas = 0.00927 arcsec.
This gives a distance of:
1/0.00927 = 108 pc
so Betelgeuse is somewhere between
108 and 167 parscecs distance.
A key measurement in astronomy is the distance to the nearest
open or galactic cluster, the Hyades. Hipparcos determined
this to be 46.34±0.27 pc. How far is this in light
years and in kilometers?
Hyades is at a distance of 46.34 pc.
This corresponds to 46.34 x 3.26 ly = 151 ly.
1 parsec = 3.086 x 1013km
so distance to Hyades in km = 46.34 x 3.086 x 1013km
= 1.430 x 1015km.
How distant is a star at the limit of Hipparcos' detection?
Hipparcos can measure parallax to a precision of 1 milliarcsecond, ie
0.001 arcseconds.
This corresponds to a distance of
d = 1/0.001 = 1000 pc.
Note that in reality the uncertainties in the measurements increase with
distance so that the error range ofro more more distant stars is much
greater than that for closer stars. (See answer to question 5).
Why was Hipparcos, with its relatively small aperture, able
to obtain more precise results than ground-based observations
utilising larger telescopes?
Ground-based measurements and observations are hindered by the effects
of the atmosphere. Apart from weather these include refraction due to
angle of star relative to horizon (which can be calculated and corrected
for), refraction due to turbulence in atmosphere and scattering. The refraction
due to turbulence results in "twinkling" where the star appears
to move around. This can be partly be corrected for using adaptive optics
techniques bt these are not in widespread use yet. Statistical techniques
through repeated measurements can help reduce uncertainties in measurements.
Space-based observations such as Hipparcos are free of the effects of
the atmosphere. They can also observe continuously and repeatedly. Two
problems though are the relative expense of space missions and their limited
lifespan.
The next generation of space-based astrometric missions includes
ESA’s GAIA mission. It will be able to observe V=15
mag stars to an accuracy of 11 micro-arcseconds. What distance
does this correspond to?
GAIA will be accurate to 11 microarcecs.
This is equal to 1.1 x 10-5arcsecs.
So distance = 1/1.1x10-5 = 90,900 parsecs.
If the Earth orbited the Sun at twice its current distance,
what impact would this have on the accuracy of our ground-based
astrometry?