Astrometry Questions

1. 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

2. 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.
3. 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

4. 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.
5. 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.

6. 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.

7. 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).

8. 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.
9. 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.
10. If the Earth orbited the Sun at twice its current distance, what impact would this have on the accuracy of our ground-based astrometry?