Physics 427
Introduction to Astrophysics
Problem Assignment #6
Due: Monday, September 30, 2013
THE DETERMINATION OF PARALLAX AND PROPER MOTION
Mean Coordinates
Time of Observation
18 June 1975 (1975.463)
7 January 1989 (1989.019)
24 April 2003 (2003.312)
14 October 1976 (1976.784)
α
hr
m
δ
s
13 44 07.3042
-23° 55' 05.017"
hr
m
s
13 44 08.5135
-23° 55' 20.255"
hr
m
s
13 44 09.7395
-23° 55' 36.139"
At this time on this date λ* = λ.
(1) The table above gives three positions for a hypothetical star, accurate to 0.0005" in
both α and δ, all referred to a common equinox. Find for the hypothetical star:
(a) λ and β, the celestial longitude and latitude for each of the three observations,
(b) π ["], the trigonometric parallax,
(c) r [pc], the distance to the star,
(d) µ ["/yr], the annual proper motion,
(e) θ [°], the "position angle" of the proper motion direction with respect to the
north ecliptic pole direction.
(f) µλ ["/yr], the annual proper motion in λ,
(g) µβ ["/yr], the annual proper motion in β,
(h) µα [s/yr], the annual proper motion in α,
(i) µδ ["/yr], the annual proper motion in δ.
(In a more realistic context you would have a large number of observed positions of
considerably lower accuracy, which could be exploited collectively by linear least
squares techniques to yield optimum results for π, µ and θ.)
(Suggestions: Use the uniform value ε = 23° 26' 38.055" in transforming from (α, δ)
to (λ, β). Use the relationships:
(λ - λo) cos β = μ sin θ (t - to) + π {sin [360°(t - t)] - sin [360°(to - t)]},
β - βo = μ cos θ (t - to) - π sin β {cos [360°(t - t)] - cos [360°(to - t)]},
where π is the parallax, to solve for π, µ and θ. You should obtain two independent
results for π, one each for λ and β, which you can compare for consistency and average for
your “best” answer. Remember that to and t are different variables, t being the time at
which λ* = λ .)