Measuring Angular Distances In Astronomy

On giving the first ever technical workshop at the Harry Bayley Observatory in 1978, one of the first tasks  of attendees was to build an effective cross staff to measure angles in the sky. This ancient instrument was probably first used as long ago as 400 B.C. by the Chaldeans to make basic angular observations and computations.  One such measurement was to get the zenith distance (φ)  of the pole star, and thereby to obtain latitude. Thus, if  φ = (90 - Lat) then:  Lat =  90 - φ.


The device is relatively simple to construct, as indicated in the link below for any who might be interested:

http://www.phy6.org/stargaze/Scrostaf.htm


But before one can use it one must become familiar with the system of angular measure. Say that one wishes to get the distance between the Moon and Saturn such as depicted in the star map (from my planetarium program) below:


It is useless to use a linear measure such as meters, feet or inches because these have no meaning when referred to the sky, or celestial objects in it and distances between them. So we use degrees. But given degrees are only one unit, and many objects (e.g. double stars) are much nearer, say fractions of a degree, one must be able to use smaller measures too.

As for the cross staff shown in the image above, that is mainly used to measure degrees.  For  a basic cross staff one can generally measure from 15 degrees to 60 degrees with a fair amount of accuracy.  For angular measures up to 15 degrees, and as low as one degree angular measure, one's extended hand - in addition to the stars in the Big Dipper  - provide a useful basis, e.g.


Thus, your pinky finger extended at arm's length equals a 1 degree span. Half of that yields the angular diameter of the full Moon or 1/2 degree. Three fingers extended at arm's length yields a span of 5 degrees, which is also the angular separation between the two stars at the front end of the Big Dipper's 'bowl'.  A fist extended at arm's length yields 10 degrees. (Of course, there will be some  variations, generally + 1 degree due to natural disparity in the size of hands. For a small woman, then, the angular degrees measured as shown will be smaller.)

Let's return to the angular width of the full Moon, and seek out smaller angular units based upon it. An arcminute is 1/30 the width of the full Moon. Hence, we conclude that 1 deg = 60 arcmin.  The arcminute is further divided into 60 arcseconds, which is typically used to measure the distance between components of a binary star, e.g.














Where the separation is in seconds of arc or arcsec. Conveniently, astronomers have learned that if  the semi-major axis of the true relative orbit (e.g. the one displayed if the system is seen face-on) has an angular distance of a" (seconds of arc) and if the system is at a distance d parsecs, then the semi-major axis in astronomical units is:

a = (a" x d)

It should come as no surprise that planetary widths - given they are tiny - would also be registered in arcseconds or ". Thus, the giant planet Jupiter can be seen up to 50" in diameter. Mars will reach 24" in 2018 at a close opposition. Neptune is 2" and Uranus is 4".  (Pluto's angular width is barely 0.1").  To fix ideas, to magnify Pluto's disk to one arcminute width (i.e. 1/30 of the full Moon's diameter) would require a magnification of:

1 arcmin/  0.1 arcsec  =  60 arcsec/ 0.1 arcsec =  600 x

For completeness another angular measure used is the radian.

For example the Sun has an angular radius of a = 959.63 "

But this must be in radians before one can compute the solar constant, for example.

One radian (1 rd) can first be converted into arcsec as follows, given there are 3600 arcsec per degree.:

1 rd = 57.3 degrees = 57.3 deg/rad x (3600"/ deg)= 206 280 "

Then: a  (rd) = 959.63"/ 206 280"/ rad = 0.00465 rad


Problems for the budding astronomer:


1) Two observers using cross staffs obtain zenith distances from their respective locations of   φ = 45 degrees, and  φ =  35 degrees. How far apart in latitude are their locations?

2) Consider the system Epsilon Ursae Majoris which semi-major axis subtends an angle of  2½" and for which the parallax of the system is 0."127. Find the  semi-major axis in astronomical units. (Hint: p"  = 1/d)

3)  What telescope magnification would be required to observe the planet Uranus as a disk 2 arcminutes in diameter?


4) Find the Moon's angular diameter in radians.


Project:

Construct your own cross staff using the directions in the above link, then use it to measure the altitude of the star Deneb in the constellation Cygnus.   Hint: see the star map at:

http://www.space.com/22915-deneb.html