# Problem on setting of sun

I came across this question from the book "A Problem book in Astronomy and Astrophysics by Aniket Sule":-

At Brazil’s National Observatory, located at the city of Rio de Janeiro $$(22◦54′ S, 43◦12′ W)$$, there is a sundial above the door of the dome of the $$32$$ cm telescope, facing to the north. The dial lies on the plane East-Zenith-West and the rod is parallel to the Earth’s axis. For which declinations of the sun and during what period of the year (months and seasons) the clock (i) does not work during, at least, some fraction of the day? and (ii) does not work at all during the day? (I12 - T01 - B)

I thought that for (i) the upper culmination should be greater than $$90◦$$ (i.e., it must culminate north of zenith) since the Sun must pass through the meridian atleast once so that the shadow become zero on the sundial, plugging the values (according to the southern hemisphere convection):-
$$H_u = 90◦ - \phi + \delta \geq 90◦$$
$$\delta \geq \phi$$

comparing my answers, it seems I am going fundamentally wrong somehwere, and this does explain why I can't get how to solve (ii)

Please give hints to solve this question, I would be grateful, also please edit/suggest for edits if you find any

• Consider what happens when the Sun's declination is further south than the observatory's latitude. Commented May 2 at 16:14
• @PM2Ring I added "lower culmination" just incase the second bit needed it to solve. I am removing it though Commented May 3 at 1:25
• Yes, as John says, the azimuth of the Sun has to be within 90° of north for the Sun to shine on the vertical north-facing dial plate. So between the September and March equinoxes there are times during the day when the Sun is too far south. Commented May 5 at 7:02
• I have a program here that shows the azimuth & altitude of the Sun in 3D. The Rio observatory has site code 880, and it's in the UT-3 timezone. Commented May 5 at 7:36
• ssd.jpl.nasa.gov/api/… Commented May 6 at 8:55