Retain horizon-grazing cases instead of forcing a visibility label

1. Define the frame and evidence

A fixed celestial direction traces a daily circle around a celestial pole as Earth rotates. Depending on latitude and declination, that circle can cross the horizon, stay entirely above it or stay entirely below it. Circumpolar describes the observer-dependent geometry, not a special physical type of star.

2. Connect the mechanism

For a point source, sin h=sinφ sinδ+cosφ cosδ cosH. Its upper and lower culminations occur at H=0 and 180°. Their signs determine whether the path crosses the geometric horizon. Exact equality deserves its own grazing label: touching at one limiting position is not the same as staying strictly above or below.

When crossings occur, the magnitude of the horizon hour angle is acos(−tanφ tanδ). Twice that angular interval gives the fraction of a rotation above the horizon; multiplying by 24 reports sidereal hours. This is not a civil-clock rise/set table. Refraction, terrain and finite source size can change a marginal real observation.

The accessible investigation compares complete numerical extrema and duration rather than relying on a sky animation. A candidate observation plan should state observer latitude, coordinate reference, geometric versus apparent horizon and what evidence would distinguish an unobserved source from a geometrically excluded one.

3. Reconstruct the worked case

At latitude 60° and declination 30°, the lower culmination touches altitude 0° and upper culmination reaches 60°. The model labels grazing, with the path above the horizon except at the limiting touch. At declination 60° the lower altitude is 30° and the source is strictly circumpolar.

4. Predict and investigate

Predict first: Why should a source that just touches the horizon be treated differently from a robustly circumpolar one?

At latitude 60°, compare declinations−60°,−30°,0°,30° and 60°. Retain both extrema and note the two grazing boundaries. Change latitude without changing declination and explain which evidence belongs to the observer.

5. Test a changed case

Apply the evidence to a changed case

At the equator, how long is a fixed δ+60° point direction above the ideal horizon, and what are its extrema?

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

It rises and sets, spending 12 sidereal hours above the horizon. Upper and lower altitudes are+30° and−30°. The same declination becomes circumpolar for a sufficiently northern observer; geometric visibility is not intrinsic to the object.

6. Retain an observation or image-comparison plan

Preserve the source, units, coordinate epoch/frame, orientation, first prediction and limitations. The screen-based source-image pathway is complete without outdoor viewing. No direct Sun viewing or instrument aiming is requested. Original teaching models are not operational pointing, eclipse or weather predictions; public practice is not independent observing competence.

Primary mechanism or source reference, checked 30 October 2026. Explanations, numerical examples, questions and geometric SVG are original. OpenStax is linked for factual scope, not copied artwork or text. Source NASA media retain their own credits and reuse terms; NASA has not reviewed or endorsed the AI-assisted course.