(a) Circumpolar stars are stars that never set below the...

(a) Circumpolar stars are stars that never set below the horizon of the local observer or stars that are never visible above the horizon. After sketching a diagram similar to Fig.12(a), calculate the range of declinations for these two groups of stars for an observer at the latitude $L$ FIGURE $12 \quad$ (a) The diurnal path of the Sun across the celestial sphere for an observer at latitude $L$ when the Sun is located at the vernal equinox (March), the summer solstice (June), the autumnal equinox (September), and the winter solstice (December). NCP and SCP designate the north and south celestial poles, respectively. The dots represent the location of the Sun at local noon on the approximate dates indicated. (b) At what latitude(s) on Earth will the Sun never set when it is at the summer solstice? (c) Is there any latitude on Earth where the Sun will never set when it is at the vernal equinox? If so, where?

(a) Circumpolar stars are stars that never set below the horizon of the local observer or stars that are never visible above the horizon. After sketching a diagram similar to Fig.12(a), calculate the range of declinations for these two groups of stars for an observer at the latitude $L$ FIGURE $12 \quad$ (a) The diurnal path of the Sun across the celestial sphere for an observer at latitude $L$ when the Sun is located at the vernal equinox (March), the summer solstice (June), the autumnal equinox (September), and the winter solstice (December). NCP and SCP designate the north and south celestial poles, respectively. The dots represent the location of the Sun at local noon on the approximate dates indicated.
(b) At what latitude(s) on Earth will the Sun never set when it is at the summer solstice?
(c) Is there any latitude on Earth where the Sun will never set when it is at the vernal equinox? If so, where?

Key Concepts

Celestial Sphere and Coordinate System

The celestial sphere is an imaginary sphere surrounding Earth onto which all celestial objects are projected. It uses a coordinate system analogous to the geographic latitude and longitude system, where declination is similar to latitude, and right ascension is similar to longitude. This framework enables astronomers to describe the positions and movements of stars, the Sun, and other celestial bodies across the sky.

Circumpolar Stars

Circumpolar stars are stars that, depending on an observer’s latitude, remain perpetually above or below the horizon. Their constant visibility or invisibility is determined by their declination relative to the observer’s position on Earth. In the context of the celestial sphere, these stars revolve in circles around the celestial poles without setting or rising, making their range of declinations a function of the observer’s latitude.

Observer's Latitude and Celestial Pole Altitude

The observer’s latitude dictates the altitude of the celestial poles above the horizon. For an observer in the northern hemisphere, the north celestial pole appears at an altitude equal to the observer’s latitude, and similarly for the southern hemisphere. This relationship is crucial when deriving conditions for stars to be circumpolar, as it establishes the geometric limits within which stars will always remain above or below the horizon on a given night.

Seasonal Variation of Solar Declination

The Sun’s apparent position on the celestial sphere changes throughout the year due to the tilt of the Earth’s axis. This results in the solar declination varying with the seasons, reaching extreme values at the solstices and being zero at the equinoxes. Understanding this variation is essential for determining the Sun's diurnal path and whether it remains above the horizon for an observer at a given latitude.

Midnight Sun and Polar Day

The phenomenon of the Midnight Sun occurs at polar latitudes during the summer months, when the Sun does not set and remains visible for 24 hours. This effect is a direct consequence of the Earth’s axial tilt and has practical implications when calculating the conditions under which the Sun remains above or below the horizon. It also illustrates how the interplay between latitude and solar declination can lead to extended periods of daylight or darkness.

Transcript

00:01 All right, let's talk about circumpolar stars.

00:03 So circumpolar stars are stars that either stay in the sky all the time.

00:08 They never set below the horizon or vice versa.

00:10 They never rise above the horizon.

00:13 So are never visible in the sky.

00:15 In order to help us figure out what declinations are going to be needed in order for a star to be circumpolar for an observer at latitude l.

00:23 I'm going to draw ourselves a diagram.

00:24 So here is our celestial sphere.

00:30 And we'll have earth in the middle.

00:32 Of the celestial sphere and we got our observer who's going to be standing right there at the top of earth and let's say that the north celestial pole is like here that's the north celestial pole so there's north which means that the south celestial pole is down here and for our observer who's standing on earth if they're standing right there and they look towards north they're looking that way so here is the north end of their horizon from their viewpoint.

01:07 And here's the south end of their horizon.

01:10 So the plane of their horizon looks something like that, pretend that i'm better at drawing ovals.

01:16 So this would be the east end of their horizon and the west end of their horizon is over there.

01:21 Okay.

01:22 Now let's put, well, there's the center of earth.

01:25 That's going to be helpful as well.

01:26 Let's put our star so that is just barely circumpolar.

01:33 Just barely circumpolar for one that's going to stay in the sky all the time means that it'll dip down to where it touches the horizon, but it's never going to dip below the horizon.

01:44 So to help us figure out how to draw this, i'm going to draw the edges of my equatorial plane would intersect with the celestial sphere there and there.

01:54 So we need to have the declination of the star be that angle.

02:00 It's going to be the same angle on both sides.

02:05 Okay, so our star is in the sky, in the sky, in the sky, in the sky, touches the horizon, and then goes back up in the sky.

02:14 Never actually crosses that blue line for the horizon.

02:18 Now, furthermore, since this is the equatorial plane here, and this is the north celestial pole over there, then we know that this total angle from the equator to the north celestial pole, going all the way up there, that would be 90 degrees.

02:38 Furthermore, the way that we measure latitude, here's our line going from the north to the south edge of the horizon.

02:46 We know from the picture in the text that this angle between north, for the observer and the north celestial pole, that's the latitude l.

02:57 So if we put those together, we can see that 90 degrees is equal to the declination of the star plus the latitude of the observer.

03:10 Now, we also know that the declination doesn't have to be exactly this variable.

03:16 It could be an even higher declination.

03:18 For example, if we were going to go totally to the extreme, we could draw a star that has a really, really close to 90 degree declination.

03:34 So that star would be like here, right? so that star would for sure be circumpolar because it would not.

03:39 Never cross the horizon.

03:41 So solving for the declination, we get that the declination has to be greater than or equal to 90 minus l degrees.

03:57 So that's for the first part.

03:59 That's for stars that always stay in the sky.

04:02 The opposite end of that would be stars that are never in the sky, which is they stay below the horizon by that same amount, right there.

04:12 So they would intersect with the horizon right there.

04:17 So that would be their path through the sky or the lack thereof because they never come above the horizon.

04:24 So in that case, their declination is the same thing just negative.

04:28 So basically the absolute value of the declination has to be bigger than 90 minus l in order for a star to be circumpolar...

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