# Solstice

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This is not an easy article to follow. It is similar to the mathematical part of the Equinox article, but on steroids. Unfortunately it is necessary to do the math, even though I am not a mathematician. If you can follow my thought process through the numbers then more power to you. I’ve done my best to make it as presentable and easy to understand as possible, but it still requires effort. Let’s start with the simpler stuff first.

## General

### June Solstice at noon

The summer solstice data (June 21st for the northern hemisphere) differs to the equinoxes in the following way. (source: timeanddate.com)

 ```June 21st 2013 Latitude Location Sun at Noon Sunrise Sunset 82.5 Alert, Canada 30.9 - 64.84 Fairbanks, Alaska, USA 48.6 15 345 60.17 Helsinki, Finland 53.3 34 326 40.71 New York City, USA 72.7 58 302 23.7 Dhaka, Bangladesh 89.7 64 296 23.6 Muscat, Oman 89.8 64 296 10.66 Port of Spain, Trinidad and Tobago 77.2 66 294 10.50 Caracas, Venezuela 77.1 66 294 -0.02 Pontianak, Indonesia 66.5 67 293 -11.66 Lubumbashi, Dem of Congo 54.9 66 294 -23.55 Sao Paulo, Brazil 43 65 295 -41.28 Wellington, New Zealand 25.3 59 301 -53.15 Punta Arenas, Chile 13.5 50 310 ```

Looking at the noon position first, we see that Pontianak is exactly 66.5°. The sun is seen to travel in a northern arc by an observer on the equator on June 21st. This means the Sun is seen at 23.5° from the 90° (which is directly overhead) in the northern sky.

 Anyone at the equator sees the Sun rise, peak and set in the northern hemisphere at about 23.5° away from a point straight above the observer’s head (90°).

At the equinoxes, the Sun at the noon position shows its real location if it were in the center of the Earth space. If this still holds true for the summer solstice data (above table), then the Sun could be 23.48° higher on the vertical axis towards the north pole (it is 23.48° rather than 23.50° because of Pontianak’s -0.02° latitude).

 With the information so far, the Sun could be 23.48° above the horizontal axis on June 21st.

This 23.48° movement roughly computes for all the latitudes. Take New York for example where the Sun is 72.7° in the southern sky at noon on June 21st. NY’s latitude is 40.71°, so 40.71° from 90° is 49.29°. This means that at the equinoxes the Sun should be only 49.29° in the southern sky (actual position is 49.4°). Add 23.48° for the summer solstice position of the Sun and we get 72.77°. The actual noon position is 72.70°, 0.07° further south than it’s calculated one. (Remember, a lower degree in the southern sky is further south, away from the straight overhead angle – 90°).

Again, looking at Helsinki, the Sun is 53.3° in the southern sky at noon. Helsinki’s latitude of 60.17° from 90° is 29.83°. This means that at the equinoxes the Sun should be only 29.83° in the southern sky (actual position is 29.8°). Add 23.48° for the summer solstice position of the Sun and we get 53.31°. That is just about bang on the actual noon Sun position. Caracas‘s latitude is 10.5°, which puts it 12.98° south of the 23.48° Sun, which means the noon position should be 77.02° on the northern side of 90°. The actual location is 77.1° in the northern sky, putting it only 0.08° further south than it should be. Dhaka at 23.7° latitude should have a noon Sun position of 89.78° (23.7° – 23.48° = 0.22° away from 90°) in the southern sky. The actual position of 89.7° which is only 0.08° further south (although timeanddate don’t publish data to 2 decimal places so the true discrepancy, if any, is unknown).

Except for Helsinki, the other three examples above have a constant slightly further south (0.08°) noon Sun than calculated. Maybe the Sun is not at 23.48° after all. If we take Dhaka or Muscat as a reference point, then we get 23.40° instead. Maybe it is 23.43°, which is the heliocentric angle of the tilting Earth. Let’s compare those three angles and see which one makes more sense. A (-) symbol under the “Sun pos” columns means that the actual noon Sun position is further south than this calculated one, and vice verse for (+).

 ```June 21st Sun pos Sun pos Sun pos Latitude Location Sun at Noon 23.48° 23.43° 23.40° 82.5 Alert, Canada 30.9 -0.08 -0.03 0.00 64.84 Fairbanks, Alaska, USA 48.6 -0.04 +0.01 +0.04 60.17 Helsinki, Finland 53.3 -0.01 +0.04 +0.07 40.71 New York City, USA 72.7 -0.07 -0.02 +0.01 23.7 Dhaka, Bangladesh 89.7 -0.08 -0.03 0.00 23.6 Muscat, Oman 89.8 -0.08 -0.03 0.00 10.66 Port of Spain, Trinidad and Tobago 77.2 -0.02 +0.03 +0.06 10.50 Caracas, Venezuela 77.1 -0.08 -0.03 0.00 -0.02 Pontianak, Indonesia 66.5 0.00 +0.05 +0.08 -11.66 Lubumbashi, Dem of Congo 54.9 -0.01 +0.01 +0.04 -23.55 Sao Paulo, Brazil 43 -0.03 +0.02 +0.05 -41.28 Wellington, New Zealand 25.3 -0.06 -0.01 +0.02 -53.15 Punta Arenas, Chile 13.5 -0.13 -0.08 -0.05 ```

Although the examples above are very limited in number, we can see above that a Sun position at 23.48° north of the center of the Earth cavity gives an actual Sun position that is further south than this calculated one for all of the above latitudes bar Pontianak. For 23.40°, most latitudes show an actual Sun position that is further north than calculated. Only 23.43° gives a very narrow “bandwidth” around zero discrepancy between actual and calculated positions. This makes 23.43° the likeliest Sun position, especially as the mainstream claim the same angle for their heliocentric Earth tilt.

 Four Sun arcs on June 21st.

Despite not being well depicted in the above illustration, the sunset and sunrise on June 21st are at increasingly northern angles because the northern hemisphere has a higher number of daylight hours the further north the latitude, until the Sun doesn’t set above +66.5° latitude at all – the midnight Sun; and vice verse for the southern hemisphere below -66.5° latitude – the polar night. This issue creates a problem if we put the rotating Sun at 23.43° north of the center of the Earth cavity on June 21st.

### Sun’s wobble

As already stated, on the summer solstice the Arctic Circle is said to experience 24 hours of daylight (above 66.5° latitude) – the midnight Sun. Also, the Sun never appears at the South Pole during this same period either. In other words, the area below -66.5° (Antarctic Circle) is in 24 hour darkness on June 21st – the polar night.

 The Sun never sets above 66.5° latitude on June 21st. (Click to animate)

This means the Sun cannot be 23.43° higher on the vertical axis. Instead, it must tilt at this angle at (or very near) the center of the Earth space and rotate around the vertical axis, but not the 23.43° one. In other words, the Sun rotates in a cone shape, i.e. a wobble.

 The Sun spins (wobbles) on its axis around a “cone” shape with a 23.43° upward angle very near the center of the Earth space on June 21st. On the summer solstice, the Sun spins in a similar fashion to this matchstick in a water vortex with its shining side looking upwards at the water surface. Unlike the matchstick however, we will see later that as the Sun falls, its tilt angle increases. (Click to animate) A much more appropriate comparison in both shape and the way the Sun tilts is a coin slowing down, wobbling on its axis. (Click to animate) (Click to animate). The precession of the Sun is in some way similar to a spinning gyroscope that has slowed down and tilted (the Sun’s face would be pointing horizontally, however).

With a bit of visualization, we can see how the Sun’s wobble accounts for both the midnight Sun/polar night as well as the increasing/decreasing amounts of daylight at the various latitudes. We can also see this when looking at the dawn and dusk angles.

### June Solstice at dawn and dusk

Up to around the 23.5° latitudes in the above table the rising and setting angles are very roughly 23.5° off 90°. The higher the latitude, the more the rising and setting angles are further north, until we get to Alert in Nunavut where the Sun never sets at all. It’s easier to visualize these dawn and dusk angles of the Sun at the summer solstice with the below timeanddate illustrations.

 The “bowl” shape of sunlight on June 21st on a 2D flat map, which is really a straight line at an angle which closes up at the South Pole and opens up at the North one. Also look at the position of the noon sun, “At this location, the Sun will be at its zenith (directly overhead) in relation to an observer.“ If we use Photoshop’s fantastic 3D globe effect we see that it is the same straight vertical line as at the equinoxes but at around the 23.5° angle. This illuminates the North Pole and puts the South one in darkness.

The Sun’s tilt upwards causes the Sun to rise and set in the northern part of the sky, as the sunlight angle at these times now always comes from the north above at dawn and goes back to the north at dusk. Now what about the December solstice?

### December Solstice at noon

Because summer in the northern hemisphere is winter in the south, the December 21st solstice should be the exact opposite of the June one; and it is, almost:

 ```December 21st 2013 Latitude Location Sun at Noon Sunrise Sunset 82.5 Alert, Canada -16 - 64.84 Fairbanks, Alaska, USA 2 155 205 60.17 Helsinki, Finland 6.5 141 219 40.71 New York City, USA 25.9 121 239 23.7 Dhaka, Bangladesh 42.9 115 245 23.6 Muscat, Oman 43 115 245 10.66 Port of Spain, Trinidad and Tobago 55.9 114 246 10.50 Caracas, Venezuela 56.1 114 246 -0.02 Pontianak, Indonesia 66.6 113 247 -11.66 Lubumbashi, Dem of Congo 78.2 114 246 -23.55 Sao Paulo, Brazil 89.9N 116 244 -41.28 Wellington, New Zealand 72.2 123 237 -53.15 Punta Arenas, Chile 60.3 133 227 ```

If we take the Sun’s tilt by looking at the Pontianak noon Sun at the equator, we get (90 – 66.6 S) = 23.4°, and then add 0.02° for the extra tilt because Pontianak is just below the equator, we get a Sun tilt of 23.42°. If we look at Sao Paulo instead, where the noon Sun is directly overhead, we get the noon Sun at 89.9° in the northern sky above it. Take off the 0.01°, and the tilt of the Sun is 23.45°. Let’s compare these two possible tilts and also the heliocentric 23.43° angle to see which could be more likely, at least so far in this journey.

A (-) symbol under the “Tilt” columns means that the actual noon Sun position is further south than this calculated one, and vice verse for (+).

 ``` December 21st Tilt Tilt Tilt Latitude Location Sun at Noon 23.42° 23.43° 23.45° 82.5 Alert, Canada -16 no Sun no Sun no Sun 64.84 Fairbanks, Alaska, USA 2 +0.26 +0.27 +0.29 60.17 Helsinki, Finland 6.5 +0.09 +0.10 +0.12 40.71 New York City, USA 25.9 +0.03 +0.04 +0.06 23.7 Dhaka, Bangladesh 42.9 +0.02 +0.03 +0.05 23.6 Muscat, Oman 43 +0.02 +0.03 +0.05 10.66 Port of Spain, Trinidad and Tobago 55.9 -0.02 -0.01 +0.01 10.50 Caracas, Venezuela 56.1 +0.02 +0.03 +0.05 -0.02 Pontianak, Indonesia 66.6 0.00 +0.01 +0.03 -11.66 Lubumbashi, Dem of Congo 78.2 -0.04 -0.03 -0.01 -23.55 Sao Paulo, Brazil 89.9N -0.03 -0.02 0.00 -41.28 Wellington, New Zealand 72.2 -0.06 -0.05 -0.03 -53.15 Punta Arenas, Chile 60.3 -0.03 -0.02 0.00 ```

We can see that the 23.45° tilt has the actual Sun further north than it is in the northern hemisphere and closer to the calculated figure in the southern one, albeit with only a few samples. The 23.43° tilt looks to be the most balanced with the least deviation around the calculated figure. It could be any of those numbers, but 23.43° is the probably the most likely. Does it have to be exactly the same as the summer solstice tilt? No, not at all. In fact, it is extremely unlikely that they are the absolute exact same figure to many decimal places, or possibly even that they are both 23.43° (accurate to one hundredth of a degree); although I’ll take the 23.43° as a rough best guess for the Sun’s tilt at both solstices when looking into the Sun’s position in more depth in the second half of this article.

### December Solstice at dawn and dusk

Of course, the rising and setting times of the Sun on December 21st are the exact opposite of those at June 21st as these familiar timeanddate illustrations below show:

 On December 21st, the North Pole is in complete permanent darkness with the South Pole never experiencing night. Photoshop’s 3d globe effect shows that the dusk/dawn line is tilted the opposite way to the June 21st solstice, downwards towards the left.

On December 21st, the Suns’ downward tilt creates the 90° sunlight angle at dawn and dusk to come from the south, thereby the Sun rises and sets in the southern part of the sky dome.

 Four Sun arcs on December 21st shows the sun to rise and set in the southern part of the sky-dome.

So far, according to the calculations of timeanddate.com, we have the Sun tilted upwards at around 23.43° on June 21st and roughly the same angle downwards on December 21st. That’s the easy stuff over with. Is it possible to go into further detail and see where the Sun is on the Earth cavity’s vertical axis on the solstices? On June 21st, is the Sun below the central point or above it, or even on it?

There is a way to find this out, but it requires us to take a closer look at the midnight Sun and polar night; neither of which start to occur exactly at +66.5°/-66.5° latitude from either the north or south pole. This information is mostly not available from timeanddate.com, so the calculations from www.dateandtime.info has been used instead; although the two websites don’t match exactly, but close enough.
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## Exact

### North pole midnight Sun – June 21st

It is said that on June 21st, the Sun never sets above 66.5° latitude, which should mean that the Sun tilts at 23.5°. This just about matches our estimate of 23.43° deducted from timeanddate’s information (not to be confused with the DMS figure of 23° 26′ 21”). However, when we actually check cities around the arctic circle (66.5°), we find that this isn’t the case.

For example, Kemi in Finland is located at 65.73° (24.27° away from the north pole) and yet has 2 days when the sun never sets (4 days according to dateandtime.info). With only 2 (or 4 days) of continuous daylight, Kemi must be just about situated very close to the maximum latitude of continuous daylight on the summer solstice. Akureyri in Iceland is situated at 65.69° latitude (24.31° away from the North Pole) and experiences 45 minutes of darkness on 20th/21st June (although dateandtime.info says it is 25 minutes 27 seconds). This means that the latitude where one day of 24-hour daylight occurs lies somewhere between these two latitudes, which is between 24.27° and 24.31° away from the north pole, not 23.5°!

Can we narrow it down any further? The only way is to calculate the difference in latitude and the number of days with 24-hour daylight against other cities and see if we can see a pattern and work on from there. The towns chosen hopefully are far enough and equally enough away from each other in latitude to give an average representation. Let’s go further north than Kemi and see if we can detect a pattern.

Kuusamo is at 65.97° latitude (24.03° from the North Pole) with 17 days of 24-hour daylight. The difference between Kuusamo and the further south Kemi is 0.018° per day of 24-hour daylight. Kemijärvi is at 66.73° latitude (23.27° from the North Pole) with 34 days of 24-hour daylight. The difference between Kuusamo and Kemijärvi is 0.045° per day of 24-hour daylight; Gällivare is at 67.14° latitude (22.86° from the North Pole) with 41 days of 24-hour daylight. Between Kemijärvi and Gällivare the ratio is 0.059° per day. Verkhoyansk in Russia at 67.54° (22.46° from the North Pole) experiences 46 days of 24-hour daylight. The difference between Gällivare and Verkhoyansk is 0.080° per day of 24-hour daylight.

If we put these differences in order of latitude we get 0.018°, 0.045°, 0.059°, and 0.080° which suggests the closer the location to the North Pole, the greater the difference. We can even break this down further and see the relationship between the distances:

 ```Locations Difference per day Difference change (as a fraction of 1) Kemi and 24.28°??? 0.00216°??? 0.08??? Kuusamo and Kemi 0.018° 0.40 Kuusamo and Kemijärvi 0.045° 0.76 Kemijärvi and Gällivare 0.059° 0.74 Gällivare and Verkhoyansk 0.080° ```

We can see the difference in days per 24-hour sunlight increase the further towards the north pole we travel. Whereas the rate of difference increases dramatically the furthest from the north pole we are (Kuusamo and Kemis’ difference – 0.18° is less than half that of Kuusamo and Kemijärvis’ – 0.045°), but seems to stay the same-ish nearer the north pole.

We can take the same dramatic increase in rate of difference between Kuusamo and Kemi, and Kuusamo and Kemijärvi (although following this pattern, it is probably less), which is (0.76-0.40 = 0.36; 0.44-0.36) = 0.08; and apply it further from Kemi to the location where there is just one day of 24-hour daylight. This is 0.018° difference multiplied by 0.12 which is 0.00144°. There are 4 days of 24-hour daylight at Kemi (24.27° away from the north pole). In order to get one day, we would have to multiply 0.00144° by 3 and add it to 24.27° which is 24.27432°, or rounded down to 24.27°.

Of course, this means the Sun should tilt at 24.27° instead of 23.43°. However, it should also tilt at 23.43° in order to get the accurate noon Sun readings at all the different latitudes to around 0.1 of a degree. Obviously this is a contradiction. Now let’s look at the polar night at the south polar on June 21st and see if there is the same discrepancy.

### South pole polar night – June 21st

It is said that on June 21st all latitudes above -66.5°, i.e. latitudes closer to the south pole, experience 24 hours of darkness. However, this isn’t so. Dumont d’Urville Antartic Research Station is based at a latitude of -66.66°, or 23.34° above the South Pole. It is 0.16° above 66.5° and so shouldn’t experience any daylight at all on June 21st… and yet there are 2 hours 2 minutes and 41 seconds of daylight on that shortest day of the year.

The next research station at a latitude further towards the South Pole is Rothera Research Station at -67.56° latitude, or 22.44° above the South Pole. This station experiences 13 days of 24-hour darkness. This means that the location where only one day of continual 24-hour darkness occurs exists somewhere between these two latitudes. The only way to be more specific is to see a pattern at higher latitudes and extrapolate from there. It isn’t great, but it’s the only option with the information available. All stations chosen are roughly one latitudinal degree away each other so as to get a good average.

Davis Station is at 68.58° latitude (21.42° from the South Pole) with 37 days of 24-hour darkness. The difference between
Rothera and Davis Station is 0.043° per day. Law-Racoviţă Station is at 69.39° latitude (20.61° from the South Pole) with 48 days of 24-hour darkness. The difference between these two stations is 0.074° per day of 24-hour darkness. Georg von Neumayer Station Station is at 70.62° latitude (19.38° from the South Pole) with 62 days of 24-hour darkness. The difference between Law-Racoviţă Station and Georg von Neumayer is 0.088° per day. Lastly, SANAE IV is at 71.67° latitude (18.33° from the South Pole) and experiences 72 days of 24-hour darkness. The difference here is 0.110° per day of 24-hour darkness.

Just like the North Pole and its 24-hours of daylight, if we put these differences in order of latitude, we get a consistent increase of degrees per day: 0.043°; 0.074°; 0.088°; and 0.110°.

 ```Locations Difference per day Difference change Rothera and 22.61°??? 0.01376°??? 0.32??? Rothera and Davis Station 0.043° 0.58 Davis Station and Law-Racoviţă Station 0.074° 0.84 Law-Racoviţă Station and Georg von Neumayer 0.088° 0.80 Georg von Neumayer and SANAE IV 0.110° ```

Again, the rate of difference increases dramatically the furthest from the south pole, and is about the same nearer the south pole. Let’s take the same difference between Rothera and Davis Station, and Davis Station and Law-Racoviţă Station, in order to apply it to the one day of 24-hour darkness. That is (0.84-0.58 = 0.26; 0.58-0.26) = 0.32. Take that as a fraction of the difference 0.043° which is 0.01376° and multiply it by 12 to get one day of 24-hour darkness (as Rothera station has 13 days) which is 0.16512°. Add this figure to Rothera station’s distance from the south pole of 22.44° and we get 22.61°. This means that all latitudes up to 22.61° away from the south pole (-67.39° lat) experience at least 1 day of 24-hour darkness.

### Sun’s location on June 21st

So far we have the Sun at the summer solstice (June 21st) shining at 23.43° above the equator with a full day of 24-hour daylight at around +/-24.27° below the North Pole (+65.73° lat) and a full day of 24-hour darkness at around +/-22.61° above the South Pole (-67.39° lat).

 These three angles are a contradiction.

How can these angles be reconciled with each other? In a convex Earth they say it is refraction because there is 0.84° more light than there should be at the north pole and roughly 0.82° more light at the south pole. The total angle of the Sun discrepancy from pole to pole is (24.27° – 22.61° =) 1.66°; half that is 0.83° which we’ll take as the extra amount of light at each pole. The exact middle between 24.27° and 22.61° is 23.435°, which is remarkably close to the Sun’s tilt of 23.43°. The figure in the middle “should” be the Sun’s tilt. 24.27° is the most accurate figure because there are only 4 days of 24-hour daylight at 0.08° per day to extrapolate. It is so little that the angle can’t be any other than 24.27° away from the north pole where one day of 24-hour daylight is experienced. 24.27° – 23.43° = 0.84°. This is the real extra amount of sunlight at both poles. Of course this is only to 2 decimal places, and maybe the Sun tilts at 23.435° instead of 23.430°?

 The straight-traveling light hitting a convex Earth could account for the 0.84° extra latitude of 24-hour daylight at both poles on June 21st. In a concave Earth refraction would cause light to fall a bit short causing more darkness at both poles – instead the opposite happens.

If not refraction, what is causing the difference in the 3 angles? In a concave Earth, the answer can only be that the position of the Sun in the Earth cavity is not dead center on June 21st. So where could it be? It can’t be in front of dead center because both poles would receive less light than 23.43°, the more the Sun moved right on the horizontal axis; whereas just the opposite occurs. It can’t be above dead center for the exact same reason. It isn’t behind dead center on the horizontal axis only, as although it would add more light to both poles, this position increases the 23.43° angle on the crust. It isn’t below the dead center on the vertical axis only for the same reason except the 23.43° angle on the crust is decreased instead. The only position that can work is taking a little bit of each of the last two positions, i.e. both below and behind the dead center. The Sun’s tilt angle can remain at 23.43° and increase the latitude of daylight at both the poles as observed.

 On June 21st in a concave Earth, the Sun’s location is below and behind the dead center if it is to agree with observed pole daylight latitudes and its 23.43° tilt.

Using simple trigonometry calculated for us by this website, we can narrow the Sun’s location down even further. If AB = 1, then the ratio of the other sides of the triangle AC and CB are 0.918° and 0.398° respectively as long as the Sun’s tilt angle is 23.43°.

 We can use a right-angled triangle to find how far the Sun is behind and below the dead center of the Earth cavity. The website freemathhelp.com gives us the correct ratios. Using the Sun angle of 23.43° and an example AB length of 1, the above ratios for the two other lengths are given.

But what is the AB figure in a concave Earth? If AB were to extend to the crust, then it would be the radius of the Earth. The Sun shines an extra 0.84° at the poles (not including refraction). How does this figure equate to the AB side of the triangle? If my maths is right, then we can use the simple math of a circle.

The circumference of a circle is π x the diameter. This means that half the circumference is π x the radius. If the radius is 1, then half the circumference is 3.14 and a quarter of the circumference is 3.14/2 = 1.57. This is the ratio between the extra sunlight angle at the poles and the AB side of the triangle. 0.84°/1.57 is 0.535° which is the length of the AB side. Now we can use this data to calculate the other two sides which make AC = 0.49088744° and CB = 0.21273111°. This means that without refraction, the Sun is 0.49° behind the dead center of the Earth cavity and 0.21° below it on June 21st.

The exact amount of refraction is unknown. Let’s take the same amount of estimated refraction at dawn and dusk that we used at the equinoxes, which was an additional 0.5°. If we add this to the 0.84° of extra sunlight we get 1.34°. So 1.34°/1.57° is 0.8535031847133758° which is the AB side. AC is now 0.78312895° and CB is 0.33937697°. So the Sun is roughly 0.783° behind the center and 0.339° below on June 21st.

 On June 21st, the Sun is about 0.783° behind the dead center of the Earth cavity and 0.339° below it, if 0.5° refraction is added.

Like the equinoxes, the above figures are only an estimate because we don’t know the true refractive angle at dawn and dusk; it could be 0.7° instead of 0.5° for example.

Interestingly, when using timeanddate.com’s data of the amount of daylight on the equinoxes at the equator, and adding the same 0.5° refraction, the Sun’s position on the March equinox was calculated as 0.780° behind the center of the Earth cavity. The September equinox Sun position was calculated at 0.776°. This is very nearly the same figure as the one we have just estimated for June 21st.

Now let’s do the same for December 21st and compare.

### North pole polar night – December 21st

Bodø in Norway is at the 67.28° latitude which is 22.72° from the North Pole, and yet experiences 53 minutes 17 seconds of daylight on December 21st 2013. Verkhoyansk in Russia at 67.54° (22.46° from the North Pole) experiences 12 days of 24-hour darkness. This means that the one day of 24-hour darkness lies between these two latitudes. As with the summer solstice data, we’ll have to extrapolate between other 24-hour darkness latitudes to estimate this location.

Let’s look at more examples with roughly equally spaced latitude that are located further towards the North Pole. The next town is Narvik at 68.43° latitude (21.57° away from the North Pole) with 33 days of 24-hour darkness. The difference between Verkhoyansk and Narvik is 0.042° per day of 24-hour darkness. Norilsk at 69.34° latitude (20.66° away from the North Pole) has 45 days of 24-hour darkness. The difference betwween Narvik and Norilsk is 0.076° per day of 24-hour darkness. Hammerfest at 70.68° latitude (19.32° away from the North Pole) has 59 days of 24-hour darkness; making this a 0.096° per day difference between Norilsk and Hammerfest. Tiksi at 71.58° latitude (18.4° away from the North Pole) has 68 days of 24-hour darkness. The last difference is 0.102° per day of 24-hour darkness.

If we put these differences in order of latitude we get 0.042°, 0.076°, 0.096°, and 0.102° which suggests the closer the location to the North Pole, the greater the difference.

 ```Locations Difference per day Difference change Verkhoyansk and 22.60°??? 0.01302°??? 0.31??? Verkhoyansk and Narvik 0.042° 0.55 Narvik and Norilsk 0.076° 0.79 Norilsk and Hammerfest 0.096° 0.94 Hammerfest and Tiksi 0.102° ```

As before, the trend is an ever higher fraction of difference the further away from the north pole. Let’s take the last difference between Verkhoyansk and Narvik and between Narvik and Norilsk, which is 0.24. It is likely to be a lower figure than this because the trend is an ever increasing one; but unfortunately I can’t guess. The same difference gives us 0.31 (0.55-0.24) as a fraction of 0.042°. This is 0.01302°. Verkhoyansk at 22.46° has 12 days of 24-hour darkness, therefore one day is 22.46 + (0.0126 x 11) = 22.60° away from the north pole.

### South pole midnight Sun – December 21st

At the other end of the cavity at the South Pole, Casey Station is 24 days in daylight around December 21st with a latitude of -66.28° (23.72° from the South Pole). The next station further away from the pole is Akademik Vernadsky Station at -65.24° latitude (24.76° from the South Pole) and has 1 hour 35 minutes of darkness on December 21st. This means so far that the one day of total daylight exists somewhere between 23.72° and 24.76° away from the South Pole (closer to the 24.76° than 23.72°). We will never know exactly where, but let’s jump into the calculated unknown anyway and fish around in the dark by extrapolating again.

Rothera Research Station at -67.57° latitude (22.43° from the South Pole) with 44 days of 24-hour daylight. The difference between Casey Station and Rothera is 0.061° per day of all-day sunlight. Davis Station at -68.58° latitude (21.42° away from the South Pole) has 55 days of 24-hour daylight. This difference between Rothera and Davis Station is 0.092° per day of 24-hour darkness. Leningradskaya Station at -69.50° latitude (20.50° away from the South Pole) has 64 days of 24-hour daylight. The difference between Davis Station and Leningradskaya is 0.102° per day. Neumayer Station at -70.65° (19.35° away from the South Pole) has 73 days of 24-hour daylight. The difference is 0.127° per day of 24-hour sunlight.

The differences in order of latitude are 0.061°, 0.092°, 0.102°, and 0.127°. Again this also suggests that the number of degrees per day increases, the closer the location to the South pole.

 ```Locations Difference per day Difference change Casey Station and 24.31°??? 0.026°??? 0.42??? Casey Station and Rothera 0.061° 0.66 Rothera and Davis Station 0.092° 0.90 Davis Station and Leningradskaya 0.102° 0.80 Leningradskaya and Neumayer Station 0.127° ```

This data looks the same as June 21st’s with a steady differential nearer the poles and a sudden increase in change nearer its furthest point away from the south pole. Without guessing, let’s take the same difference between Casey Station and Rothera, and Rothera and Davis Station, which is 0.24 and multiply that as a fraction of 0.061°. This gives us a 0.02562° difference per day of 24-hour daylight from Casey Station (23.72° away from the south pole) with 24 days of full sunlight to the location where only one full sunlight day occurs. This location is at (0.02562° X 23) + 23.72° = 24.31°.

So we have a possible +/-22.60° away from the North Pole of 24-hour darkness and a possible +/-24.31° away from the South Pole of 24-hour daylight. This gives us a 1.71° difference between the two poles. Split in half, we come to 0.855° extra sunlight at both poles than there should be. 23.455° is the exact angle in the middle of 22.60° and 24.31°, which is also very close to 23.43°. But again, it should be 23.43° (to 2 decimal places), rather than 23.445°. The north pole has an estimated difference per day of 0.013° per day over 11 days and the other is 0.26° over 23 days, making the north pole a more accurate figure; however, not accurate enough. So let’s split the difference (23.445° – 23.43°) = 0.015° between the two poles. This gives us 1.695° instead of 1.71°. Half of that is 0.8475° instead of 0.855°, which is the extra amount of daylight at both poles – 0.0075° more than the summer solstice.

### Sun’s location on December 21st

So where is the Sun calculated to be on December 21st? Using the same calculations, trigonometry website freemathhelp.com, and the 0.5° added refraction which we used before, the AB angle is 0.8582802547770701°. This makes the Sun 0.7875° behind the center and 0.341° above on December 21st.

 On December 21st, the Sun is 0.7875° behind the dead center of the Earth cavity and 0.341° above it, if 0.5° refraction is added.

It must be stated that all this data is only based off dateandtime.info and timeanddate.com calculations (which very slightly differ between themselves) and not iron-clad observed reality. This is demonstrated by a Dutch travel photographer Jan Van der Woning who reported that Vernadsky Research Base has 24-hour daylight for a brief period in the winter (at least until December 23rd) instead of the 1 hour 35 minutes of darkness it is supposed to have on December 21st 2000, according to dateandtime.info.

 “This is a sundown and rise, as in this part of the world and at this date the sun does set anymore and daylight stays for 24 hours. The picture was taken on 23 December 2000 at 12 at night just before it started to snow for 24 hours, from Winter Island in direction of Skua Island across the Skua creek. Camera was a Seitz Roundshot 220 VR camera equipped with a 35 mm perspective control Nikon lens and shot on Kodak Portra NC 160 film. Location: Faraday Base, Marina Point, Galindez Island, Argentine Islands, Antarctica. 65 15′ S 64 16′ W.” Photo link.

It could have been suggested that the Dutch photographer may have witnessed the geometric center of the Sun setting below the horizon, but not the whole Sun and so 24-hours of daylight were possible; but the above photo seems to show otherwise. Also note how extremely elliptical the Sun appears. However, an elliptical Sun at extreme latitudes only seems to be present in some photos and videos taken at these locations, such as the Top Gear video in Lapland (05:30 secs). Also, sometimes the ellipse is both horizontal and vertical.

There may be other Antarctic locations where this anomaly is present if readers wish to look further; however dateandtime.info seems to be accurate concerning the northern cities. For example, from Oulu in Finland the Sun has been observed to just dip below the horizon on June 21st, and Bodo in Norway has no observed polar night on December 21st.

### Sun’s location overall

So far we have the following data as an estimation for the Sun’s location inside the Earth cavity.

 ```Solstice Tilt *X-axis *Y-axis March 20th 0° -0.780° 0° June 21st 23.43° -0.783° -0.339° September 20th 0° -0.776° 0° December 21st 23.43°+ -0.7875° +0.341° *0.5° refraction added ```

This estimate of the degrees per day makes this particular info less accurate than when the equinox Sun position was calculated using the amount of daylight hours at the equator. We also don’t know how accurate the 23.43° tilt figure really is, as it is only to 2 decimal places. Then there is the latitude of cities, which has to be used for the solstices, unlike the equinoxes. Latitude relies on calculation models (see Earth ellipsoid) and may not be 100% accurate. Some cities have a large sprawl and may not have one exact latitude to two decimal places as well. What about longitude? Is the Sun at its absolute highest or lowest point for all longitudes? It seems to be. The Sun remains at a 23.43° tilt for a few days around the solstices as mentioned in the next article, so we can rule longitude out as a possible discrepancy.

What is the reason behind the plus (+) figure in 23.43°+ tilt for December 21st? The December solstice will very likely tilt a touch more than the June one, because it is a touch more away from the center than the summer solstice Sun. This is later theorized to be caused by the stronger magnetic attraction/repulsion of the geographic south pole hole. So further out, means more tilt. This degree of tilt difference is in the thousandths, or even ten thousandths of a degree.

There is further support for these ratios, and that is the differing lengths of the solar day throughout the year.
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### Solar analemma

The length of a true solar day is normally not exactly 24 hours long. A day takes longer than 24 hours at the solstices and shorter at the equinoxes.

Earth’s rotation period relative to the Sun (true noon to true noon) is its true solar day or apparent solar day… Currently, the perihelion and solstice effects combine to lengthen the true solar day near December 22 by 30 mean solar seconds, but the solstice effect is partially cancelled by the aphelion effect near June 19 when it is only 13 seconds longer. The effects of the equinoxes shorten it near March 26 and September 16 by 18 seconds and 21 seconds, respectively.

+++

The longest day is the December solstice at 30 seconds more than 24 hours, then the summer solstice at 13 seconds longer. The shortest day is the September equinox at 21 seconds less than 24 hours, closely followed by the March equinox at 18 seconds. The maximum difference from shortest to longest is 51 seconds.

In a concave Earth, the rotation of the Sun around the center of the Earth cavity is responsible for day and night. If we look at the Sun’s respective positions at the different times of the year, we can clearly see an approximate correlation with the distance behind the center point of the cavity and its speed of rotation.

 ```Solstice Tilt *X-axis *Y-axis Day length March 20th 0° -0.780° 0° -18s (26th) June 21st 23.43° -0.783° -0.339° +13s September 20th 0° -0.776° 0° -21s (16th) December 21st 23.43°+ -0.7875° +0.341° +30s *0.5° refraction added ```

We can see now that the further the Sun is away from the center of the Earth cavity, the longer the solar day. This differing length of day is half responsible for the vertical figure of eight pattern that the Sun makes in the sky at noon throughout the year. This is the solar analemma.

 A solar analemma graph. The figure of eight is usually seen in a photo at a slant because the photographer didn’t take the pictures of the Sun at the exact time of solar noon and of course used a wide angle lens to accommodate both the Sun’s highest point and the horizon.

The vertical movement is the height of the Sun in the sky, which in this theory is due to the tilt of the Sun, and varies by about 46.86° in total. The horizontal movement is the accumulation of the differing times of the apparent (true) solar day, which is the same as mainstream theory.

The ellipticity of Earth’s orbit causes the actual solar time to first get ahead of, and then fall behind, mean solar time. This makes the Sun appear to slide back and forth across the vertical axis of the eight, forming the rest of the figure.

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Notice that the solar day becoming shorter or longer doesn’t equate to the Sun being on the right or left side of the figure eight. Instead, the figure eight is due to an accumulation. The true solar day is exactly 24 hours around the end of May, half way through July, the start of November and at beginning of February – the edges of the figure of eight. From November to February the true solar day is more than 24 hours (slower) and hence moves to the left, although it slows down until December 22nd and then speeds up again, but it always more than 24 hours. From February to May it is always less than 24 hours, hence its movement to the right and so on. The figure of eight is fatter around December and does not intersect perfectly in the middle because the Sun is a full 17 seconds slower than its June counterpart.

### Climate

The cause of climate and temperature differences around the world in a concave Earth is the tilt of the Sun rather than the heliocentric tilt of the Earth. The effect though is the same. The official reason for differing temperatures is based on the height of the Sun in the sky which is the same for everyone no matter what model explains it. The higher the Sun in the sky, the more direct the sunlight is shone on to the crust. This means there is more sunlight on less crust than there would be if the Sun were low in the sky.

 The higher the Sun, the more sunlight hits less Earth.

Climate is more of an add-on to the article as there isn’t anything new for me to say on this subject.

### Summary

• In the concave Earth model, on June 21st, the noon Sun is near 90° straight above our heads if viewed from Dhaka and Muscat. Also, its noon location as viewed from Pontianak on the equator is 66.5° in the northern sky. This means that the Sun is shining directly on the locations at 23.5° latitude.
• This has been narrowed down to 23.43° as that has the narrowest margin of error between observed noon Sun positions and those calculated from a hypothetical Sun location. This is also the mainstream tilt angle of the Earth.
• On June 21st, the 24-hour daytime within the Arctic Circle, and 24-hour night at the Antarctic Circle can only mean that the Sun tilts by 23.43° upwards and its rotational movement is a horizontal precession around the center of the Earth cavity, i.e it wobbles.
• This upward tilt causes the sunlight to always come from, and leave to, the north as observed at dawn and dusk on June 21st.
• The December 21st solstice is the exact opposite of the June one. The noon Sun figures coupled with the north polar night and south pole midnight sun as well as sunlight coming from, and leaving to, the south at dawn and dusk, show the Sun to tilt downwards by 23.43°.
• On June 21st, there is at least one day of 24-hour daylight up to 24.27° from the north pole, and one day of 24-hour darkness up to 22.61° from the south pole. This is a nearest estimate to within 0.005° accuracy, as 23.435° is the exact middle of the two latitudes and the Sun is tilting at 23.43°.
• On a convex Earth this extra sunlight is said to be caused by refraction. In a concave one, refraction causes light to fall short.
• The only way to reconcile the three angles – 24.27°, 22.61°, and 23.43° is if the Sun is below and behind the center of the Earth cavity.
• The exact position can be calculated using simple trigonometry by finding out the ratio of the curve of the crust on which the extra 0.84° of sunlight shines in comparison to a straight line (radius). With an estimated added refraction of 0.5° thrown into the calculation, the Sun is 0.339° below and 0.783° behind the center of the Earth cavity on June 21st.
• The equinox is 0° on the vertical axis and also calculated as 0.780° (March) and 0.776° (September) behind the center using the amount of sunlight on the equator (a previous article) and the same 0.5° of added refraction.
• On December 21st, there is at least one day of 24-hour daylight up to 24.31° from the south pole, and one day of 24-hour darkness up to 22.60° from the north pole. Similar to June 21st, this is a nearest estimate to within 0.015° accuracy as 23.445° is the exact middle of the two latitudes and the Sun is tilting at 23.43°.
• With an estimated 0.5° of refraction added, the Sun is calculated to be 0.341° above and 0.7875° behind the center of the Earth cavity on December 21st.
• The data comes from calculations for the year 2013 from dateandtime.info and dateandtime.com. The Sun wasn’t measured and observed at noon by an instrument to obtained this data. One anomaly was found by an observer in the Antarctic were the Sun was photographed fully above the horizon at midnight on December 23rd, 2000 to give 24 hour daylight. According to dateandtime.info, there should have been an hour and a half of darkness at this time and place.
• The Sun’s position behind the central vertical axis of the Earth cavity at the four times of the year matches the varied length of the apparent solar day at these times. The further away the Sun is behind the central point of the cavity, the longer the day.
• The figure-of-eight pattern (Sun’s analemma) made by the Sun in the sky at noon throughout the year is explained by its tilt (vertical axis) and its precessional (rotational) speed (horizontal axis).
• The reason for the varied climate is due to the height of the Sun in the sky which determines how direct it shines on to the crust – no different than what is taught today.

The Sun moving vertically as well as tilting 23.43° up and then down describes the observations, but it doesn’t explain why. What could be the mechanism behind this movement? First we need to explain how the Sun is powered and the rest will then fall into place. What is the power source of the Sun as a sulfur lamp? In a concave Earth, part of the answer looks to be magnetism flowing through the holes near the poles.

### 8 Responses to Solstice

1. Wise One says:

With all the knowledge I have learned about seasons, climate and the tilt of the earth, I cannot seem to accept or comprehend that the sun is closer to me in the Winter than it is in the Summer. It is my understanding that objects high in the sky are closer to the observer than objects that are lower in the sky. I’m not referring to objects that are on earth at the horizon, I specifically mean the SKY portion of the horizon.

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• Wild Heretic says:

Very marginally further away in summer using a straight line distance. It is the same distance when the bend of light is accounted for. In fact, in June the Sun is very, very slightly closer to the center of the cavity than December. Therefore when taking the bend of light as the measuring distance, the Sun’s light is a tiny fraction closer to the northern hemisphere in June. Therefore temperatures around the Tropic of Cancer should on average be very slightly hotter than at the Tropic of Capricorn in December, which is what we find I think (although that would need further research on this point).

I initially tried all other positions, but mathematically, only what is presented works.

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2. JMAC1978 says:

Spirit levels will always show you what ever degree you want based on your location. It will never change.

The earth is round overall but its full of flat, concave, convex etc. sections. So the only way to truly do a rectilinear experiment is in the ocean

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• Wild Heretic says:

Look. Plausible deniability is a path anyone can take, no matter how outlandish or far-fetched. If he feels safe believing that only that four and one eighth mile stretch of sand just so happened to be curved in the exact dimensions of a concave Earth, let him. The odds against it are astronomically high. Just think about it.

And it was measured against the level ocean parallel to that four and one eighth mile section of beach. They forced a straight line against the level (ocean) over that distance and the beams lowered into that level in the exact same dimensions as an earth curved as a cavity. That is it in a nutshell.

It is being repeated now in Texas, so astronomical odds will become godlike odds.

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• JMAC1978 says:

thanks WH, im slowly moving him in the right direction..

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3. Jason McCormack says:

Great stuff… funny, had a ball earther ask me TODAY!!! about position of the sun and how it worked with relation to 24 hr days in North and South… this basically answered most of his questions…

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• Wild Heretic says:

Yeah. Just because someone can’t be bothered or isn’t able to think of an alternative model to fit the new experimental data (rectilineator), doesn’t mean the data is wrong 🙂 It is the failing of the mind to join the dots. It isn’t easy to connect info together let alone finding it in the first place.

The data says we live inside the Earth. It is up to us to try and make a “mind model” to quell the curious mind. That part is the speculative part. There are a few variations within the concave community itself. But that is fine as the model isn’t important. It is only in our heads anyway.

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4. Wild Heretic says:

I have a new article now out on the “exact” location of the Sun at the solstices. It is heavy on the numbers, but the maths is necessary. The mechanics is next, which is easy to understand and is more fun.

Steven can ignore this article as it conflicts with his Sun position, but that is ok.

Now, I have a problem with the solar analemma that I didn’t mention in the article because I feel I am probably missing something simple. The length of the apparent solar day seems to conflict with the apparent solar year (it is the same problem no matter what Earth/Sun model is describing it).

The figure of eight is definitely determined by the position of the Sun at 12 (noon) every day. The solar year takes 365 and a quarter days which is determined by the position of the Sun at 12 (noon). This can only be caused by an accumulation of slower days overall throughout the year, but I don’t see a 6-hour slow down overall in the analemma over the year. Am I reading it wrong?

Maybe on the 366th day (Jan 1st again) at 12 noon, the Sun is a quarter way behind itself and the Jan 1st noon position of the previous year? Even so, is that slow down in the analemma data? I’m a bit confused by that, but I’ll let that go for now and see.

WH

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