旭日落日因何而扁

祝玉華，石鳳良，王賀清

（唐山師范學(xué)院 物理系，河北 唐山 063000）

If you had been to the seashore to watch the sunrise,perhaps you saw the situation shown in figure 1, the rising sun was obviously “squashed”.But do you know why the sun is oblate and why the phenomenon only appears when the sun is rising or setting? And do you pay attention to what about the ratio is between the solar vertical diameter and the horizontal one?

Fig.1 The oblate rising sun

1 Two pre-requisite conclusions

Conclusion 1 If the incident ray with the angle of αis refracted continuously by multilayer planes composed of different media, the direction of the final refracted ray is determined by the refractive index of the final medium.Proof is as follows: we suppose that the incident ray AB enters to the first medium from the air as shown in figure 2, then the refracted ray BCwith the angle of β travels to the second medium, the angle of the final refracted ray CD is γ.Because n0sin α=n1sin β and n1sin β = n2sinγ ,then n0sinα = n2sinγ.Namely, the second medium deter- mines the final angle of refraction γ, irrespective of whether the first one exists or not.This example of refraction in a two-layer planes can be obviously extended to the refraction in the multilayer planes of media.

Fig.2 The ray’s refraction in the two-layer planes of media

Conclusion 2 As shown in figure 3, the difference θ between the incident angle αand the angle of refraction γ changes along with the change of α in the plane of incidence.Namely, as the incident angle αincreases, the difference angle θ also increases, and vice versa.Therefore, the maximum value of γ appears when ray A is at grazing incidence(α =900).We had already proved it in a paper of "Physical Bulletin"[1].

Fig.3 The differenceθbetween the incident angleαand the angle of refractionγchanges along with the change ofα

2 Simplified atmospheric model

We suppose that the atmosphere is composed of many layers of different densities and every layer is a isopycnic plane because the thickness of the atmospheric layer is far smaller than the Earth’s radius.The incident solar ray from the exosphere ( n0=1) will be refracted continuously by so many planar layers before it reaches the ground.But we know that the direction of the final refracted ray is only determined by the refractive index of atmospheric plane most nearing to the ground (n=1.00029) according to the conclusion 1.Therefore we may establish a simplified atmospheric model as follows:The ground is one plane, the homogeneous atmosphere covers the surface whose refractive index is slightly bigger than 1(1.00029), and above the atmosphere is vacuum.What would the rising sun look like according to the simplified atmospheric model? As observed from the Earth's surface, the measured angular diameter of the sun "u" is 31'39.3" which is approximately 32' or 0.533o.

We introduce an xyz coordinate system with xz defining a horizontal plane at the top of the atmosphere.The x-axis is in the direction of the observer.The y-axis is normal to the plane.Above the xz pane is a vacuum (n0= 0) and below the xz plane is the homogeneous atmosphere (n=1.00029).The sun's lower edge is tangent to the xz plane at the point (x2, 0,0). The ray from x2passes through the origin O and makes a 90o angle of incidence with the normal.We call this ray SlowerO .Since n0sin 900= nsin∠NOA (OAis the refracted ray of SlowerO ),then sin ∠ NOA = 1/1.00029, so ∠N OA = 88037.2′.The solar angular diameter is 32′ in the direction of oy.Namely,the incident angle of SupperO is 89028′, so the angle of the refracted ray OBmay be worked out ∠NOB =88031.2′.The reverse projection of OA is S 'lowerO .The included angle between the incident ray SlowerO and its refracted rayOAis θlower, so the point Slowerappears elevated by the angle θlower,and

Likewise, the point Supperappears elevated by the angle

From formula (2) minus the formula (1)

it is clear that sun’s apparent angular diameter in vertical direction is given by:

or

Substituting data to the formula (4), we obtain

But the two points Sleftand Sright, being on the sun along the direction OZ, are elevated by the same angle θleft=θright, also there is no change in the oz direction because the incident angles are consistent.As a result of atmospheric refraction the sun’s angular diameter in the vertical direction appears to be u ′=6′, but in horizontal direction the sun's angular diameter appears to be u ′=32′.So our calculation gives a vertical to horizontal ratio of 6/32= 0.187.But the actual observation of the ratio between the two diameters is about 0.8, why is this?

Fig.4 The angle being elevated to the point of the sun’s edge

3 Deficiencies in this simplified atmosphere model

If the Earth’s radius were really infinite, then the oblate display in the rising sun or setting sun would be obvious.Because of the Earth’s radius is finite, the atmosphere is better modeled as being divided into many spherical layers concentric with the Earth's center and the gas refractive index is equal in each spherical layer.In order to facilitate the discussion, we only draw two layers in figure 5.

Fig.5 The ray’s refraction in the atmosphere

The incident rayAB enters the first layer with incident angle ∠ABN(the normal is CN), and its refracted ray continues to the second layer where it becomes the incident ray at D into layer 2 (the normal is CN′).If ∠ABN approaches 900then the angle of refraction is also approaches 900which causes arc FD to be wide.Since the Earth’s radius is anything but infinite, CNand that of CN′are not parallel. If we still regard the atmosphere to be composed of many planar layers then a large error results from this simplified model at the grazing incidence.The astronomical observations indicate that the sun’s ray is elevated by an angle of 35' from the angle of grazing incidence (we know this is the widest angle from the conclusion 2).But we obtained according to the formula (1).In other words, much difference exists between the simplified model and the actual atmosphere when the sun is at a grazing angle .The strict explanation is very complex so as to be found in specialized astronomy textbooks.When the angle ∠CBD is not very close to 900, the arcwill be very close to straight line FD in figure 5(tan 89.5o = 114.6, tan 89.0o = 57.29, tan 85.0o = 11.4, the values decline rapidly).Namely, the direction of CNand CN′approach colinearity.So the simplified model is quite reasonable except for the grazing incidence.We suppose that the sun leaves the axis ox slightly in figure 4.For example, if the sun’s lower edge sends out the ray with incident angle of 88040′, then the angle of refraction is 8804.9'=,therefore the point Slowerwill be “elevated” by the angle= 35 .1'= 35'6''.But this time sun's upper edge sends out the ray with incident angle of 88040′-32′=8808′, then it is easy to calculate that its refraction angle is 87040.7'= 87040'42'', therefore the pointwill be“elevated” by the angle

Then from the formula (5) we can obtain

(or obtains it directly by using 8804.9′- 87040.7′).So the ratio is now 24.2'/32'=0.76 between the vertical angular diameter and the horizontal one, which approaches to the actual observation.

4 Why is only the rising or setting sun oblate?

Only transforming formula (5) slightly, we can explain why only the rising or setting sun is oblate.In figure 2 , if the upside is vacuum( n0=1), the underside is atmosphere(n=1.00029), then

θ is very small.Its maximum value is 35' or 0.01 radians,therefore from the small angle approximation (θ =sinθ =tanθ for θ ＜100), then sinα =n sinα -nθc osα .We can obtain

(the unit is radian) or

(the units are minutes of arc) (6).Formula (6) is a very efficient way to rapidly estimate the degree of elevation of the incident ray.

If the sun’s lower edge sends out the ray with incident angle ofα, then it will be “elevated” by the angleθlower= tan α;the upper edge sends out the ray with incident angle of α-u,therefore it will be “elevated” by the angle θupper= tan(α -u),therefore the formula (5) becomes u′=u+ tan(α- u) - tanα.But the tangent function's characteristic is that only when the independent variable is close to 900, the very small change(u=0.5330= 32′) of the independent variable will cause the function value to change greatly, namely tan(α-u ) and tan αwill have the obvious difference, then u′and u will have the wide difference so that the flattening of the rising sun or setting sun can be seen.Once the incident angle (independent variable) is much smaller than 900, the very small change(u=0.5330= 32′) of the independent variable will not have caused the function value to change greatly, namely,tan(α-u )≈ tanα, u′≈ u, the human eye is unable to differentiate the difference between u′ and u, so we see the sun as circular.