The solar constant is the same for the area of the sun's sphere (4πr²) as for the area is its circle (πr²)

 

Author; Rogelio Perez C

Abstract

The amount of energy that comes from the sun to the earth is measured with a solar constant. Which has a value of 1364w/m², this energy is the radiation emission value of the sun for the area of a sphere 4πr²,  at the distance of an area of a sphere 4πr² of an astronomical unit, as the earth only receives sunlight the area of a circle πr², then 4 parts of that energy are not reaching the earth, so the value of the constant is divided by four, to know how much energy emitted from the sun enters the earth, but this work shows that the value of the solar constant does not change when we measure the energy emitted by the sun, in an area of the solar circle (πr²), between the value of the area of a circle (πr²), of an astronomical unit, which is the area that looks to the earth, so the division of the solar constant to know how much energy enters the earth in the area of a circle, should not be divided by 4, as is currently done.

Introduction;

To know the value of the energy that enters the planet, we must know the solar constant, but as the part of the energy that enters the planet is in the area of a circle (πr²), and the solar constant is the energy value for the area of the sphere (4πr²) of the Sun, then the value of the solar constant has to be divided into 4 parts, because the value of the sun's energy in its circle (πr²) is one quarter of that of its sphere. But this work shows that the energy of the solar constant is the same when we take the area of the sun sphere, as when we take the area of the sun circle, so the solar constant should not divide into 4 parts to know the value of the sun's energy entering the earth at a distance from an astronomical unit.

Theory;

The solar constant, SO, is the total amount of solar radiation (including all wavelengths of the solar spectrum) per unit of area that affects in a plane normal to the direction of the solar rays on the outside of the earth's atmosphere at the average distance between the sun and tierra.1

The average annual solar radiation reaching the upper part of the Earth's atmosphere (1361W/m²) represents the power per unit of solar irradiation area through the spherical surface surrounding the sun with a radius equal to the distance to Earth (1 AU).

This means that the Earth's approximately circular disk, as seen from the sun, receives approximately stable 1361W/m² at all times. The area of this circular disk is πr², in which r is the radius of the Earth. Because the Earth is approximately spherical, it has a total area of 4πr² which means that solar radiation reaching the top of the atmosphere,, averaged over the entire surface of the Earth, is simply divided by four to obtain 340 W/m².2

Development;

What is the value of the solar constant for the area of a sphere 4πr²?

The radiant energy that reaches us from the sun will decrease inversely to the area formed by a sphere of radius of 1 U.A., that is:

An astronomical unit =1.496E 11M

Astronomical unit in M²=

m²=1.496E 11M x 1.496E 11M= 2,238022E 22m²

Astronomical unit for the radius of a sphere;

=2,23802E+22m² x  4π 3,14 =2,23802E+22m² x π 12.56=

=2,81095E+23.

Total energy from the sun by the area of a sphere 4πr²;

Formula; E = σ * A * T⁴

E = sun-radiated heat (Luminosity).

Σ= Stefan-Boltzmann constant.

A =Area of a body (Sun)

T⁴=Temperature raised to 4 power.

Result;

σ =    5,67E-08   W/(M²K4)

A=     6,06679E+18     m²

T⁴=   1,11458E+15     Kelvin4

E =    3,834E+26         Wm²

An area 6.066C10 18m² with a temperature of 5778°k at the fourth power, multiplied by the Stefan-Boltzmann constant, has an energy of 3.834E 26wm²

Then;

The value of the solar constant= to the energy emitted by the sun in the area 4πr² of a sphere= 3.834E 26w/m², between the astronomical unit in the area of a sphere 4πr²=2, 81095E+23m²

3.834E 26w/m² /2, 81095E+23 = 1363.953709w

The value of the solar constant is equal to 1364wm²

What is the solar constant value for the area of a circular disc πr²?

An astronomical unit =1.496E 11M

Astronomical unit in M²=

m²=1.496E 11M x 1.496E 11M= 2,238022E 22m²

Astronomical unit for the radius of a circle πr²;

=2,238022E 22m² x π 3.14 = 7,027E 22m²

Energy emitted by the sun for the area of a circle πr².

Formula; E = σ * A * T⁴

E = sun-radiated heat (Luminosity).

Σ= Stefan-Boltzmann constant.

A =Area of a body (Sun)

T⁴=Temperature raised to 4 power.

Result;

σ =    5,67E-08   W/(M²K4)

A=     1,5167E+18m²,  m²

T⁴=   1,11458E+15     Kelvin4

E =    9,58501E+25 wm²,

An area 1,5167E+18m², with a temperature of 5778°k at the fourth power, multiplied by the Stefan-Boltzmann constant, has an energy of 9,58501E+25 wm²,  

Then;

The value of the solar constant= to the energy emitted by the sun in the area of a circle (πr²).= 9,58501E+25 wm², between the astronomical unit in the area of a circle (πr²).= 7,027E 22m²

9,58501E+25 wm²/ 7,027E 22m² = 1363.953709w

Value of the solar constant for the circle (πr²). of the sun is equal to 1364wm²

Conclusion;

The solar constant for the Earth is equal to the energy emitted by the sun for a radius of a sphere, than for the radius of a circle, so the temperature that reaches us from the sun to the top of the atmosphere, should not be divided by 4, since the Solar constant does not change the energy value.

Bibliography;

1- file:///C:/Users/MASTER/Desktop/radiacion.pdf

2-https://en.wikipedia.org/wiki/Solar_irradiance