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The Tel Aviv (Jaffa) Equation and a Theory for the Atom and Solar System Linking Microscales to

Macroscales; Where Cosmology and Archaeology Meet

Ian Beardsley

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Contents!

The Tel Aviv (Jaffa) Equation and a Theory for the Atom and

Solar System Linking Microscales to Macroscales; Where

Cosmology and Archaeology Meet

Abstract………………………………………………………………………..3!

Signiﬁcance of Tel Aviv……………………………………………………….3!

List of Constants, Variables, and Data in this Paper………………………4!

The Jaﬀa Equation…………………………………………………………….5!

Theory For The Solar System And Atom’s Proton

Important To Understand…………………………………………………….11!

Introduction……………………………………………………………………12!

Delocalization Time of the Earth…………………………………………….14!

The Solar System’s Quantized Structure Encodes !

Its Birth Cloud’s Mass And Mirrors The Atom’s Proton………………….20!

The Unfolding Of The Rudiments For A Theory Of Everything………….23!

Theory For Inertia…………………………………………………………….31!

The Solar Solution……………………………………………………………42!

Jupiter and Saturn……………………………………………………………46!

Modeling Star Systems With The Theory………………………………….49!

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Abstract

My theory for the atom and for the Solar System that both have a Schrödinger wave equation solution

based on a characteristic time of 1-second suggests that there should be a place on Earth whose climate

and historical cultural history since ancient times should be pivotal to the equation for their designs. This

place should be climatically central to the theory for the Solar System because it determines the

habitability of the Earth, and historically related to the location because the second comes to us from the

ancient origins of the second we have today, which are in the beginnings of agriculture because the

climate of the region lead to agriculture which lead to the invention of mathematics and writing from

which we got the unit of a second to measure time in ancient Mesopotamia. The location of this cultural,

and climatic hub is in Tel Aviv Israel (What was called Jaffa). We can distill our equation for Earth

climate to something being centered around the climate of Tel Aviv, an ancient sea port city on the

Mediterranean coast that in ancient times supplied cedar wood to both Sumer (where civilization began

and that gave us the second for measuring time) and supplied cedar wood to Egypt as well, that was used

in building the pyramids. Jaffa was a central sea port trading hub in ancient times, and is a cultural hub

today. We will present the Jaffa equation in this paper then follow up with a paper on the theory from

which it was derived, titled Theory For The Solar System And The Atom’s Proton; Linking Microscales

To Macroscales.

Signiﬁcance of Tel Aviv (Deep Seek Summary)

Jaffa stands as one of the world's oldest seaports, a legendary gateway on the Mediterranean for over

4,000 years. Its natural harbor made it a vital strategic and economic prize, controlled by Egyptians,

Canaanites, Phoenicians, Romans, Crusaders, and Ottomans. It was the primary port for the coveted cedar

trade and a crucial entry point for pilgrims, armies, and immigrants throughout history.

As a cultural hub, Jaffa was a melting pot where ancient Egyptian, Mesopotamian, Levantine, and later

Greek and Roman inﬂuences converged. This legacy is etched into its stone alleys, from its Egyptian

conquests to its role as the "White City's" ancient counterpart in modern Tel Aviv.

In the Bible, Jaffa is a cornerstone of sacred narrative. It was the port where the cedars of Lebanon arrived

for Solomon's Temple, and where the prophet Jonah famously ﬂed from God, only to be swallowed by a

great ﬁsh. In the New Testament, it was in Jaffa that the Apostle Peter had a visionary dream, pivotal to

the expansion of the early Christian church.

Thus, Jaffa's story is a unique tapestry, weaving together the threads of global commerce, ancient

mythology, biblical prophecy, and continuous human civilization on a single, storied shore.

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List of Constants, Variables, And Data In This Paper

(Proton Mass)

(Proton Radius)

(Planck Constant)

(Light Speed)

(Gravitational Constant)

1/137 (Fine Structure Constant)

(Proton Charge)

(Electron Charge)

(Coulomb Constant)

(The Author’s Solar System Planck-Constant)

(Earth Mass)

(Earth Radius)

(Moon Mass)

(Moon Radius)

(Mass of Sun)

(Sun Radius)

(Earth Orbital Radius)

(Moon Orbital Radius)

Earth day=(24)(60)(60)=86,400 seconds. Using the Moon’s orbital velocity at aphelion, and Earth’s

orbital velocity at perihelion we have:

(Kinetic Energy Moon)

(Kinetic Energy Earth)

: 1.67262E − 27kg

: 0.833E − 15m

h : 6.62607E − 34J ⋅ s

c : 299,792,458m /s

G : 6.67408E − 11N

α :

: 1.6022E − 19C

: 8.988E 9

ℏ

⊙

: 2.8314E 33J ⋅ s

: 5.972E 24kg

: 6.378E6m

: 7.34767309E 22k g

: 1.7374E6m

⊙

: 1.989E 30kg

⊙

: 6.96E 8m

: 1.496E11m = 1AU

: 3.844E 8m

K E

(7.347673E 22k g)(966m /s)

= 3.428E 28J

K E

(5.972E 24kg)(30,290m /s)

= 2.7396E 33J

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The Jaffa Equation I have found that the unit of a second of time that is in use today is the characteristic

time in the fundamental particles that make up the atom: the proton, the electron, and the neutron and the

characteristic time in the Solar System, centrally in the Earth/Moon/Sun system in my wave equation

solution of it. I found that the radius of a proton is given by:

is the ﬁne structure constant, the golden ratio, is Planck’s constant, is the universal

constant of gravitation, and is the speed of light. is the radius of a proton and is its mass. These

two equations yield an accurate equation for the radius of the proton

The ground state of the solar system we found is based on the Moon orbiting the Earth with a

characteristic time of 1 second:

Where is a Planck-type constant for the Solar System and is the mass of the Moon. This comes

from the solution of the wave equation for hydrogen describing the ground state as

We have

Which is close to the orbital radius of the Moon around the Earth. We ﬁnd our Planck-type constant for

the Solar System is given by the kinetic energy of the Earth in its orbit and a characteristic time of 1-

second. We have

So we see the Moon plays a central role in the quantization of the Earth/Moon/Sun system. The unit of a

second comes to us from the ancient Sumerians, who were the ﬁrst to settle down from wandering,

hunting, and gathering to invent writing, agriculture, and mathematics. They divided the Earth day into 12

units (what we call today hours) and the Earth night into 12 units called hours, dividing the Earth day into

24 hours. From their base 60 counting they divided the hour into 60 minutes, and the minute into 60

(

6 α

4πh

G c

)

⋅

= 1secon d

ϕ ⋅

π r

G m

⋅

= 1secon d

ϕ = 0.618

= ϕ ⋅

ℏ

⊙

⋅

= 1secon d

ℏ

⊙

ℏ

k e

ℏ

⊙

(2.8314E 33)

(6.67408E − 11)(7.34763E 22kg)

= 3.0281E 8m

ℏ

⊙

= (1secon d )KE

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seconds. This is how we have the duration of the second today. The Moon is curious because as seen from

the Earth it near perfectly eclipses the Sun and it makes life so successful on Earth because its orbit holds

the Earth at its inclination to its orbit of 23.5 degrees allowing for the seasons and preventing temperature

extremes. I found

the kinetic energy of the Moon, the kinetic energy of the Earth, and EarthDay is 24 hours

(86,400 seconds, the rotation period of the Earth). Earth’s tilt to its orbit is . The puts

the Earth’s rotation in the plane of its orbit into the equation. I realized I could solve star systems in

general if I had an equation for the radius of the Earth.

In order to apply this to other star systems, we have to be able to predict the radius of the habitable planet,

presumably in the n=3 orbit. I found the answer to be in the Vedic literature of India. They noticed that the

diameter of the Sun is about 108 times the diameter of the Earth and that the average distance from the

Sun to the Earth is about 108 solar diameters, with 108 being a signiﬁcant number in Yoga. So I wrote the

equivalent:

radius of the Sun, is the Earth orbital radius, and is the Earth radius.The surprising result I found

was, after applying it to the stars of all spectral types from F through K, with their different radii and

luminosities (the luminosities determine , the distances to the habitable zones), that the radius of the

planet always came out about the same, about the radius of the Earth. This may suggest optimally

habitable planets are not just a function of the distance from the star, which determines their temperature,

but are functions of their size and mass probably because they are good for life chemistry atmospheric

composition, and gravity when they are the size and mass of the Earth.

In order to get , the distance of the habitable planet from the star, we use the inverse square law for

luminosity of the star. If the Earth is in the habitable zone, and if the star is one hundred times brighter

than the Sun, then by the inverse square law the distance to the habitable zone of the planet is 10 times

that of what the Earth is from the Sun. Thus we have in astronomical units the habitable zone of a star is

given by:

10.

the luminosity of the star, and the luminosity of the Sun. It is at this point we want to ﬁgure out if

the unit of the second comes from some design of the Solar system, and if it has an archaeological origin

for some reason. That is we look at what these equations imply for the design of the Earth astronomically,

then see if we can locate a place on Earth that might be a clue the origins of the second. Whether it was

known to any ancient peoples to be a natural part of the Universe. To do this we star with equation 9 by

multiplying both sides of it by , that is by the Earth’s angular velocity, :

K E

(Ear th Da y)cos(θ ) = 1.0secon d s

K E

θ = 23.5

∘

cos(θ )

= 2

⊙

planet

⋆

⊙

⋆

⊙

2 π /24hrs

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11.

We now consider equation 8, combining it with equation 11 (EarthDay=24hrs):

Yields;

12.

We use the kinetic energies of the Earth and the Moon given by , where the orbital

velocities of the Earth and the Moon are:

and.

This gives us:

13.

is the surface area of the Sun, and is the equatorial rotational velocity of the Earth. This

equation (13) is the design of the Earth/Moon/Sun system that yields the second as Natural. Since the

gravitational acceleration at the surface of the Sun is given by we have:

14.

(rotational velocity of Earth at the equator).

We have our equation yields close to a second:

⊙

⋅

2 π

24hrs

K E

(Ear th Da y)cos(θ ) = 1.0secon d s

1secon d = 4π

⊙

⋅

K E

⋅

cos(θ )

K E = (1/2)Mv

⊙

1secon d =

4π R

⊙

⋅

⋅ cos(θ )

4π R

⊙

= ω

⊙

1secon d =

⊙

⋅

4π GM

⋅ cos(θ )

4π GM

4π (6.674E − 11)(7.34767E 22k g)

3.844E 8m

= 1.5918E5

= ω

2 π

86400s

(6.378E6) = 463.82m /s

⊙

(6.674E − 11)(1.989E 30kg)

(6.96E 8m)

= 274.03m /s

cos(23.5

∘

) = 0.917

⊙

⋅

4π GM

⋅ cos(θ ) =

(274m /s

)

⋅

(463.82m /s)

⋅ (1.5918E5m

)(0.917) = 1.14857secon d s

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This is where we can get a geographical location on Earth. We consider

And, write…

We have a velocity and we notice it is in the range of the speeds of sound on Earth. The speed of sound

varies with temperature and pressure. We can do a search with AI to ﬁnd out where this is. Here I used

Deep Seek. This is the speed of sound for a temperature of 68 deg F (20 deg C) which is the lower limit to

room temperature, and the pressure at sea level, 1 atmosphere. The place on Earth that has this annual

average temperature of 20 deg C and that is at sea level, is a sea port of Israel, Jaffa. Jaffa today is called

Yafo in Yiddish, which is a suburb of Tel Aviv. Tel Aviv and Yafo merged to make Tel Aviv-Yafo. We will

refer to it as Jaffa because that is the old ancient name that appears in the Bible. This sea port was the

center of commerce linking Egypt and Israel with Mesopotamia, the Levant, bringing Lebanon cedar

wood to the Sumerians and Egyptians, a crucial resource. They got the wood from the Phoenicians who

ﬂoated the logs along the coastlines to Jaffa, from where it was distributed, among other commodities.

This velocity of 343 m/s is exactly the speed of sound in Tel Aviv and Jaffa. It is Mach 1 there, the

velocity a jet must go to break the sound barrier in Jaffa, or Tel Aviv in general. We have:

15.

The atmospheric pressure in Jaffa was about the same 2000 years ago or so because Jaffa is at sea level.

The temperature might have been a little cooler because of the Urban Heat Island Effect. This effect

makes Jaffa a little warmer today than it was then because all of the concrete and asphalt there today

raises the temperature a little due to absorbing and reradiating heat more efﬁciently. But, we can say with

conﬁdence that the annual average temperature then was not much less than 0.5 to 1.5 degrees cooler, So

then it may have been about 18.5 to 19.5 deg C. The formula for the speed of sound is mostly dependent

on temperature and is given by

Where T is the temperature in degrees centigrade. So, today is

Ancient Jaffa might have been

= ω

2 π

86400s

(6.378E6m) = 463.82m /s

4π GM

4π (6.674E − 11)(7.34767E 22kg)

3.844E 8m

= 1.5918E5

⋅

4π GM

463.82 m /s

⋅ 1.5918E5m

= 343.2 m /s

Ja f f a, Isr a el Mach1

SolarSur fa ceGr a vit y

cos(θ ) = 1secon d

v = 331.4 + 0.6T

v = 331.4 + (0.6 × 20) = 343.4m /s

331.4 + (0.6 × 19) = 342.8m /s

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Which closely rounds to our value of 343 m/s because today the value is actually 343.4, not 343.0 that we

rounded it to in this paper. For all practical historical purposes, the sound barrier in Jaffa has been a

constant physical feature for millennia.

The sound barrier was ﬁrst broken on October 14, 1947 by U.S. Air Force Captain Chuck Yeager in the

Bell X-1 experimental rocket-propelled aircraft. Israel was founded on May 14, 1948 exactly seven

months later. There is no data available for the annual average temperature in Jaffa (Tel Aviv) for this

year, just for the average over all of Israel, which was 25.6 deg C (78.1 deg F). The average annual

temperature in Tel Aviv this year (2025) has been 20.5 deg C (68.9 deg F) with a average high of 24 deg

C (75 deg F) and an average low of around 15 deg C ( 59 deg F) according to Google AI Overview.

The Sun's mass determines its radius, luminosity, and surface gravity. Its mass and radius, through the

requirements of hydrostatic and thermal equilibrium, determine its internal structure and surface

temperature. The resulting luminosity, in turn, is the dominant factor setting the temperature on Earth. The

speed of sound in a location is directly and fundamentally related to its climate. In fact, climate is the

primary determinant of the local speed of sound in air. Thus, the equation

relates the climate of the Earth to the mass of the Sun. The annual average temperature of the Earth is 15

deg C (59 deg C), Jaffa on the average is a great example of one of Earth’s more desirable climate zones.

This Mediterranean climate in not just good but is of the world’s most productive and valuable climate

zones for agriculture. Good agricultural conditions were key to civilizations forming, which gave us the

unit of a second that we use today.

Ja f f a, Isr a el Mach1

SolarSur fa ceGr a vit y

cos(θ ) = 1secon d

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Theory For The Solar System And The Atom’s Proton; Linking Microscales !

To Macroscales

Version 3.0

Ian Beardsley

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Important To Understand One can speak of the structure of the long term structure of the solar system.

The whole object of developing a theory for the way planetary systems form is that they meet the

following criterion: They predict the Titius-Bode rule for the distribution of the planets; the distribution

gives the planetary orbital periods from Newton’s Universal Law of Gravitation. The distribution of the

planets is chieﬂy predicted by three factors: The inward forces of gravity from the parent star, the outward

pressure gradient from the stellar production of radiation, and the outward inertial forces as a cloud

collapses into a ﬂat disc around the central star. These forces separate the ﬂat disc into rings,

agglomerations of material, each ring from which a different planet forms at its central distance from the

star (they have widths). In a theory of planetary formation from a primordial disc, it should predict the

Titius-Bode rule for the distribution of planets today, which was the distribution of the rings from which

the planets formed.

Also, the Earth has been in the habitable zone since 4 billion years ago when it was at 0.9 AU. Today it is

at 1AU, and that habitable zone can continue to 1.2 AU. So we can speak of the distance to the Earth over

much time. The Earth and Sun formed about 4.6 billion years ago. As the Sun very slowly loses mass over

millions of years as it burns fuel doing fusion, the Earth slips microscopically further out in its orbit over

long periods of time. The Earth orbit increases by about 0.015 meters per year. The Sun only loses

0.00007% of its mass annually. The Earth is at 1AU=1.496E11m. We have 0.015m/1.496E11m/

AU=1.00267E-13AU. So,

The Earth will only move out one ten thousandth of an AU in a billion years. Anatomically modern

humans have only been around for about three hundred thousand years. Civilization began only about six

thousand years ago.

The unit of a second becomes important in my theory. We got the second from the rotation period of the

Earth at the time the moon came to perfectly eclipse the Sun. The Moon slows the Earth rotation and this

in turn expands the Moon’s orbit, so it is getting larger as the Earth loses energy to the Moon. The Earth

day gets longer by 0.0067 hours per million years, and the Moon’s orbit gets 3.78 cm larger per year.

That is as the Earth’s day gets longer and the lunar orbit grows larger, we got the second at the time that

the Earth day was what it is during the epoch when the Moon perfectly eclipses the Sun, 24 hours.

The near perfect eclipse is a mystery in the sense that it came to happen when anatomically modern

humans arrived on the scene, even before that, perhaps around Homo Erectus and the beginning of the

Stone Age. The Earth day was 18 hours long, long before that, 1.4 billion years ago. Homo Erectus is

around two to three million years ago. In our theory, we suggest conditions for life may be optimized

when the earth day is about 24 hours long, what it is today, because the earth day of 24 hours has a

characteristic time of 1 second in our theory to be presented.

(1.00267E − 13A U/year)(1E 9years) = 0.0001AU

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Introduction

In this version 3, we show the Earth/Moon/Sun system is a solution of the Schrodinger wave equation by

looking at a Gaussian wave packet and showing it predicts the delocalization time of the Earth, six

months if we quantize in terms of the Earth’s Moon, and a characteristic time of one second with the

Earth’s orbital kinetic energy. My theory predicts the radius of a proton, has a model for inertia that

works, and shows the Solar System to be a direct analog of the wave equation solution for the hydrogen

atom. It shows a common thread between microscales (atom's proton) and macroscales (the Solar

System). The theory is used to model star systems in general for a wide array of stellar spectral types. It

would seem the Earth orbit intersects with the hydrogen atom in a mutual solution of the Schrödinger

wave equation. I think this paper might suggest that life is part of a Universal Natural Process with

consciousness the end product of the Universe, and if what applies here to our Solar System is a more

general theory for star systems in general, might point to where life can be found and what its Nature is.

In this third version we expand the proton theory to the electron and neutron.

I report here the ﬁnding that the atom’s proton and a wave structure of the Solar System lend themselves

to a unifying characteristic time of a duration of 1.0 seconds in common. This is interesting in that the

duration of a second came down to us historically from the ancient Sumerians and Babylonians because

we inherited their base 60 counting. We develop a theory for inertia from this for the proton. We ﬁnd at

the outset that the connection between the atom’s proton and the Solar System is in the Planck length and

that the Solar System’s quantized structure encodes its birth cloud’s mass. The ﬁnding here open the door

to some profound archaeological questions. I further ﬁnd that the Earth lends itself to base 60 counting in

terms of its mass, size, and rotational period by looking at its rotational angular momentum. This opens

the door to suggesting the possibility that the phenomenon of the Moon perfectly eclipsing the Sun may

be a condition for optimally habitable planets and that habitable star systems can be modeled under a

common idea that might imply that life is part of a Universal natural process.

I am particular about story telling, especially short stories. In the second section I present a short story

about how the rudiments for a theory of everything unfolded. I hope by relating this story you can share in

the joy of discovery that came in this adventure which took place in the mind and on paper. It is a journey

into the microcosmos, and out into the vast expanse of the Universe. What took shape here instills a sense

of mystery in me in that it formulates the magic that is in the Universe in elegant symbolic language,

mathematics.

Some of the intriguing things we will ﬁnd is that the fundamental particle that makes up atoms, the

proton, has a characteristic time of a second. We will ﬁnd the ground state of our Solar System is

described by the Earth’s moon and has a characteristic time of one second. We will ﬁnd that the Planck-

type constant for the Solar System is given by the Earth’s orbital kinetic energy multiplied by 1 second,

where Earth is the planet in the Solar System optimized for life. We will ﬁnd that the 24 hour day (the

rotation period of the Earth) which is what it is today, has a characteristic time of one second. We will see

that the Earth orbital energy state as a solution to the quantum mechanical wave equation uses the Earth’s

moon to normalize the Sun’s size, giving it a size of 400. We will see that the condition for the Moon

orbiting the Earth such that it perfectly eclipses the Sun may be a condition for optimally habitable

planets in the habitable zones of stars in general. And, we will see the mass, size, and rotational frequency

of the Earth is conducive to the base 60 counting of the ancient Sumerians which is responsible for the

unit of the second we have today as our base unit to measure time. All of this I see as quite magical and

seems to me to suggest that the Earth/Moon/Sun system is this way for a reason, that there may be a

mysterious force behind it. Let’s begin with a quote from Carl Sagan…

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The surface of the Earth is the shore of the cosmic ocean. From it we have learned most of what we know.

Recently, we have waded a little out to sea, enough to dampen our toes, or at most, wet our ankles. The

water seems inviting. The ocean calls. Some part of our being knows this is from where we came. We long

to return. These aspirations are not, I think, irreverent, although they may trouble whatever gods may be.

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Delocalization Time of the Earth In order to suggest that the Solar System has a quantum

wave solution we have to show our hypothesis is right, that the Planck-type constant for the

Solar System, , is given by the orbital kinetic energy of the Earth times 1-second. And, the

ground state is given by the Moon orbiting the Earth, by its mass, not that of Mercury, planet 1.

That is we have:!

Which is close to the orbital radius of the Moon around the Earth. This comes from the wave

solution of the hydrogen atom, which is!

Because we interestingly ﬁnd that!

So, we guess!

In order to show that our hypothesis is right, we solve the wave equation for a Gaussian wave

packet and determine the delocalization time, . If it is about six months, the time it takes the

Earth to delocalize (travel its orbital diameter), using the Moon playing the role of the mass of

an electron and our as above to describe the Earth, then the hypothesis can be taken as

correct, and we can solve the whole system for the Earth/Moon/Sun system from the rest of

the equations in the hydrogen atom solution to the Schrodinger wave equation, which is in

spherical coordinates:!

The delocalization time of a particle, molecule, or mass in general, , is the time it takes a

particle to delocalize. If we want to apply our wave equation theory of the Solar System to this

concept, then the delocalization time should be the time for the Earth to travel the diameter of

it’s orbit, which would be half a year (about six months). In order to derive the delocalization

time we must consider a Gaussian wave packet…!

The Gaussian wavefunction in position space is!

It’s Fourier wave decomposition is!

ℏ

⊙

ℏ

⊙

(2.8314E33)

(6.67408E − 11)(7.34763E 22kg)

= 3.0281E8m

ℏ

⊙

⋅

= 1secon d

ℏ

⊙

= (1secon d )KE

ℏ

⊙

−

ℏ

[

∂

∂r

(

∂

∂r

)

sinθ

∂

∂θ

(

sinθ

∂

∂θ

)

sin

∂

∂ϕ

]

ψ + V(r)ψ = E ψ

ψ (x,0) = Ae

−

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We use the Gaussian integral identity (integral of quadratic)!

We ﬁnd via the inverse Fourier transform. It is!

Substitue :!

ψ (x,0) = Ae

−

∫

2π ℏ

ϕ( p)e

ℏ

∫

∞

−∞

−a x

+bx

d x =

ϕ( p)

ϕ( p) =

∫

∞

−∞

d x ψ (x,0)e

−

ℏ

ψ (x,0)

ϕ( p) = ϕ( p) = A

∫

∞

−∞

−

ℏ

[ px]

d x

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This is of the form:!

, !

Using!

We have to ﬁnd the normalization constant, A, because the probability has to be 1 at its

maximum. We have!

We now consider the evolution of a free particle. For a free particle!

The time evolution in free space is!

∫

{−a x

−bx}

d x

a =

b =

ℏ

∫

∞

−∞

−a x

+bx

d x =

ϕ( p) = A

1/(2d

)

exp

−

ℏ

4 ⋅

= −

2ℏ

ϕ( p) = Ad 2πexp

(

−

2ℏ

)

∫

∞

−∞

ψ (x,0)

= 1

ψ (x,0)

−x

∫

∞

−∞

{−x

2}d x

= d π

A = (πd

)

−1/4

H =

ϕ( p, t) = ϕ( p)e

−i

t/h

= ϕ( p)e

−i

2mℏ

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Substitute:!

Factor out the term and we have!

The integral is then,!

, Re(a)>0!

, !

, , , , !

We determine the probability density . We have!

, , !

ψ (x, t) =

∫

2π ℏ

ϕ( p)e

{

ℏ

px}

{−i

2mℏ

}

ϕ( p) = Ad 2πexp

(

−

2ℏ

)

ψ (x, t) = A

d 2π

2π ℏ

∫

∞

−∞

dpexp

[

−

2ℏ

− i

2mℏ

ipx

ℏ

]

α =

2ℏ

2mℏ

2ℏ

[

itℏ

]

∫

∞

−∞

{−αp

+bp}

dp =

/(4a)

a = α

b = i x /ℏ

d 2π

2π ℏ

ℏ

2α

ψ (x, t) =

ℏ 2α

exp

[

−

/ℏ

4α

]

τ =

m d

ℏ

m d

ℏ/m

itℏ/m

ψ (x, t) =

ℏ 2α

exp

[

−

(1 + it /τ)

]

ψ (x, t)

ℏ 2α

exp

[

−

⋅ 2Re

(

1 + it /τ

)

]

exp(−Bx

)

= exp(−2ReBx

)

B =

(1 + it /τ)

ReB =

⋅

1 + t

/τ

of 18 64

Because we multiplied the top and bottom of by and took its real part. We have!

is the delocalization distance, which for instance could be the width of an atom. is the

delocalization time, the average time for say an electron to traverse the diameter of the atom and

even leave it, to delocalize. If we substitute for our , and say that the delocalization distance

uses for the Moon, the width of the Earth orbit, we should get a half a year for the delocalization

time, the time for the Moon and Earth to traverse the diameter of their orbit around the Sun. We

have

Where is the mass of the Moon, and is the orbital radius of the Moon. We have

Now let’s compute a half a year…

(1/2)(365.25)(24)(60)(60)=15778800 seconds

So we see our delocalization time is very close to the half year over which the Earth and Moon

travel from one position to the opposite side of the Sun. The closeness is

Conclusion We see we the quantization of the Earth/Moon/Sun system is with a characteristic

time of one second and the mass of the Moon. We can now find all of the solutions of a wave

equation solution to the Earth/Moon/Sun system by direct analog to their solutions for the

hydrogen atom. This may give us clues as to what is going on with quanta too small to observe

2ReB = 2 ⋅

(1 + t

/τ

)

(1 + t

/τ

)

1 − it /τ

1 + i

⋅

1 − i

1 +

ψ (x, t)

∝ exp

[

−

(1 + t

/τ

)

]

ψ (x, t)

[

−

⋅

(1 + t

/τ

)

]

τ =

m d

ℏ

⊙

τ =

moon

(2r

moon

)

ℏ

⊙

moon

τ = 4

(7.34767E 22kg)(3.844E 8m)

2.8314E 33J ⋅ s

= 15338227secon d s

15338227

15778800

100 = 97.2 %

of 19 64

by the uncertainty principle with any cause and effect or determinism. The interesting thing is

that in the solution for the Solar System, the mass of the Moon of the Earth is the the mass of an

electron to solve the orbit of the Earth and the characteristic time happens to be about 1 second.

The value of used above will be determined in the following sections, though we have said

here how it is determined. This analog to the hydrogen atom has to have a nuance like this

because planets are different than say protons, for one thing they are not the same sizes as one

another or the same masses. We will go into this more, as we expand the theory in following

sections.

The second comes from the rotation period of the Earth and dividing it up with ancient

Sumerian base 60 counting. The rotation of the Earth is gradually slowing as it loses energy to

the Moon, which is slipping further out into its orbit. In this paper we have the solution is for

when the Earth day is 24 hours, what it is today and what it was during the time of the ancient

Sumerians, which gives a quantization of 1-second. We have this is given by the following

equation:

Where EarthDay=24 hrs and the ratio is the kinetic energies between the Moon and the Earth

and is the tilt of the Earth to its orbit. All of this will be explained in the following sections.

Meanwhile, if the wave solution is from the rotation of the cloud that collapsed into the

protoplanetary disc under its rotation, from which the planets and Sun formed, then perhaps

that cloud mass is measured by the Earth, Moon, and Sun…

ℏ

⊙

K E

(Ear th Da y)cos(θ ) = 1.0secon d s

of 20 64

The Solar System’s Quantized Structure Encodes Its Birth Cloud’s Mass And Mirrors The

Atom’s Proton We have established the possible connection of the atom’s proton to the Solar

System through the characteristic time of a second, a connection between the microcosmos

and the macrocosmos. We want to see what that connection is mathematically. The radius of a

proton to its mass determines the characteristic time of a proton as:!

1. !

2. !

is the ﬁne structure constant, the golden ratio, is Planck’s constant, is the

universal constant of gravitation, and is the speed of light. is the radius of a proton and

is its mass. These two equations yield an accurate equation for the radius of the proton!

3. !

The ground state of the solar system we found is based on the Moon orbiting the Earth with a

characteristic time of 1 second:!

4. !

Where is a Planck-type constant for the Solar System and is the mass of the Moon.

This comes from the solution of the wave equation for hydrogen describing the ground state as!

5. !

We have!

6. !

Which is close to the orbital radius of the Moon around the Earth. We ﬁnd our Planck-type

constant for the Solar System is given by the kinetic energy of the Earth in its orbit and a

characteristic time of 1-second. We have!

7. !

(

6α

4πh

)

⋅

= 1secon d

ϕ ⋅

πr

⋅

= 1secon d

ϕ = 0.618

= ϕ ⋅

ℏ

⊙

⋅

= 1secon d

ℏ

⊙

ℏ

⊙

(2.8314E33)

(6.67408E − 11)(7.34763E 22kg)

= 3.0281E8m

ℏ

⊙

= (1secon d )K E

of 21 64

Combining equation1, 4, and 7 we have using as well the kinetic energy of the Earth

, and the orbital velocity of the Earth :!

8. !

9. !

10. !

2.6566-E16kg/m=2.889E-16kg/m=92% accuracy!

We used at perihelion. Where is the mass of the Sun, is the mass of the Earth, is

the Earth’s orbit (1AU), and is the mass of a proton. is the cross-sectional area of a

proton and is the Planck length given by!

11. !

Where is the reduced Planck constant. This equation (equation 10) is in kilograms

per meter . We have!

12. !

This is on the order of the mass of a star-forming nebula from which a smaller one breaks away

and the protoplanetary disc and the star form to form the Solar System. We see it’s inverse

multiplied by the square of the Earth’s orbital radius is on the left. We see that the Planck

length action on a mass in terms of the Earth, Moon, and Sun gives the mass of a proton to its

radius. It would seem the Solar System’s quantized structure encodes its birth cloud’s mass

and mirrors the atom’s proton. The Planck length may be the smallest unit of space below

which space is no longer described by classical physics, that space does not have an inﬁnite

number of points, but rather has pixels per square meter determining its deﬁnition or degree of

resolution of space.!

Star systems form from giant molecular clouds (GMC’s). A cloud that is a portion of the GMC

collapses under its rotation and ﬂattens out into a rotating disc from which the planets and

central star form. Good examples of GMC’s from which star systems form that have masses on

the order of what is given in equation 12:!

= (1/2)M

= GM

⊙

2π

⋅

⊙

3α

⋅

πr

. ℓ

⊙

= π

12α

⋅

. ℓ

⊙

⋅

ℓ

6 2

(5.972E 24kg)

(1.989E30kg)

(7.34767E 22kg)

(1.6162E − 35m)

(1.471E11m)

(8.485)

1.67262E − 27kg

0.833E − 15m

(18769)

⊙

πr

ℓ

ℏG

ℏ = h /2π

⊙

= 3.5568E41kg

of 22 64

are,!

The Orion Molecular Cloud Complex!

Total Mass: ~ (~ )!

Includes: Orion A (M42, the Orion Nebula ~ ) and Orion B (~ )!

NGC 604 —- Triangulum Galaxy (M33)!

Mass: ~ (~ )!

The characteristic time of 1 second should be in the protoplanetary disc from which the planets

formed. I would guess it would be in the time between collisions of particles in the

protoplanetary disc. Ultimately, the planets form from these collisions. The time between

collisions in the protoplanetary disc are given by!

1. Particle number density n ( the number of particles per unit volume).!

2. Relative velocity between particles .!

3. Particle cross-section (related to particle size).!

For micron to millimeter sized grains in a dense inner region of the protoplanetary disc (like

about 1 AU from the star, which is the Earth orbit) the range of these values are:!

1. particles per meter cubed (from disc models)!

2. Particles sizes are meters.!

3. Relative velocities of particles are as driven by Brownian motion,

turbulence, and gas drag.!

We can imagine a scenario where this yields 1 second for the time between collisions of

particles in the formation of planets by using typical values!

It could be that the Earth/Moon/Sun system is a fossil of the protoplanetary disc’s initial

conditions.#

⊙

= 3.5568E41kg

200,000M ⊙

4E41kg

100,000M ⊙

80,000M ⊙

150,000 − 200,000M ⊙

3 − 4E41kg

rel

σ = π r

n ≈ 10

− 10

r ≈ 10

−6

− 10

−3

rel

≈ 1 − 10m /s

≈

nσ v

rel

(0.32 × 10

−3

)(π(10

−6

(1m /s)

= 1secon d

of 23 64

The Unfolding Of The Rudiments For A Theory Of Everything

It all started when I was trying to work out a theory for inertia, that property of matter to resist

change in motion: when you push on it, it pushes back. The more of it there is, the more it

pushes back. I had decided to start with the constants, like the gravitational constant, because

I ﬁgured they measured the properties of space and time. I eventually wrote an expression that

to my surprise was equal to 1 second:!

the radius of a proton, its mass, the constant of gravitation, the speed of light,

Planck’s constant, and the ﬁne structure constant. I found that interesting and ﬁgured if the

proton was characterized by the second, and the second came from the ancient Sumerians

dividing-up the rotation period of the Earth into 24 hours, each hour into 60 minutes, and each

minute into 60 seconds from their base 12 and base 60 mathematics, that it had to have

something to do with the celestial motions and periods they observed in the sky, that if the

second was natural, then it would be in the motions of the Earth, Moon, Sun, and stars. My ﬁrst

guess, which panned out, was that the kinetic energy of the Moon to the kinetic energy of the

Earth times the 24 hour day, should be one second, or close to it. At ﬁrst I found it was close to

it, but then I made an adjustment for the Earth’s tilt to its orbit of and it came out

exact for all practical purposes. I got!

the kinetic energy of the Moon, the kinetic energy of the Earth, and EarthDay is 24

hours (86,400 seconds, the rotation period of the Earth). I then thought this was quantum

mechanical and that I should make a Planck-type constant for the Solar System, . I found it

was in this very equation because it is in joule-seconds which could be the kinetic energy of

the earth times one second in the above equation, so I had:!

I then thought I don’t need to solve the Schrodinger Wave equation of quantum mechanics for

the Solar System, but just look at the equations of Niels Bohr for the Bohr model of the atom,

which he wrote down before the Schrodinger equation existed from suggesting the proton had

discreet orbitals for the electrons and was quantized by , the Planck constant. He didn’t know

why or how it quantized like this by integer multiples of , but he found it worked. He was

inspired to do this by the emission spectra of hydrogen for diﬀerent energies, he suggested

after the electron jumped from one orbital to another by adding energy, that when it fell back in

it would emit light of a particular frequency. So I looked at his equation for the energies of

orbitals and their orbital distances :!

(

6α

4πh

)

⋅

= 1secon d

θ = 23.5

∘

(Ear th Day)cos(θ) = 1.0second s

ℏ

⊙

ℏ

⊙

= (1secon d )K E

ℏ

= −

)

2ℏ

of 24 64

It later turned out for the hydrogen atom the case of Z=1 (1 proton) that the Schrodinger equation had

these as the solutions. That equation in spherical coordinates is:

This equation can be solved for the hydrogen atom giving the results of the the Bohr equations, but that is

for Z=1, for other elements, like Z=2,3,4…The equation becomes too difﬁcult to solve. But Bohr’s

equation works for these other values of Z.

I then looked at Bohr’s equations and changed the Coulomb constant to the gravitational constant ,

which changed charge (electron charge) into mass , put in my Planck-type constant for the Solar

System, , did the dimensional analysis and wrote:

They worked correctly to better than 99% for the Earth/Moon/Sun system, which was the n=3 orbit. I

knew I couldn’t use , which would ridiculously be the number of protons in the Sun, so since this was

gravity not electric ﬁelds, I used the radius of the Sun, . But, since the Moon of the Earth was

mysteriously appearing in my equations, I used it to normalize the radius of the Sun, which gave it a

radius of 400 (400 moon radii). This made it work. I then realized since we were dealing with the Earth,

the planet most optimally suited for life in the Solar System, and thought since this was because the Moon

optimizes the conditions for life on Earth by holding the Earth at its inclination to its orbit around the Sun

allowing for the season’s and preventing temperature extremes, that the Earth perfectly eclipsing the Sun

as seen from the Earth had meaning, and I wrote the conditions for a perfect eclipse, which are the radius

of the Sun to the radius of the Moon equals the orbital radius of the Earth to the orbital radius of the

Moon. I then thought this might be a condition for habitable planets to be optimally successful and wrote

it:

Later I used this in my solutions for habitable star systems by spectral type of the stars. Finally, I wanted

to formulate the ground state for the solar system and used the equation from the Bohr atom, which is

ℏ

Z k

−

ℏ

2 m

[

∂

∂r

(

∂

∂r

)

si n θ

∂

∂θ

(

si n θ

∂

∂θ

)

si n

∂

∂ ϕ

]

ψ + V(r)ψ = E ψ

ℏ

⊙

K E

= n

⊙

⋅

2ℏ

⊙

2ℏ

⊙

⋅

⊙

⋅

⊙

planet

moon

star

moon

ℏ

k e

≈ 0.529E − 10 m

of 25 64

Thus writing it with the mass of the Moon, .

Which is about the Moon’s orbital distance (3.844E8m). Noticing this in meters, is about the speed of

light in meters per second, if I multiplied by one over , the speed of light, it would be 1 second:

It was at this point that I was sure I was onto something. Later, playing with the equations, I ended up

with another equation for a second using the proton. It was

Where is the golden ratio (0.618). Equating the left of this with left of my ﬁrst equation got the ball

rolling

I found it yielded the radius of the proton as

It was at this point that I realized I had the rudiments for a kind of theory of everything that bridged the

microcosms to the macrocosmos. In the next section we will put forth a theory for the mass of a proton.

With what we have so far we can almost apply it to stars of other spectral types (masses, sizes, colors, and

temperatures) for formulating how other stars might structure the mechanics of their habitable planets.

In order to apply this to other star systems, we have to be able to predict the radius of the habitable planet,

presumably in the n=3 orbit. I found the answer to be in the Vedic literature of India. They noticed that the

diameter of the Sun is about 108 times the diameter of the Earth and that the average distance from the

Sun to the Earth is about 108 solar diameters, with 108 being a signiﬁcant number in Yoga. So I wrote the

equivalent:

radius of the star.The surprising result I found was, after applying it to the stars of all spectral types

from F through K, with their different radii and luminosities (the luminosities determine , the

distances to the habitable zones), that the radius of the planet always came out about the same, about the

radius of the Earth. This may suggest optimally habitable planets are not just a function of the distance

ℏ

⊙

(2.8314E 33)

(6.67408E − 11)(7.34763E 22kg)

= 3.0281E 8m

ℏ

⊙

⋅

= 1secon d

ϕ ⋅

π r

G m

⋅

= 1secon d

(

6 α

4πh

G c

)

⋅

= 1secon d

= ϕ ⋅

planet

= 2

⋆

planet

⋆

planet

of 26 64

from the star, which determines their temperature, but are functions of their size and mass probably

because they are good for life chemistry atmospheric composition, and gravity when they are the size and

mass of the Earth.

In order to get , the distance of the habitable planet from the star, we use the inverse square law for

luminosity of the star. If the Earth is in the habitable zone, and if the star is one hundred times brighter

than the Sun, then by the inverse square law the distance to the habitable zone of the planet is 10 times

that of what the Earth is from the Sun. Thus we have in astronomical units the habitable zone of a star is

given by:

the luminosity of the star, and the luminosity of the Sun. We see our theory has applications to

archaeology because the second came to us historically from the ancient Sumerians because they divided

the Earth day (rotation period) into 24 hours, and, because each hour and minute got further divisions by

60 because their base 60 counting system was inherited by the ancient Babylonians who were the ultimate

source of dividing the hour into minutes and the minutes into seconds. I have found this system is given

by the rotational angular momentum of the Earth, , in terms the solar system Planck-type constant,

because, as I already pointed out:

This base 60 counting combined with dividing the day into 24 units is mathematically optimal because

the rotational angular momentum incorporates not just the day (rotation period of the Earth) but the mass

and size of the Earth. We can say we are touching on archaeoastronomy. This is because 60/24=2.5 and

the Scottish engineer, Alexander Thom, found ancient megalithic (stone) observatories throughout Europe

may have been based on a unit of length he called the megalithic yard and that the separations between

stones, that align with celestial positions and cycles, are recurrently separated by 2.5 megalithic yards.

Like in Stonehenge.

You will notice the 4/5 in the above equation can be written as 2/5, meaning the equation becomes:

the rotational frequency of the Earth, Since , where is the angular velocity of the Earth, we

have:

We notice , the ancient Sumerian factors deﬁning the second from the rotation period of the

Earth, We see the rotation, mass and radius (size) of the Earth are optimally formulated with ancient

planet

⋆

⊙

⋆

⊙

earth

ℏ

⊙

24 = 60

earth

π M

earth

2 π f

= ω

earth

2/5 = 24/60

of 27 64

Sumerian base 12 and base 60 mathematics (they divided the day into 12 hours, and the night into 12

hours).

I have found that the pressure gradient of the protopanetary disc, as a function of radius, that gave birth to

our solar system, is given by:

the radius of the disc where pressure is . This is the solution to:

The protoplanetary disc that evolves into the planets has two forces that balance its pressure, the

centripetal force of the gas disc due to its rotation around the protostar and the inward gravitational

force on the disc from the protostar , and these are related by the density of the gas that makes

up the disc.

I have computed my Planck-type constant, , as such:

Where

This characteristic time of one second is not just in the Solar System, and atom’s proton, but in the basis

of life chemistry, carbon, and the hydrocarbons, the skeletons of life chemistry. I found

is carbon (C)

is hydrogen (H)

Which is to say that six protons, which is carbon, the basis of life as we know it, has a characteristic time

of one second because in the ﬁrst equation above, we have a mass divided by the mass of a proton, times

seconds, giving six protons times a second (6 proton-seconds) which means 6 protons (carbon, the basis

P(R) = P

(

)

−

earth

ℏ

⊙

d P

= − ρ

(

⋆

−

)

⋆

ℏ

⊙

ℏ

⊙

= (hC )K E

hC = 1secon d

C =

⋅

π r

G m

ℏ

⊙

= (hC )K E

earth

= (1.03351s)(2.7396E 33J ) = 2.8314E 33J ⋅ s

6proton s

⋅

4πh

G c

= 1secon d

1proton

⋅

4πh

G c

= 6secon d s

of 28 64

of life) has a characteristic time of one second. This means that 1 proton, hydrogen, has a characteristic

time of six seconds. Hydrogen is the most fundamental element in the periodic table of the elements

which was theoretically created in the so-called big bang that gave birth to the universe, and is the

element from which all of the other heavier elements were made by stars. This six-fold symmetry that is

in hydrocarbons, the skeletons of biological chemistry, is fundamental to deﬁning the periodic table of the

elements because it has been found that the six protons of carbon and their respective charges, interact

with its six electrons, their respective charges, to balance to make carbon the most stable element

mathematically in which to describe the rest of the atoms in the periodic table. This is no doubt related to

the regular hexagon, a six-sided polygon which tessellates (tiles a surface without leaving gaps) because it

has its radii equal in length to its sides. This hexagonal tessellating property is used by bees to make their

honeycombs. So we see our theory now goes beyond the atom and the solar system. That it goes to

biological chemistry. But, it does not stop there. It seems to go into cosmology, the study of the origin and

fate of the universe. We see this because my equations link proton properties to 1-second, and protons

were ﬁxed in the universe at 1 second after it, meaning we could be seeing a universal clock that has

inﬂuenced everything since the Big Bang.

The idea is that neutrino decoupling (neutrinos stop interacting with one another) happens when the

reaction rate of weak interactions falls below the Hubble parameter, the expansion rate of the universe

. The reaction rate per particles is given by

is the Fermi constant is about , and is the temperature of the Universe. The

expansion rate of the universe is given by

Where is the Plank mass is about 1.22E19GeV. and have units of inverse time ( ). Neutrino

decoupling happens when

This is when the number of protons in the universe was set in place which, as it would turn out, is close to

one second in rough estimate.

The expansion rate of the Universe is governed by the Friedmann equation

Where is the energy density of the Universe. It is

The Hubble expansion rate is

Γ ≈ G

1.166E − 5G eV

−2

H ≈

−1

8π G

ρ ∝ T

of 29 64

Since

we have

We said protons and neutrons are set in the universe when it has cooled in its expansion to about 1MeV.

We have

This was done in Planck units where time can be expressed in inverse energy. Since in Planck units

we have

This theory seems, then, to have applications at the core of cosmology, astrobiology (the study of life in

the universe in general), solar system mechanics, particle physics, theories of common structure between

micro-scales and macro-scales, and biology.

To compute the moon’s orbital radius I just use

Where Ag is the molar mass of silver and Au is the molar mass of silver, a connection to the 1.8 that

appears in our Solar System. We use this because we know it works for our Solar System.

This theory has a magical, mystical feel to it in that it involves the condition for the Moon perfectly

eclipsing the Sun as seen from the Earth, the ratio the Solar radius to the lunar orbital radius being in the

ratio of a gold atom mass to a silver atom mass, where gold and silver have been the metals of ceremonial

jewelry on Earth since ancient times, the Sun gold in color, the Moon silver in color, and in that the basis

unit of time in the theory is the second, which came to us from the ancient Sumerians who ﬁrst settled

H ∝

≈ 2.4E18G eV

t ∝

2.4E18G eV

(1E − 3G eV )

= 2.4E 24G eV

−1

1G eV

−1

= 5.39E − 25s

t ≈ (2.4E 24)(5.39E − 25)

t ≈ 1.3secon d s

= R

⊙

⋆

(1.8)

of 30 64

down from following the herds, and crafting spearpoints from stone to hunt, to invent agriculture,

architecture, mathematics, and writing.

Perhaps one day, humans will travel to other star systems and ﬁnd other worlds whose mathematical

relationships are as beautiful and exciting as they are here in our star system, the Solar System. Perhaps

we will learn of other civilizations on other planets, and learn their ancient histories and how they became

who they are today, what kind of music and art they have, and did they build pyramids or tipis in that

these things may be universal in their geometries.

of 31 64

Theory For Inertia I had two equations that gave the radius of a proton with characteristic times of one

second each. I had to break down the equations in terms of their operational parameters as described by a

geometric model. This is what I came up with, a proton is a 4d hypersphere who's cross-section is a

sphere. Of course occupying the dimension of time (4th dimension in drawing) is the vertical component

of the drawing. I have to draw this 3d cross-section as a circle (we cannot mentally visualize four

dimensions). The proton is moving through time at the speed of light (vertical component in the drawing)

it is a bubble in space. The normal force holding it in 3d space is proportional to the inertia

created by the pliability of space measured by G. So when we push on it (Force applied in drawing) there

is a counter force explaining Newton's action/reaction.

I think you could look at this another way: the cross sectional area of the proton moving against space is

in the opposite direction of the force applied and h is the granularity of space, G still its pliability. That is

to say, the ﬂux of a normal force to a hemisphere is over the area of the cross-section of the sphere.

It is the goal of this opening section to provide a theory for inertia, that quality of a mass to resist change

in motion. We want the the theory to include not just the quantum mechanics constant for energy over

time Planck’s constant, but to include the universal constant of gravitation G, the constant the speed of

light from relativity, and the ﬁne structure constant for theories of electric ﬁelds so as to bring together

the things that have been pitted against one another: quantum mechanics, relativity, classical physics,

electric ﬁelds, and gravitational ﬁelds. Towards these ends we will suggest a proton is a 3D cross-section

of a 4D hypersphere held in place countering its motion through time by a normal force that produces its

inertia (measured in mass in kilograms) much the same way we model a block on an inclined plain

countered by friction from the normal force to its motion. The following is the illustration of such a

proton as a cross-sectional bubble in space:

To get the ball rolling, I had found a wave solution to the Earth/Moon/Sun system where the Earth

orbiting the Sun is like an electron orbiting a proton with a quantum mechanical solution. I found this

solution had a characteristic time of one second. But, I found as well, I could describe the proton as

having a characteristic time of one second, and that this yielded the radius of a proton very close to that

obtained by modern experiments. So, it is now before me to come up with a theory for the proton in terms

of these characteristic times before I present my theory for a wave solution of the Solar System.

= h /(ct

)

of 32 64

The expressions for the characteristic times of 1-second for the proton that I found, were:

Where is the golden ratio, is the radius of a proton, and is the mass of a proton. We ﬁnd

these produce close to the most recent measurements of the radius of a proton, if you equate the left sides

of each, to one another:

To derive this equation for the radius of a proton from ﬁrst principles I had set out to do it with the Planck

energy, , given by frequency of a particle, and from mass-energy equivalence, :

We take the rest energy of the mass of a proton :

The frequency of a proton is

We see at this point we have to set the expression equal to . So we need to come up with a theory for

inertia that explains it:

The radius of a proton is then

(

6α

4πh

G c

)

⋅

= 1secon d

ϕ ⋅

π r

G m

⋅

= 1secon d

ϕ = 0.618

= ϕ

= 0.816632E − 15m

E = h f

E = m c

E = h f

E = m

⋅

= ϕ =

= ϕ ⋅

of 33 64

In order to prove our theory for the radius of a proton as incorporating , we will apply our model

outlined involving a normal force, to a 3d cross-section of a 4d hypersphere countering its direction

through time, t. We begin by writing equation 1 as:

Where , the constant of gravitation measures the pliability of space, and the granularity of space, and

c the speed of propagation. measures the inertia endowed in a proton. We write equation 2 as:

We now say that and that the normal force is

This gives us:

Since , we have

This gives

10.

is the cross-sectional area of the proton countering the normal force, , against its motion through

time, this is measured by the constant of gravitation. It is to say that

11.

= ϕ ⋅

6α

4πh

G c

⋅

1secon d

1 =

⋅

π r

G m

⋅

c(1secon d )

⋅

= 1secon d

1 =

⋅

π r

G m

⋅

⋅ F

9α

⋅

(

)

= ϕ

1 =

9α

⋅

3α

⋅

π r

∝

Ar eaCr ossSect ion Pr oton ⋅ F

of 34 64

And, the coupling constant is

12.

Let us see if this is accurate:

We used the experimental value of a proton . And we have demonstrated that our

model of a proton as a 3D cross-section of a 4D hypersphere countering the normal force against its

motion through time gives its inertia that can counter a force at right angles to its motion through time and

the normal force.

It is thought that the proton does not have an exact radius, but that it is a fuzzy cloud of subatomic

particles. As such depending on what is going on can determine its state, or effective radius. It could be

that the proton radius is as large as

Which it was nearly measured to be before 2010 in two separate experiments. Or as small as

Which is closer to current measurements, which have decreased by 4% since 2010, and could get smaller.

In which case the characteristic time, , could be as large as

Using 2/3 as a ﬁbonacci approximation to . Or, it could be as small as

C =

3α

6.62607E − 34J ⋅ s

(299,792,458m /s)(1s

)

= 2.21022E − 42N

18769

⋅

π (2.21022E − 42N )

6.674E − 11N

(0.833E − 15m) = 1.68E − 27kg

= 0.833E − 15m

⋅

6.62607E − 34

(299,792,458)(1.67262E − 27)

= 0.88094E − 15m

= ϕ ⋅

= 0.816632E − 15m

⋅

π r

G m

⋅

= 1.03351secon d s

ϕ ⋅

π r

G m

⋅

= (0.618)

(352275361)π (0.833E − 15m)

(6.674E − 11)(1.67262E − 27kg)

⋅

6.62607E − 34

299792458

of 35 64

=0.995 seconds

Or perhaps more often it is in the area of:

But, what this tells us is that the unit of a second might be a natural constant. And, since the second comes

from dividing the Earth rotation period into 24 hours, and each hour into 60 minutes, and each minute

into 60 seconds, which ultimately comes to us from the ancient Sumerians who ﬁrst settled down from

hunting, wandering, and gathering and ﬂaking stones into spearpoints to invent agriculture, writing, and

mathematics, that this might be related to the mechanics of our Solar System. We ﬁnd if we take the

second as natural we have a wave mechanics solution to our Solar System with a characteristic time of

one second that is connected to the characteristic time of the proton, thus connecting macro scales (the

solar system) to micro scales (the atom).

Why Is Used In The Equation For The Radius Of A Proton

We ask why the golden ratio is used to derive the radius of a proton. We start with our equation

1:!

This can be written!

13. !

Where . We notice is the force between two protons separated by the

radius of a proton. Of course two such protons cannot overlap by current theories. But it would

seem this gives rise to the proton’s inertia. We will call it . We also notice is the

normal force that gives rise to the proton’s inertia, . We have!

14. !

Now we look at equation 2. It is!

6α

h 4π r

G c

= 1.004996352secon d s

(

6α

4πh

)

⋅

= 1secon d

⋅

4π

36α

= 1secon d

⋅

= F

⋅

4π

36α

of 36 64

It can be written!

15. !

We see that is the inverse of the potential energy between the two protons

separated by the radius of a proton, we will call such a potential energy . We write 15 as!

16. !

Where !

Is the normal potential.!

17. !

Where is the golden ratio. Now we notice from equations

14 and 16 that!

18. !

Or!

19. !

And this should explain it. The gravitational force and its potential is in the normal force and

normal potential in time . The golden ratio is to divide a line such that the whole is to the

greater part as the greater part is to the lesser. What that means is the normal and the action

ϕ ⋅

πr

⋅

= 1secon d

(

⋅

ϕπ

)

(

)(

⋅

)

= 1

(

)

(

)

(

)

(

⋅

ϕπ

)

= 1

(

⋅

)

4π

36α

ϕπ

= Φ

Φ = 1/ϕ = ( 5 + 1)/2 = 1.618...

= Φ

(

)(

)

(

) (

)

of 37 64

are in the ratio of which being the most irrational number, there is no repetition over cycles,

there is minimal interference between the two for equation 19. Verifying using :!

The Electron We had equation 10 for the mass of a proton:

10.

Where is the cross-section of a proton and . We have

The electron has no radius, it is a one-dimensional point, all of its rest mass is in its electrostatic energy.

But we can speak of a characteristic size of the electron, not a size it has, but that it acts like its has. We

will call it . We will distinguish this from a characteristic size it already has, the classical electron

radius, which we will call . It is given by

20.

Where is the charge of an electron, is the permittivity of space, is the mass of an electron, a well

established natural constant, and c is the speed of light. Equation 10 for the electron becomes

21.

Putting in the mass of an electron we have the characteristic radius of an electron:

22.

We notice

23.

We consider equation 1:

2.7E − 34N

2.21E − 42N

⋅

2.92E − 57J

2.24E − 49J

= 1.6 = Φ

3α

⋅

π r

= h /ct

6.62607E − 34J ⋅ s

(299,792,458m /s)(1s

)

= 2.21E − 42N

eClassical

4π ϵ

⋅

= m

πF

⋅ 3α

(9.11E − 31kg)

(6.674E − 11)

π (2.21E − 42)

⋅

18769

4.5E − 19m

(4.5E − 19m)

3α

= (4.5E − 19m)

18769

= 2.8E − 15m = r

eClassical

of 38 64

24.

We have

25.

This should equal one second for the electron:

26.

Since we have from equation 21;

21.

We have

27.

Which is correct. Where we have lost the factors of in equation 21 when converting from a

characteristic electron radius to the classical electron radius, which is a characteristic length as well

because the electron has no size, all of its mass-energy is bound up in electrostatic energy. We have the

characteristic times for the proton and electron are each 1-second:

28.

(

6α

4πh

G c

)

⋅

= 1secon d

(

6α

4πh

G c

)

⋅

= 1secon d

18769

4π (6.62607E − 34)

(6.674E − 11)(299,792,458)

⋅

4.5E − 19m

9.11E − 31kg

= 0.9968secon d s ≈ 1secon d

= m

πF

⋅ 3α

eClassical

= m

πF

(9.11E − 31kg)

(6.674E − 11)

π (2.21E − 42)

= 2.82E − 15m

(

6α

4πh

G c

)

⋅

= 1secon d

4πh

G c

⋅

eClassical

= 1secon d

of 39 64

Finally,…

29.

% accuracy

This recurrent is the scaling factor in equation 10

10.

Where is the cross-second of a proton and is the normal force that gives rise to inertia.

The Neutron The neutron mass is approximately 1.675E-27 kg close to the proton’s of 1.67262E-27 kg.

And, its radius is anywhere from 0.8E-15m to 1.2 E-15m depending on the interpretation and method of

measuring it. Both size and mass are close to that of a proton. Let us suggest a radius for the neutron

based on our theory for inertia. Let the mass of a neutron be , and its radius be . First, we have

equation 1:

We write, then, for the characteristic time of a neutron:

30.

The characteristic time of a neutron is thus a 1-second. The mass is given by

31.

4π (6.62607E − 34)

6.674E − 11)(299,792,458]

⋅

2.82E − 15m

9.11E − 31kg

= 0.998secon d s ≈ 1secon d

3α

eClassical

18769

⋅

0.833E − 15m

1.67262E − 27kg

2.82E − 15m

9.11E − 31kg

3.0955E15

3.1157858E15

100 = 99.35

1/3α

3α

⋅

π r

= h /ct

(

6α

4πh

G c

)

⋅

= 1secon d

(

6 α

4πh

G c

)

⋅

= 1secon d

18769

4π (6.62607E − 34)

(6.674E − 11)(299,792,458)

⋅

0.834E − 15

1.675E − 27

= 1.004779secon d s

3α

⋅

π r

of 40 64

32.

Not much to do here where the neutron is concerned because it all follows from everything we did with

the proton.

Conclusion We have the characteristic times of the proton, the electron, and the neutron are all one

second:

And, that their masses and radii can be predicted by a theory for inertia where mass in space is due to

motion through time, a normal force ;

, ,

18769

⋅

π (2.21022E − 42N )

6.674E − 11N

(0.834E − 15m) = 1.68E − 27k g

= m

πF

⋅ 3α

(1.675E − 27kg)

(6.674E − 11)

π (2.21E − 42)

⋅

18769

= 0.830E − 15m

(

6α

4πh

G c

)

⋅

= 1secon d

4πh

G c

⋅

eClassical

= 1secon d

(

6α

4πh

G c

)

⋅

= 1secon d

3α

⋅

π r

eClassical

3α

⋅

π r

= ϕ ⋅

= m

πF

⋅ 3α

= m

πF

⋅ 3α

eClassical

= m

πF

of 41 64

Where

And,

In the case of an electron, it has no size and all of its rest mass is bound up in its electrostatic energy and

is the ﬁne structure constant, which measures the pliability or impedance of space. It was our goal in

this paper to explain the proton, electron, and neutron in terms of a theory based on a characteristic time

of one second, so we could not only have the basic components of atoms described in terms of a second,

but a theory for the atom that shows the larger scale of the Universe, the Solar System, is described in

terms of a characteristic time of a second and is a related idea on a larger scale. We want a theory for

atoms, H, He, C, N, O,.. also, because the planets formed from the collapse of a cloud made of these

elements. For my theory about the Solar System and its connection to the atom’s proton, which can now

include the electron and neutron, and thus atoms.

In the end we will want extend this theory to the rest of the particles of the standard model of particle

physics.

eClassical

4π ϵ

⋅

α =

4π ϵ

⋅

ℏc

of 42 64

The Solar Solution Our solution of the wave equation for the planets gives the kinetic energy of the

Earth from the mass of the Moon orbiting the Earth, but you could formulate based on the Earth orbiting

the Sun. In our lunar formulation we had:

We remember the Moon perfectly eclipses the Sun which is to say

Thus equation 1 becomes

The kinetic energy of the Earth is

Putting this in equation 3 gives the mass of the Sun:

We recognize that the orbital velocity of the Moon is

So equation 5 becomes

This gives the mass of the Moon is

Putting this in equation 1 yields

K E

= 3

⊙

⋅

2ℏ

⊙

K E

= 3

⋅

2ℏ

⊙

K E

⋅

⊙

= 3r

⋅

ℏ

⊙

= 3r

⋅ v

⋅

ℏ

⊙

ℏ

⊙

K E

⊙

⋅

⊙

of 43 64

We now multiply through by and we have

10.

The Planck constant for the Sun, , we will call , the subscript for Planck. We have

We write for the solution of the Earth/Sun system:

11.

We can write 11 as

12.

Where we say

Let us see how accurate our equation is:

We have that the kinetic energy of the Earth is

K E

⊙

⋅

⊙

ℏ

⊙

= r

= (1.496E11m)(1022m /s)(5.972E 24kg) = 9.13E 38kg ⋅

= r

= 7.4483E 77J ⋅ m

⋅ kg = 8.3367E 77kg

⋅

K E

⊙

⋅

⊙

K E

⊙

⋅

⊙

2ℏ

⊙

ℏ

⊙

= 9.13E 38J ⋅ s

⊙

= 2 π ℏ

⊙

= 5.7365E 39J ⋅ s

K E

⊙

⋅

⊙

(6.67408E − 11)

(5.972E 24kg)

(1.9891E 30kg)

2(8.3367E 77kg

⋅

)

⊙

(6.759E 30J )

⊙

6.957E 8m

1737400m

= 400.426

K E

= 2.70655E 33J

of 44 64

Our equation has an accuracy of

Which is very good.

Let us equate the lunar and solar formulations:

This gives:

13.

We remember that

And since,

14.

Equation 14 becomes

K E

earth

(5.972E 24kg)(30,290m /s)

= 2.7396E 33J

2.70655E 33J

2.7396E 33J

= 98.79 %

K E

= n

⊙

⋅

2ℏ

⊙

K E

⊙

⋅

⊙

2ℏ

⊙

⋅

2ℏ

⊙

⋅

⊙

⋅ ℏ

⊙

ℏ

⊙

= (hC )K E

hC = 1secon d

K E

(1secon d )

⊙

(5.972E 24kg)

(1.9891E 30kg)

(7.34763E 22k g)

(1.732)

= 321,331.459 ≈ 321,331

of 45 64

15.

The condition of a perfect eclipse gives us another expression for the base unit of a second. is another

version of the Planck Constant, which is intrinsic to the the solar formulation as opposed to the lunar

formulation. We want to see what the ground state looks like and what its characteristic time is, if it is 1

second like it is for the lunar formulation. Looking at the equation for energy:

We see the ground state should be:

16.

And, it is equal to 1 second. You will notice where in the derivation for the energy we lost , we

have to put it in the ground state equation. The computation is:

1secon d = 2r

⊙

K E

⊙

⋅

⊙

⋅

= 1secon d

n = 3

(9.13E 38J ⋅ s)

(6.674E − 11)(5.972E 24kg)

(1.989E 30kg)

⋅

= 1.0172secon d s

of 46 64

Jupiter and Saturn We want to ﬁnd what the wave equation solutions are for Jupiter and Saturn because

they signiﬁcantly carry the majority of the mass of the solar system and thus should embody most clearly

the dynamics of the wave solution to the Solar System. We also show here how well the solution for the

Earth works, which is 99.5% accuracy.

I ﬁnd that as we cross the asteroid belt leaving behind the terrestrial planets, which are solid, and go to the

gas giants and ice giants, the atomic number is no longer squared and the square root of the the orbital

number moves from the numerator to the denominator. I believe this is because the solar system here

should be modeled in two parts, just as it is in theories of solar system formation because there is a force

other than just gravity of the Sun at work, which is the radiation pressure of the Sun, which is what

separates it into two parts, the terrestrial planets on this side of the asteroid belt and the gas giants on the

other side of the asteroid belt. The effect the radiation pressure has is to blow the lighter elements out

beyond the asteroid belt when the solar system forms, which are gases such as hydrogen and helium,

while the heavier elements are too heavy to be blown out from the inside of the asteroid belt, allowing for

the formation of the terrestrial planets Venus, Earth, and Mars. The result is that our equation has the

atomic number of the heavier metals such as calcium for the Earth, while the equation for the gas giants

has the atomic numbers of the gasses. We write for these planets

So, for Jupiter we have (And again using the maximum orbital velocity which is at perihelion):

Jupiter is mostly composed of hydrogen gas, and secondly helium gas, so it is appropriate that .

Our equation for Jupiter is

Where is the atomic number of hydrogen which is 1 proton, and for the orbital number of

Jupiter, . Now we move on to Saturn…

E =

2ℏ

⊙

K E

(1.89813E 27kg)(13720 m /s)

= 1.7865E 35J

E =

(6.67408E − 11)

(1.89813E 27kg)

(7.347673E 22k g)

2(2.8314E 33)

E =

(3.971E 35J ) = Z

(1.776E 35J )

1.7865E 35J

1.776E 35J

= 1.006proton s ≈ 1.0proton s = h ydr ogen(H )

Z = Z

2ℏ

⊙

n = 5

K E

(5.683E 26kg)(10140m /s)

= 2.92E 34J

of 47 64

The equation for Saturn is then

It is nice that that Saturn would use Helium in the equation because Saturn is the next planet after Jupiter

and Jupiter uses hydrogen, and helium is the next element after hydrogen. As well, just like Jupiter, Saturn

is primarily composed of hydrogen and helium gas.

The accuracy for Earth orbit is

=2.727E33J

The kinetic energy of the Earth is

Which is very good, about 100% accuracy for all practical purposes. The elemental expression of the

solution for the Earth would be

Where

E =

(6.67408E − 11)

(5.683E 26kg)

(7.347673E 22)

2(2.8314E 33)

2.45

(3.5588E 34J ) = Z(1.45259E 34J )

Z(1.45259E 34J ) = (2.92E 34J )

Z = 2proton s = Heliu m(He)

2ℏ

⊙

= n

⊙

2ℏ

⊙

6.96E 8m

1737400m

= 400.5986

= (1.732)(400.5986)

(6.67408E − 11)

(5.972E 24kg)

(7.347673E 22k g)

2(2.8314E 33)

K E

(5.972E 24kg)(30,290m /s)

= 2.7396E 33J

2.727E 33J

2.7396E 33J

100 = 99.5 %

= 3

2ℏ

⊙

of 48 64

In this case the element associated with the Earth is calcium which is Z=20 protons.

⊙

→ Z

of 49 64

Modeling Star Systems With The Theory The best way to solve star systems with our theory would be

to use

1.0 , 3.0

2.0. , 4.0.

Where here . Equation 2.0 becomes

5.0.

Where for Earth p=2.5, the exponent in the pressure gradient for its protoplanetary disc. From this we get

. We now get the characteristic time, , from

by using

6.0. and 7.0.

And we have . We can now put in equation 1.0

Which here is 1 second for the Earth, to get the mass of the moon, . But to use equation 5, we need

from equation 3.0. This requires the mass of the Earth, the frequency of the earth, which we get

from the planet’s day (Its rotation rate) and the radius of the planet. We have all of these values for

habitable planets in a K2V star and an M2V star, but they are tidally locked. We have the frequencies

because the planets are tidally locked, so their planets rotation periods are equal to their orbital periods. If

we can’t measure the planet’s radius in another star system, we might obtain it from:

8.0.

Which works for the Earth. We can get the orbital radius of our Moon from

9.0.

It is given by the ratio of silver (Ag) to gold (Au) by molar mass is equal to . The radius of the

planet’s moon we suggested is given by a perfect eclipse:

ℏ

⊙

⋅

= 1secon d

earth

π M

P(R) = P

(

)

−L

earth

ℏ

⊙

ℏ

⊙

= t

K E

= 1secon d

earth

ℏ

⊙

= p

ℏ

⊙

ℏ

⊙

= t

K E

earth

ℏ

⊙

⋅

= 1secon d

earth

⊙

= R

⊙

⋆

(1.8)

⊙

of 50 64

10.0.

We have already applied our theory to the Earth/Moon/Sun System and it worked out nicely. Now we

want to apply this ideal approach we just outlined, to it, so we can test it. We start with the angular

momentum of the Earth. It is given by

Or,…

The orbital velocities and kinetic energies of the Earth are given by:

We can now determine :

This is correct for our solar system’s Planck constant. We have the characteristic time is

Which is correct as well. Now we compute the mass of our moon…

This is also very accurate (actual value: 7.347673kg. Now we compute the orbital radius of the Moon…

This is accurate too (actual value: 3.84E8m). From this we have the radius of the Moon:

⋆

earth

π M

π (5.972E 24kg )

(86400secon d s)

(6.378E6m)

7.07866672E 33J ⋅ s

⋆

(6.674E − 11)(1.989E 30k g)

(1.496E11m)

= 29,788.24m /s

K E

(5.972E 24kg)(29,788.24m /s)

= 2.65E 33J

ℏ

⋆

ℏ

⋆

(7.07866672E 33J ⋅ s)

2.5

= 2.831467E 33J ⋅ s

ℏ

⋆

K E

(2.831467E 33J ⋅ s)

2.65E 33J

= 1.068secon d s ≈ 1secon d

(2.831467E 33J ⋅ s)

(6.674E − 11)(299,729,458m /s)(1.068secon d s)

= 7.213E 22k g

= R

⋆

= R

⋆

/(1.8) =

6.957E 8m

1.8

= 3.865EE 8m

of 51 64

This is pretty accurate, too. The actual value is 1.7374E6m

Now to get the density of the Moon…

This is good, our Moon is about 3.344g/cm3. Now we want to check

So this gives the correct characteristic time for the Earth/Moon/Sun system. Let’s compute the planet day

characteristic time…

We see the system for modeling star systems works.

We can make a program to model star systems in general given the spectral class of the Star. So HR

diagrams plot mass versus luminosity to give spectral types of stars. So F Stars would be more luminous

blue stars, G stars would be yellow medium luminosity stars, and K stars would be less luminous orange

stars, and so on. There are ten divisions of each, and a “V” meaning “ﬁve” indicates the star is on the

main sequence. So our Sun is a G2V star. A medium luminosity, yellow star. Here is a my program in C

that does that for the method we just outlined.

= R

⋆

= (6.957E 8m)

3.865E 8m

1.496E11m

= 1.79738E6m

π R

π (1.79738E 6m)

= 2.432E19m

7.213E 22k g

2.432E19m

= 2.96587g /cm

≈ 3g /cm

1secon d = 2r

⊙

(6.674E − 11)(5.972E 24kg)

(3.865EE 8m)

= 1,015.5m /s

2(1.496E11m)

1,015.5m /s

(29,788.24m /s)

(7.213E 22k g)

(1.732)

(5.972E 24kg)

(1.989E 30kg)

= 1.03648sec ≈ 1secon d

1secon d ≈

K E

(Ear th Da y)

K E

(7.213E 22k g)(1,015.5m /s)

= 3.719E 28J

K E

(Pla n et Da y) =

(3.719E 28J )

(2.65E 33J )

(86,400sec) = 1.2secon d s

of 52 64

// main.c

// modelsystem

// Created by Ian Beardsley on 2/9/25.

#include <stdio.h>

#include <math.h>

int main(int argc, const char * argv[]) {

float R_p, M_p, R_s, M_s, t_c, M_m, rho_m, rho_p, PlanetDay,

V_p,StarRadius, PlanetRadius, PlanetMass, StarLuminosity, PlanetOrbit,

StarMass, r_p, T_p, p, L_p, KE_p, v_p, T_m,Tmoon, C_m;

float G=6.674E-11, hbarstar, PDCT,Tsquared,T,PlanetYear;

float r_m, R_m, V_m, MoonDensity, part1, part2, part3,v_m, KE_m;

int i;

printf ("What is the radius of the star in solar radii? ");

scanf ("%f", &StarRadius);

printf ("What is the mass of the star in solar masses? ");

scanf ("%f", &StarMass);

printf ("What is the luminosity of the star in solar luminosities? ");

scanf ("%f", &StarLuminosity);

PlanetOrbit=sqrt(StarLuminosity);

r_p=PlanetOrbit*1.496E11;

M_s=1.9891E30*StarMass;

Tsquared=((4*3.14159*3.14159)/(G*M_s))*r_p*r_p*r_p;

T=sqrt(Tsquared);

PlanetYear=T/31557600;

printf("Do you want us to compute the planet radius, 1=yes, 0=no? ");

scanf("%i", &i);

R_s=6.9364E8*StarRadius;

if (i==1)

{

R_s=6.9364E8*StarRadius;

R_p=2*(R_s*R_s)/r_p;

PlanetRadius=R_p/6.378E6;

}

else

{

printf("What is the planet radius in Earth radii?: ");

scanf("%f", &PlanetRadius);

R_p=PlanetRadius*6.378E6;

}

printf("What is the mass of the planet in Earth masses? ");

scanf("%f", &PlanetMass);

M_p=PlanetMass*5.972E24;

printf ("What is the planet day in Earth days? ");

scanf ("%f", &PlanetDay);

T_p=PlanetDay*86400;

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printf("That is %f seconds \n", T_p);

{

printf("What is p the pressure gradient exponent of the

protoplanetary disc? ");

scanf("%f", &p);

M_s=1.9891E30*StarMass;

r_m=R_s/1.8;

v_p=sqrt(G*M_s/r_p);

L_p=0.8*3.14159*M_p*(1/T_p)*R_p*R_p;

KE_p=0.5*M_p*v_p*v_p;

hbarstar=L_p/p;

t_c=hbarstar/KE_p;

part1=cbrt(hbarstar/(t_c));

part2=cbrt(1/G);

part3=cbrt(hbarstar/299792458);

M_m=part1*part2*part3;

R_s=StarRadius*6.9634E8;

R_m=R_s*r_m/r_p;

V_m=1.33333*3.14159*R_m*R_m*R_m;

rho_m=(M_m/V_m);

MoonDensity=rho_m*0.001;

V_p=1.33333*3.14159*R_p*R_p*R_p;

rho_p=(M_p/V_p)*0.001;

printf("\n");

printf("Angular Momentum of Planet: %f E33 \n", L_p/

1E33);

printf("\n");

printf("PlanetYear: %f years \n", PlanetYear);

printf("PlanetYear: %f seconds \n", T);

printf("planet orbital velocity: %f m/s \n", v_p);

printf("planet mass: %f E24 kg \n", M_p/1E24);

printf("planet mass: %f Earth masses \n", M_p/5.972E24);

printf("planet radius %f meters \n", R_p);

printf("planet radius: %f Earth Radii \n", PlanetRadius);

printf("planet orbital radius: %f E11 m \n", r_p/1E11);

printf ("planet orbital radius: %f Earth distances \n",

r_p/1.496E11);

printf("planet KE: %f E33 J \n",KE_p/1E33);

printf("planet density: %f g/cm3 \n", rho_p);

printf("\n");

printf("hbarstar: %f E33 Js \n", hbarstar/1E33);

printf("characteristic time: %f seconds\n", t_c);

printf("\n");

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printf("Orbital Radius of Moon: %f E8 m \n", r_m/1E8);

printf("Orbital Radius of Moon: %f Moon Distances \n",

r_m/3.84E8);

printf("Radius of Moon: %f E6 m \n", R_m/1E6);

printf("Radius of Moon: %f Moon Radii \n", R_m/1.7374E6);

printf("Mass of Moon: %f E22 kg \n", M_m/1E22);

printf("Mass of Moon %f Moon Masses \n", M_m/

7.347673E22);

printf("density of moon: %f g/cm3 \n", MoonDensity);

printf("\n");

v_m=sqrt(G*M_p/r_m);

KE_m=0.5*M_m*v_m*v_m;

PDCT=(KE_m/KE_p)*(T_p);

printf("Orbital Velocity of Moon: %f m/s \n", v_m);

printf("PlanetDay Characteristic Time: %f seconds \n",

PDCT);

C_m=2*3.14159*r_m;

T_m=C_m/v_m;

Tmoon=T_m*(1.0/24)*(1.0/60)*(1.0/60);

printf("Lunar Orbital Period: %f seconds \n", T_m);

printf("Lunar Orbital Period: %f days \n", Tmoon);

return 0;}}

Now we show running the program (3 examples) for a wide spread of spectral types including F, G, and

K-type stars. We will need to input in the program not just the mass of the star, its luminosity, and size,

but the pressure gradient exponent for the disc from which the star’s planets formed.

To compute the moon’s orbital radius I just use

Where Ag is the molar mass of silver and Au is the molar mass of silver, a connection to the 1.8 that

appears in our Solar System. We use this because we know it works for our Solar System. I compute the

radius of the planet using

But, give the option of putting in your own radius. I have run the program for F5V stars, through GV

stars, to K3V stars and I use this equation to compute the radii of the planets because, again, we know it

works for our star system, and further we found given the way the radius of a star varies with with

luminosity in the HR diagram, this equation always gives a planet around the size of the Earth. I feel this

size is ideal for planets with sophisticated life because of the laws of chemistry determining a functional

density for the planet having water and the right gravity. As such I always use the planet day as one Earth

day, which again I feel is optimal for life in terms of climate. So these values all constant, we only vary

star mass, size, and luminosity as they work on the HR diagram. I also vary the pressure gradient

exponent now using the average theoretical values it has for each spectral class. The trend is that it

steadily decreases on average with mass and luminosity of the star though it can go up and down

depending on the peculiarities of the system. One of the reasons is that while for a G2V star it can range

on average from p= 1.7-2.1, for our Sun, a G2V star, it is actually high, it is 2.5. However, here we will

= R

⊙

⋆

(1.8)

⊙

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model stars with everything constant, as we said, but the pressure gradient will gradually decrease with

spectral class, and when we do a G2V star, we won’t use the Sun’s data, but the average value for G2V

stars. We will do lot’s of models, allowing no gaps in the data for a plot, so we can get a well deﬁned

curve. We will use the upper value for p in instances here. We will also use

Instead of

Thus taking the tilt of the planet to its orbital as and not what it is in particular for the Earth

, which may actually be optimal for the most habitable types of planets around stars like our

Sun. The average pressure exponents by spectral class are given in the following table…

K E

moon

K E

planet

(Pla n et Da y)cos(0

∘

) ≈ 1secon d

K E

(Pla n et Da y)cos(23.5

∘

) = 1secon d

θ = 0

∘

θ = 23.5

∘

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Now we run the program using this for examples of three spectral types…

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F5V Star

What is the radius of the star in solar radii? 1.473

What is the mass of the star in solar masses? 1.33

What is the luminosity of the star in solar luminosities? 3.63

Do you want us to compute the planet radius, 1=yes, 0=no? 1

What is the mass of the planet in Earth masses? 1

What is the planet day in Earth days? 1

That is 86400.000000 seconds

What is p the pressure gradient exponent of the protoplanetary disc? 2.4

Angular Momentum of Planet: 9.321447 E33

PlanetYear: 2.280109 years

PlanetYear: 71954776.000000 seconds

planet orbital velocity: 24888.847656 m/s

planet mass: 5.972000 E24 kg

planet mass: 1.000000 Earth masses

planet radius 7325190.000000 meters

planet radius: 1.148509 Earth Radii

planet orbital radius: 2.850263 E11 m

planet orbital radius: 1.905256 Earth distances

planet KE: 1.849692 E33 J

planet density: 3.627237 g/cm3

hbarstar: 3.883936 E33 Js

characteristic time: 2.099775 seconds

Orbital Radius of Moon: 5.676287 E8 m

Orbital Radius of Moon: 1.478200 Moon Distances

Radius of Moon: 2.042695 E6 m

Radius of Moon: 1.175720 Moon Radii

Mass of Moon: 7.107576 E22 kg

Mass of Moon 0.967323 Moon Masses

density of moon: 1.990782 g/cm3

Orbital Velocity of Moon: 837.955261 m/s

PlanetDay Characteristic Time: 1.165595 seconds

Lunar Orbital Period: 4256210.000000 seconds

Lunar Orbital Period: 49.261688 days

Program ended with exit code: 0

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G3V Star

What is the radius of the star in solar radii? 1.002

What is the mass of the star in solar masses? 0.99

What is the luminosity of the star in solar luminosities? 0.98

Do you want us to compute the planet radius, 1=yes, 0=no? 1

What is the mass of the planet in Earth masses? 1

What is the planet day in Earth days? 1

That is 86400.000000 seconds

What is p the pressure gradient exponent of the protoplanetary disc? 2.1

Angular Momentum of Planet: 7.393050 E33

PlanetYear: 0.989814 years

PlanetYear: 31236148.000000 seconds

planet orbital velocity: 29789.738281 m/s

planet mass: 5.972000 E24 kg

planet mass: 1.000000 Earth masses

planet radius 6523625.500000 meters

planet radius: 1.022833 Earth Radii

planet orbital radius: 1.480965 E11 m

planet orbital radius: 0.989950 Earth distances

planet KE: 2.649862 E33 J

planet density: 5.135297 g/cm3

hbarstar: 3.520500 E33 Js

characteristic time: 1.328560 seconds

Orbital Radius of Moon: 3.861263 E8 m

Orbital Radius of Moon: 1.005537 Moon Distances

Radius of Moon: 1.819172 E6 m

Radius of Moon: 1.047066 Moon Radii

Mass of Moon: 7.754257 E22 kg

Mass of Moon 1.055335 Moon Masses

density of moon: 3.074905 g/cm3

Orbital Velocity of Moon: 1015.987427 m/s

PlanetDay Characteristic Time: 1.304901 seconds

Lunar Orbital Period: 2387924.250000 seconds

Lunar Orbital Period: 27.638012 days

Program ended with exit code: 0

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K3V Star

What is the radius of the star in solar radii? 0.755

What is the mass of the star in solar masses? 0.78

What is the luminosity of the star in solar luminosities? 0.28

Do you want us to compute the planet radius, 1=yes, 0=no? 1

What is the mass of the planet in Earth masses? 1

What is the planet day in Earth days? 1

That is 86400.000000 seconds

What is p the pressure gradient exponent of the protoplanetary disc? 1.5

Angular Momentum of Planet: 8.340819 E33

PlanetYear: 0.435785 years

PlanetYear: 13752343.000000 seconds

planet orbital velocity: 36167.078125 m/s

planet mass: 5.972000 E24 kg

planet mass: 1.000000 Earth masses

planet radius 6929175.500000 meters

planet radius: 1.086418 Earth Radii

planet orbital radius: 0.791609 E11 m

planet orbital radius: 0.529150 Earth distances

planet KE: 3.905860 E33 J

planet density: 4.285367 g/cm3

hbarstar: 5.560546 E33 Js

characteristic time: 1.423642 seconds

Orbital Radius of Moon: 2.909435 E8 m

Orbital Radius of Moon: 0.757665 Moon Distances

Radius of Moon: 1.932263 E6 m

Radius of Moon: 1.112158 Moon Radii

Mass of Moon: 10.277222 E22 kg

Mass of Moon 1.398704 Moon Masses

density of moon: 3.400867 g/cm3

Orbital Velocity of Moon: 1170.438843 m/s

PlanetDay Characteristic Time: 1.557185 seconds

Lunar Orbital Period: 1561850.125000 seconds

Lunar Orbital Period: 18.076969 days

Program ended with exit code: 0

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Here is a plot of these results from F5V Stars to K3V stars:

We see the characteristic time decreases as a curve to intersect close

to a characteristic time of one second with a planet day

characteristic time of one second as a straight line at a G3V star, a

G5V star, and a G8V, which is a region near where our Sun is and may

have something to do with it being so optimal for life. We note here

that this uses the average value for a G2V star, and our Sun comes

closer to a second because its pressure exponent is higher than on

average, it is very high, or steep (p=2.5) which means the pressure of

its disc drops rapidly with distance.

Planet Day Characteristic Time:

Characteristic time: ,

Here we ﬁt the curves for characteristic time and planet day characteristic time. We name the spectral

types with number for input according to the following scheme.

F5V is 1.5, F6V is 1.6, F7V is 1.7,…G0V is 2.0, G1V is 2.1,…

1secon d ∼

K E

(Ear th Da y)

ℏ

⊙

= (1seco n d )K E

ℏ

⊙

⋅

= 1seco n d

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For the characteristic time we ﬁt the curve with a power law decay

For the planet day characteristic time we ﬁt the curve with a straight line

Where we have chosen in ,

We intend to ﬁt the curves in the following plot:

y = 2.8x

−

+ 1.1

y = 0.168x + 0.913595

y = m x + b

m =

G3V − F 9V

2.3 − 1.9

= 0.168

b = 1.165595 − 0.168(1.5)

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The results are

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The Author