# Physics Tutorial: Sun and Planetary Motion

In this Physics tutorial, you will learn:

• What are some basic features of the Sun (mass, radius, position)?
• Why is Sun the main source of energy for Solar System?
• How the solar energy is produced?
• How energy transmission does occur in different layers of the Sun?
• What is the structure of Sun? What features does each layer present?
• What are solar protuberances?
• What laws the planetary motion around the Sun does follow?
• What is sidereal and tropical period?
• What is ecliptics and ecliptic plane?
• What are inner and outer planet? How does the observation angle varies with position?
• How do planets rotate around their own axis?
• What are seasons/solstices/equinoxes?
• How day and night are formed? How do they vary in different seasons and locations?

## Introduction

As stated in the previous tutorial, the Sun is the source of energy of our Solar system. All planets revolve around the Sun; it determines the structure and stability of the solar system due to its enormous mass (recall what we have discussed on gravitational force in Section 8). If the Sun did not exist, none of the solar system planets would exist either.

In this tutorial, we will provide some useful info about the Sun structure. Moreover, the planetary motion discussed in Section 8 (Kepler's Laws) will get a final answer on why it is so.

## Basic Features

The Sun is a fiery giant orb of mass M = 1.9989 × 1030 kg, mostly made up of hydrogen and helium. The Sun is an average sized star formed about 4.6 billion years ago. The radius of the Sun is 6.96 × 105 km. The Sun is located at centre of our Solar System and it is the main source of energy for it. All planets are "fed" by portions of solar energy originating from the Sun.

## The Source of Solar Energy

The question that has intrigued many scientists for centuries was: "What is the source of solar energy?" It was only in the last century (by the end of 1930s) that scientists realized that all this energy is produced by means of nuclear reactions of fusion (as discussed in tutorial 20.4). Such reactions occur in the innermost part of the Sun called the "Sun core". It is a spherical-shaped region with a radius not more than 1/5 of the Sun's radius.

Because of enormous pressure exerted by upper layers, the temperature of the Sun's core reaches 10-15 million Celsius degree. The Sun's core density is more than 100 times higher than water density. This is because electrons have been removed from atoms when reducing resulting in empty spaces that exist in cold (normal) atoms. In these conditions, hydrogen nuclei (i.e. protons) have very small distances between them, resulting in the perfect conditions for nuclear fusion to occur. In fact, there is not a single type of nuclear reaction involved in this process but a number of nuclear reactions forming the proton-proton (p-p) reaction chain. From Section 21 it is undestood that the reactions which convert hydrogen to helium are accompanied by the release of large amounts of energy. In this way, the Sun's hydrogen acts like a "nuclear fuel" while the helium produced is like the "ash" produced after a burning process.

The composition of the Sun consists of 71% hydrogen and 27% helium. The remaining 2% consists of other chemical elements. The amount of hydrogen decreases and that of helium (and heavier elements) increases with time.

In one of the p-p reactions in the aforementioned chain, the weak interaction is involved - a process that brings the emission of an electronic neutrino. In Section 21 we learned that these reactions have a low probability of occuring. This is why hydrogen is consumed very slowly and the Sun produces energy at the same rate for several billions of years.

## Energy Transmission from Core to Surface of the Sun

The gamma photons produced in the Sun's core shift towards the surface of the Sun. Since temperatures are very high and matter is very compressed, the photons experience frequent collisions with matter particles. As a result, they are absorbed and re-emitted multiple times from solar matter causing them to lose energy. This random process produces a kind of Brownian motion, i.e. photons travel a long way until they get into the surface of the Sun. When this finally occurs, the photon's energy (and frequency) has decreased to the visible light level.

Another method of solar energy transmission is through convection, i.e. through gas currents. More specifically, the hot gas layers move from the core to the surface and cool down as a result, at the same time colder layers shift towards the inside and heat up.

## Structure of the Sun

The Sun structure is illustrated in the figure below.

From this figure, you can see that the Sun's core is part of radiative zone. This zone lies from centre to 3/4 of the Sun's radius, where temperature and matter density are so high that they only allow energy transmission by means of radiation, through Brownian motion. The last (outer) quarter contains the convection zone in which energy transmission is also carried out through gas currents.

The Sun is not a regular body; it has no defined surface. As stated earlier, it is a fiery orb made of very hot gases (plasma). The "surface" visible from the Earth is known as photosphere. It is a 600 km thick layer with an average temperature of 5800 K. The Photosphere has a granular appearance because of convection currents that move up and down continuously. The solar matter extends up to photosphere; the other two outer layers (chromosphere and corona) are part of Sun's atmosphere and not of its inner material structure. Hence, practically, when calculating the dimensions of the Sun, we consider the part from core to photosphere.

Sunspots represent an interesting phenomenon that occur in the photosphere. They are darker regions than the other parts of the Sun due to the lower temperature (1000 K - 1500 K colder than the neighboring regions). Their presence is explained through magnetic fields present in the Sun. The number of such spots varies periodically every 11.2 years. When the number of sunspots is at maximum, the Sun is at peak of its magnetic activity.

As stated earlier, the Sun's atmosphere lies above chromosphere and solar corona, including these two regions as well. It is very difficult to observe the solar atmosphere in normal sunny days due to the strong light emitted by the photosphere, which obscures all the other light sources. Scientists prefer to make their observations of the Sun's atmosphere during solar eclipses, when the intensity of solar radiation decreases drastically. During these events the photosphere is covered by the Moon and only the Sun's atmosphere is visible.

Chromosphere - which represents the colored layer of the Sun - has a red ring shape. It is about 6000 km thick and has a temperature of 7000 K.

Corona on the other hand, extends up to millions of km away from the Sun, or several times the Sun's radius. However, its density is very low. The temperature of corona reaches a value of 2 million Kelvin degree.

Solar protuberances are another phenomenon observed in the Sun. They are difficult to see in photos but they are more visible in high-resolution videos of the Sun. Solar protuberances are rose-colored masses on the limb of the sun which are seen to extend beyond the edge of the moon during a solar eclipse. They may be observed with a spectroscope on any clear day. They are also known as solar prominences.

Solar protuberances are flame bolts flowing out from the chromosphere and returning back into the chromosphere again. This occurs due to the effect caused by the Sun's magnetic field.

The Sun revolves around its own axis. The presence of sunspots makes the study of this phenomenon easier. The solar self-rotation period is 24.47 days at the equator and almost 38 days at the poles. The average rotation is 28 days.

The Sun is a typical representative of a star class known as main-sequence stars, which we will discuss in the upcoming tutorials. This class includes most of stars existing in the universe.

## Planetary Motion around the Sun

As discussed in other topics, our Solar System is a set of celestial bodies interacting with each other by means of gravitational forces. The Sun is the main contributor to the stability of this structure as it alone has a mass that is much greater than that of all the other celestial bodies of the solar system combined together. Therefore, when studying planetary motion at introductory level (as in this case), we neglect any gravitational interaction between celestial bodies with each other and focus only on their interaction with the Sun, as the results do not vary too much from the true ones.

Because of gravitational force (that acts as centripetal force as discussed in Section 8), planets perform a circular motion around the Sun (they also rotate around themselves and we cover this phenomenon later in this tutorial). When applying the general gravitation law, we obtain important results on a planets trajectory, movement speed in their respective orbits and period of rotation around the Sun. These results are summarized in the three Kepler's Law discussed in tutorial 8.2 (Gravitational Potential Energy. Kepler Laws):

1. Planets revolve around the Sun in elliptic orbits where the Sun is located in one of ellipse's foci.
Scientist have also discovered that all planets revolve in the same direction (anticlockwise when viewed from North of our galaxy). In addition, planetary orbits lie nearly at the same plane. The normal line to the orbits plane passing through the centre of Sun is called the solar system axis.
2. The Sun-Earth radius "wipes" equal surfaces in equal time intervals.
From this law, it is evident that planets do not revolve at the same speed around the Sun. when a planet is closer to the Sun (perihelion), planets have the greatest speed and when planets are in aphelion (the farthest point from the Sun), they revolve at minimum speed.
3. The square of planetary period (time to complete one revolution) is proportional to the cube of ellipse major semi-axis, i.e.
T2 ~ a3 or T2/a3 = constant

In popular terminology, we call a planetary period a"year" . As the distance from the Sun to the planet increases, you can easily see that the planetary period increases too.

The angle of the Earths' revolution around the Sun depends on the frame of reference. If stars are taken as reference frame, the corresponding period of revolution is known as sidereal (stellar) period, while when the Sun is taken as a reference frame, then corresponding period is known as tropical (solar) period. These periods (that determine the length of the corresponding years) are not the same; they have small deflections due to the effect of the Moon on the Earths' revolution around the Sun. Thus, the duration of one sidereal year is 365.2568 days and that of a tropical year is 365.2422 years. Such differences has led to the use of two different solar-based calendars: Julian and Gregorian ones, we will discuss this in more detail in the upcoming tutorials.

### Example 1

An ancient site is 5300 years old when calculated using the sidereal period (year). How old is this site when using the tropical period (year)?

### Solution 1

The age of the ancient site in days is:

Age(d) = Ns ∙ Ts

where Ns is the site age expressed in number of sidereal years. Substituting the known values, we obtain

Age(d) = 5300 y ∙ 365.2568 d/y
=1 935 861.04 d

This number must be divided by the duration of a tropical year in order to give the site's age in tropical years (Nt). We have

Nt = Age(d)/Tt

where Tt = 365.2422 days is the tropical year period. Thus, after substitutions we obtain for the site's age in tropical years:

Nt = 1 935 861.04 d/365.2422 d/y
≈ 5300.2 years

Hence, there is a small deflection when the duration of a year is measured by taking into account the two aforementioned periods of the Earth revolution around the Sun.

## Observation of Planetary Motion from Earth

Since all planetary orbits are nearly at the same plane when observed from Earth we can assess that the planets are aligned in a unique arc in space. The plane containing this arc is known as ecliptic plane and it divides the sky in two parts. More specifically, the ecliptic plane is defined as the imaginary plane containing the Earth's orbit around the sun. If we consider the consider only the imaginary line that connects the centre of Sun, Earth and all the other planets of they are all aligned, it is called ecliptics. Obviously, planets are in ceaseless motion even when considered from ecliptic plane viewpoint. The only restriction is that they are seen only at certain parts of the day, when they are found inside the maximum observation angle θ from the Sun as illustrated in the figure below.

Since Mercury and Venus are the only planets inside the Earth's orbit (they are closer to the Sun and their orbit is smaller than the Earth's orbit), they are called inner planets. All the other planets, from Mars to Neptune are called outer planets since their orbits include that of Earth.

The observation angle θ for Mercury is 23° and for Venus it is 47°. Let's clarify what this mean through an example.

### Example 2

How many hours a day and at what point can we see Mercury and Venus in the sky?

### Solution 2

Given that a complete angle is 360°, this means that Mercury can be observed only 23/360 of the day in the sky while Venus can be observed only 47/360 of the day. When these parts are expressed in hours, we obtain

tMercury = 23/360 × 24 h=1.53 hours

and

tVenus = 47/360 × 24 h = 3.13 hours

Hence, since this situation is repeated in both sides of the line that connects Earth, the other planet and the Sun, we can see Mercury in the sky twice a day for a maximum duration of about 1.5 h each time (one before sunrise and the other after sunset). For Venus, this duration is a little more than 3 hours each time. We can see Venus (the "morning star") twice a day for 3.13 hours before sunrise and after sunset.

The outer planets can be observed in the sky as well. They can be seen in angles θ2 varying from 0 to 180° on the other side of (away from) the Sun. Obviously, they are observable only during the night as they are in the opposite direction to the Sun. This is illustrated in the image below.

It is evident that the Sun lies on ecliptics and it moves only on it. This motion is the opposite of that of Earth in respect to the Sun (action-reaction principle). Therefore, the duration of this process is 365.2422 days too. Hence, since we observe the Sun moving from East to West, the Sun motion in ecliptics occurs from West to East, as we will see later.

The days of week (in Latin language) are named after the seven celestial bodies known in ancient times: Monday comes from Moon, Tuesday from Mars, Wednesday from Mercury, Thursday from Jupiter, Friday from Venus, Saturday from Saturn and Sunday from Sun.

## Planetary Rotation around Own Axis

In addition to orbital motion (revolution) around the Sun, planets rotate around themselves as well. This rotation occurs according an axis passing through the centre of planet, known as planetary axis. This self-rotation derives from the fact that planets are objects with large dimensions; they are not merely material points as assumed in Kepler's Laws.

Planetary axis forms a certain angle to the Solar System axis (they are not parallel). The last column of planets table in the previous tutorial gives the values in degree of this inclination angle for all planets of our Solar System. Thus, for Mercury the angle is 0°. This means Mercury equator plane is in the same direction to the plane of its orbit around the Sun. on the other hand, Earth has an inclination of 23.45°. (Recall the deflection of Earth geographic poles in respect to the corresponding magnetic poles explained in Section 16, which is exactly 23.45°. This is because magnets point towards north-south direction of our galaxy). The figure below that gives a view taken from behind the Earth clarifies this point. The dashed line represents the North-South direction of our galaxy.

From the aforementioned table we have the following values for the angle of inclination: Mercury 0°, Venus 177.3°, Earth 23.45°, Mars 25.19°, Jupiter 3.12°, Saturn 26.73°, Uranus 97.86° and Neptune 120°. Planets that have an angle less than 90° rotate in the same direction while those with angle of inclination greater than 90° rotate in the same direction with each other but in opposite direction to planets with inclination angle less than 90°.

The period of self-rotation is known as a "day" in popular terminology. The longest day is that of Mercury (which is more than 253 Earth days) and the shortest one is that of Jupiter (less than 10 hours).

## Earth Rotation around its Own Axis

The Earth completes one rotation around its own axis in 23 h 56 min, a figure computed using stars as a (stationary) reference frame. This value corresponds to the duration of an Earth day in respect to the stars. This means that Earth rotates by 360° around its own axis in one day. This rotation occurs from West to East of our galaxy. Hence, it seems that the Sun rotates in the sky (horizon) from East to West.

On the other hand, when the Sun is considered as reference frame, there is another factor to be taken into account. Thus, when one day is completed, there is also a slight movement of Earth around the Sun in addition to rotation of Earth around its axis. This causes a delay of 1/365 of a day which corresponds to (1/365) × 24 h = 0.0657 h = 4 minutes. This value is added to the day duration when starts are taken as reference frame and thus, we obtain a value of 24 h for the duration of one day when the Sun is taken as a reference frame. This is the value of day duration we use when measuring the time elapsed in our daily activities. It is known as the solar day.

## Seasons

The phenomenon of seasons occurs because of the angle formed by the rotation axis of the Earth and the orbital plane (which is 23.45° as well). When this angle is taken in reference to the orbital plane, the value of inclination becomes 90° - 23.45° = 66.55°. This inclination makes the solar radiation fall at different angles on a specific point of Earth in various period of year. Since light intensity depends on the incidence angle, the solar energy brought to Earth is not the same throughout the year. It is greater when the incidence angle of solar radiation is closer to normal (vertical). This coincides to summer in any given location. On the other hand, when the incidence angle of solar radiation is the most deflected from normal (it is not so vertical), we have winter in that location. Look at the figure.

The Northern Hemisphere is taken as example to illustrate the season formation given that about 90 percent of the global population live in this hemisphere. Thus, in the left part of figure, sunrays fall almost normal to the surface of northern hemisphere. As a result, there is summer in that part of the Earth as more energy is coming from the Sun. On the other hand, there is winter on the right part of figure, as sunrays do not fall normal to the surface. They have the lowest angle possible in that part. Intermediate seasons such as spring and autumn are shown in the upper and lower part of figure, when sunrays fall at an angle that is in-between of the above angles.

The Southern hemisphere experiences the opposite phenomenon; people living in those areas have winter on June-July-August and summer on December-January-February. This is why Australians usually celebrate Christmas on the beach.

Solstice represents the time when the angle of incidence of sunrays on a region of the Earth is the lowest or the greatest possible. The day that this angle is the lowest, is the longest day of the year in that place, and the summer begins. In the northern hemisphere, this date corresponds to June 22nd of each year. On the other hand, winter solstice (the day when the angle of solar radiation incidence is the greatest) occurs on December 22nd in northern hemisphere - a day which corresponds to the beginning of winter and when the day is the shortest possible.

An Equinox occurs when the Earths' axis is deflected laterally only (not forwards or rearwards) in respect to the orbital plane. We have two equinoxes: in the northern hemisphere there is a spring equinox on March 2121 and autumn equinox on September 23rd. On these days, the duration of day and night are equal.

## Day and Night

"Day" represents the time in which a region of Earth is directly exposed to sunrays. Night is the opposite, i.e. a region is either illuminated indirectly from the Sun (through Moon surface that reflects the sunrays on Earth) or by means of other light sources (usually artificial, such as electric bulbs).

All locations on the equatoralways have equal durations of day and night as the sunlight always falls on them at the same angle. On the other hand, the further North (or South for southern hemisphere) a location is, the more variation there is between day and night duration at solstice(s). For example, in locations near the North Pole, no sunset occurs for several months during summer solstice while the sun does not rise for several months (during winter solstice) on locations near the South Pole in the same period of year.

### Example 3

A certain location receives 1400 W/m2 energy during the summer solstice, where the angle of incidence is normal to the ground. How much solar energy per unit area does the same location receive in winter solstice if the deflection angle to the vertical in that moment is 72° (cos 72° = 0.309, sin 72° = 0.951).

### Solution 3

We denote by E0 the maximum energy falling in the given location (E0 = 1400 W/m2). When sunrays fall normal to the ground, the effect of energy transfer is at its' maximum. Thus, if we consider the incidence angle as 0° to the normal, we must multiply the value of energy by cos 0°, as this gives the maximum value (cos 0° = 1). When the angle of incidence becomes 72°, we must therefore multiply the previous value of energy by cos 72° to find the amount of solar energy falling on that location in winter. Hence, we have

E = E0 ∙ cos⁡720
= 1400 W/m2 ∙ 0.309
= 432.6 W/m2

## Summary

The Sun is a fiery giant orb of mass M = 1.9989 × 1030 kg, mostly made up of hydrogen and helium. The Sun is an average sized star formed about 4.6 billion years ago. The radius of the Sun is 6.96 × 105 km. The Sun is located at centre of our Solar System and it is the main source of energy for it. All planets are "fed" by portions of solar energy incident from the Sun.

The composition of the Sun consists of 71% hydrogen and 27% helium. The rest of 2% are other chemical elements.

The proton-proton reactions necessary to produce solar energy occur in the innermost part of the Sun called the "Sun core". It is a spherical-shaped region of radius not more than 1/5 of the Sun's radius.

The gamma photons produced in the Sun's core shift towards the surface of the Sun, where they radiate in all directions. Another method of solar energy transmission is through convection, i.e. through gas currents.

The Sun structure consists on the following layers: core (at centre), radiative zone (including the core), convection zone, photosphere, chromosphere and corona.

Sunspots represent an interesting phenomenon occurring in photosphere. They are darker regions than the other parts of the Sun due to the lower temperature (1000 K - 1500 K colder than the neighboring regions). Their presence is explained through magnetic fields present in the Sun. The number of such spots varies periodically every 11.2 years. When the number of sunspots is at maximum, the Sun is at peak of its magnetic activity.

Solar protuberances are flame bolts flowing out from chromosphere and turning back into chromosphere again. This occurs due to the effect caused by the Sun's magnetic field.

The Sun revolves around its own axis. The presence of sunspots makes the study of this phenomenon easier. The solar self-rotation period is 24.47 days at the equator and almost 38 days at the poles. The average rotation is 28 days.

All planets of our Solar System adhere to Kepler Laws of planetary motion.

The angle of the Earths revolution around the Sun depends on the frame of reference. If stars are taken as reference frame, the corresponding period of revolution is known as sidereal (stellar) period, while when the Sun is taken as a reference frame, then corresponding period is known as tropical (solar) period. These periods (that determine the length of the corresponding years) are not the same; they have small deflections due to the effect of the Moon on the Earths revolution around the Sun. Thus, the duration of one sidereal year is 365.2568 days and that of a tropical year is 365.2422 years.

Since all planetary orbits are nearly at the same plane when observed from Earth, planets are aligned with a unique arc in space. The plane containing this arc is known as ecliptic plane and it divides the sky in two parts. More specifically, the ecliptic plane is defined as the imaginary plane containing the Earth's orbit around the sun. If we consider the imaginary line that connects the centre of Sun, Earth and all the other planets of they are all aligned, it is called ecliptics.

Since Mercury and Venus are the only planets inside the Earth's orbit (they are closer to the Sun and their orbit is smaller than Earth's orbit), they are called inner planets. All the other planets, from Mars to Neptune are called outer planets since their orbits include that of Earth.

In addition to orbital motion (revolution) around the Sun, planets rotate around themselves as well. This rotation occurs according an axis passing through the centre of planet, known as planetary axis. The plane of Earth planetary axis is deflected at 23.45° to the ecliptics plane.

When stars are taken as a (stationary) reference frame, the Earth completes one rotation around its own axis in 23 h 56 min. A solar day is slightly longer; it is about 24 hours.

The phenomenon of seasons occurs because of the angle formed by the rotation axis of the Earth and the orbital plane. This inclination makes solar radiation fall at different angles on a specific point of Earth at various period of the year. Since light intensity depends on the incidence angle, the solar energy brought to Earth is not the same throughout the year.

Solstice represents the time when the angle of incidence of sunrays on a region of Earth is the lowest or greatest possible. The day in which this angle is the lowest, is the longest day of the year in that place, and the summer begins.

An Equinox occurs when the Earths axis is only deflected laterally (not forwards or rearwards) in respect to the orbital plane. We have two equinoxes: in the northern hemisphere there is spring equinox on March 21st and autumn equinox on September 23rd. On these days, the duration of day and night are equal.

Day represents the time in which a region of the Earth is directly exposed to sunrays. Night is the opposite, i.e. a region is either illuminated indirectly from the Sun (through Moon surface that reflects the sunrays on to the Earth) or by means of other light sources (usually artificial, such as electric bulbs).

All locations on the equator have always have an equal duration of day and night as sunlight always falls on them at the same angle.

## Sun and Planetary Motion Revision Questions

1. The major semi-axis of Earth is 149.6 million km. What is the average distance of Pluto from the Sun if one year in Pluto lasts as 248 Earth years?

1. 5.905 billion km
2. 4.44 billion km
3. 7.38 billion km
4. 453 784 billion km

2. How fast do the points on Sun's equator rotate? Take 1 month = 30 days.

1. 2.07 km/h
2. 7.44 × 103 km/h
3. 43.7 km/h
4. 6.07 × 103 km/h

3. How many times less energy does a unit area on March surface receive from Sun compared to the same area and for the same incident angle on Earth?

1. 1.52 times less
2. 2 times less
3. 2.25 times
4. 2.32 times