The Solar System Revision Notes

The Solar System is centred around one star, the Sun. It is a nearly perfect sphere of hot plasma that produces energy through nuclear fusion, primarily converting hydrogen into helium in its core. This energy radiates outward, providing the light and heat necessary to sustain life on Earth and influencing the orbits of all bodies within the Solar System.

The Sun contains more than 99.8% of the total mass of the Solar System, which means its gravitational pull dominates the motion of all orbiting objects. Its gravitational field keeps planets, minor planets, comets, and other bodies in orbit around it.

• The Sun is a star composed mainly of hydrogen and helium.
• It produces energy by nuclear fusion in its core.
• It holds 99.8% of the Solar System's mass.
• Its gravity governs the orbits of all Solar System bodies.

There are eight recognised planets orbiting the Sun. These planets are, in order of increasing distance from the Sun: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune.

Each planet follows an elliptical orbit around the Sun, with the Sun at one focus. The planets vary in size, composition, atmosphere, and surface conditions. The inner four planets are rocky, while the outer four are gas giants or ice giants.

The order of the planets is important for understanding their relative distances, orbital periods, and physical characteristics.

• Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune.
• Planets orbit the Sun in elliptical paths.
• Inner planets are rocky; outer planets are gaseous or icy.
• Planets have moons orbiting them.

Minor planets are smaller objects orbiting the Sun that do not meet all the criteria to be classified as full planets. This category includes dwarf planets such as Pluto, Eris, and Ceres.

Asteroids are rocky bodies, mostly found in the asteroid belt between Mars and Jupiter. They vary in size from tiny rocks to objects hundreds of kilometres across.

Dwarf planets orbit the Sun directly but have not cleared their orbital paths of other debris, which distinguishes them from the eight major planets.

• Dwarf planets like Pluto orbit the Sun but are smaller than planets.
• Asteroids are rocky bodies mainly located in the asteroid belt.
• Minor planets have irregular shapes and varied compositions.
• They orbit the Sun but are not dominant in their orbital zones.

Moons are natural satellites that orbit planets. Earth has one moon, while other planets have multiple moons; for example, Jupiter has over 70 known moons.

Moons vary in size, composition, and geological activity. They are held in orbit by the gravitational pull of their parent planet.

Moons can affect their planets through gravitational interactions, such as causing tides on Earth.

• Moons orbit planets, not the Sun directly.
• They vary widely in number and size per planet.
• Held in orbit by the planet's gravity.
• Can influence planetary conditions, e.g., tides on Earth.

Comets are icy bodies that orbit the Sun in highly elliptical orbits. When they approach the Sun, their ices vaporise, creating visible tails of gas and dust.

Natural satellites include moons but can also refer to smaller objects orbiting planets or dwarf planets.

These smaller bodies contribute to the dynamic nature of the Solar System and can provide clues about its formation and evolution.

• Comets have elliptical orbits and develop tails near the Sun.
• Natural satellites include moons and smaller orbiting bodies.
• They are composed of ice, rock, and dust.
• Their orbits are influenced by the gravity of the Sun and planets.

The Solar System consists of eight major planets divided into two groups based on their distance from the Sun and their physical properties.

The four planets closest to the Sun — Mercury, Venus, Earth, and Mars — are small and rocky. They have solid surfaces composed mainly of metals and silicate rocks.

The four planets farthest from the Sun — Jupiter, Saturn, Uranus, and Neptune — are large and gaseous. They are often called gas giants or ice giants and consist mostly of hydrogen, helium, and other gases, with relatively small rocky cores.

• Inner planets: small, dense, rocky composition, solid surfaces.
• Outer planets: large, low density, gaseous composition, thick atmospheres.

The accretion model explains how the Solar System formed from a giant cloud of gas and dust called the solar nebula.

Gravity caused the cloud to collapse, pulling material together to form the Sun at the centre and a rotating disc of material around it.

Within this accretion disc, particles collided and stuck together, gradually forming planetesimals and eventually planets through a process called accretion.

• Solar nebula: a cloud of gas and dust.
• Gravity causes collapse and central condensation.
• Rotation forms a flattened accretion disc.
• Particles collide and stick, forming larger bodies.

Gravity is the force that attracts particles of gas and dust towards each other, enabling the growth of planetesimals and planets.

The Sun’s gravity dominates the Solar System, keeping planets in orbit.

Gravity also causes the denser materials to move towards the centre of forming planets, influencing their composition and structure.

• Gravity pulls particles together to form larger bodies.
• Sun’s gravity maintains planetary orbits.
• Gravity influences planet density and layering.

The solar nebula contained a mixture of elements and compounds, including hydrogen, helium, metals, and ices.

Closer to the Sun, high temperatures caused lighter gases and volatile compounds to evaporate or be blown away, leaving heavier elements like metals and silicates to form rocky planets.

Farther from the Sun, cooler temperatures allowed gases and ices to condense and be captured by growing planets, forming large gaseous envelopes around rocky cores.

• Solar nebula contained diverse elements and compounds.
• High temperatures near the Sun prevented gas condensation.
• Heavier elements formed rocky inner planets.
• Cooler outer regions allowed gas and ice accumulation.

As the solar nebula collapsed under gravity, conservation of angular momentum caused it to spin faster and flatten into a disc shape called the accretion disc.

This rotation caused material to spread out in a plane around the forming Sun, concentrating matter and enabling planet formation within the disc.

The distribution of temperature and material within the disc influenced the types of planets formed at different distances from the Sun.

• Conservation of angular momentum causes spinning and flattening.
• Accretion disc forms around the proto-Sun.
• Material distribution in the disc affects planet composition.
• Inner disc hotter, outer disc cooler, influencing planet types.

A gravitational field is a region of space surrounding a mass where another mass experiences a force of attraction. The gravitational field strength (symbol g) at a point is defined as the gravitational force experienced per unit mass placed at that point.

Mathematically, gravitational field strength is given by:

\[ g = \frac{F}{m} \] where: F is the gravitational force (in newtons, N), and m is the mass experiencing the force (in kilograms, kg). The unit of gravitational field strength is newtons per kilogram (N/kg). This unit is equivalent to metres per second squared (m/s²), which is the acceleration due to gravity.

• Gravitational field strength (g) = gravitational force (F) ÷ mass (m)
• Units of g: newtons per kilogram (N/kg) or metres per second squared (m/s²)

The strength of the gravitational field at the surface of a planet depends directly on the planet's mass. A more massive planet exerts a stronger gravitational pull on objects at its surface, resulting in a higher gravitational field strength.

The gravitational force between two masses is given by Newton's law of universal gravitation:

\[ F = G \frac{M m}{r^2} \] where: G is the gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²), M is the mass of the planet (kg), m is the mass of the object (kg), and r is the distance between the centres of the two masses (m). At the planet's surface, r equals the planet's radius (R). Rearranging to find gravitational field strength at the surface: \[ g = \frac{F}{m} = G \frac{M}{R^2} \] This shows that gravitational field strength is proportional to the planet's mass and inversely proportional to the square of its radius.

• Gravitational field strength at surface: \( g = G \frac{M}{R^2} \)
• Greater planet mass (M) → stronger gravitational field strength (g)
• Radius (R) also affects g; larger radius reduces g

Gravitational field strength decreases as the distance from the planet's centre increases. This is because the force of gravity follows an inverse square law with distance.

At a distance r from the planet's centre (where r ≥ R), gravitational field strength is:

\[ g = G \frac{M}{r^2} \] As r increases, \( g \) decreases rapidly. For example, at twice the planet's radius (r = 2R), gravitational field strength is one quarter of that at the surface: \[ g_{2R} = G \frac{M}{(2R)^2} = \frac{1}{4} G \frac{M}{R^2} = \frac{1}{4} g_{surface} \] This explains why astronauts experience microgravity in orbit far from Earth's surface.

• Gravitational field strength decreases with the square of distance from centre: \( g \propto \frac{1}{r^2} \)
• At twice the radius, g is one quarter of surface value
• Inverse square law governs variation with distance

Calculate the gravitational field strength at the surface of a planet with mass \( 6.0 \times 10^{24} \) kg and radius \( 6.4 \times 10^{6} \) m. Then find g at a height of 640 km above the surface.

Step 1: Calculate g at surface using \( g = G \frac{M}{R^2} \)

\[ g = (6.67 \times 10^{-11}) \times \frac{6.0 \times 10^{24}}{(6.4 \times 10^{6})^2} \] \[ = 6.67 \times 10^{-11} \times \frac{6.0 \times 10^{24}}{4.096 \times 10^{13}} \] \[ = 6.67 \times 10^{-11} \times 1.4648 \times 10^{11} = 9.77 \text{ N/kg} \] Step 2: Calculate g at height h = 640 km = \( 6.4 \times 10^{5} \) m above surface, so \( r = R + h = 6.4 \times 10^{6} + 6.4 \times 10^{5} = 7.04 \times 10^{6} \) m. \[ g = G \frac{M}{r^2} = 6.67 \times 10^{-11} \times \frac{6.0 \times 10^{24}}{(7.04 \times 10^{6})^2} \] \[ = 6.67 \times 10^{-11} \times \frac{6.0 \times 10^{24}}{4.953 \times 10^{13}} = 6.67 \times 10^{-11} \times 1.211 \times 10^{11} = 8.08 \text{ N/kg} \] The gravitational field strength decreases from 9.77 N/kg at the surface to 8.08 N/kg at 640 km altitude.

• Use \( g = G \frac{M}{r^2} \) with correct radius for distance r
• Substitute values carefully with units
• Calculate surface and above-surface gravitational field strength

Gravitational field strength represents the acceleration an object would experience due to gravity at a given point. Near the surface of Earth, this is approximately 9.8 m/s², meaning objects accelerate downwards at this rate if in free fall.

Because gravitational field strength decreases with distance, satellites in orbit experience weaker gravity than objects on the surface, which affects their orbital speed and period.

The dependence on mass explains why larger planets have stronger surface gravity, affecting atmospheric retention and surface conditions.

• Gravitational field strength = acceleration due to gravity
• Decreases with altitude, affecting satellite motion
• Planet mass influences surface gravity and environment

Light travels at a constant speed in a vacuum, denoted by c, which is approximately 3.00 × 10⁸ metres per second (m/s). This speed is fundamental in physics and astronomy because it sets the maximum speed at which information or energy can travel.

Knowing the speed of light allows us to calculate how long light takes to travel between objects in space, such as from the Sun to Earth or between planets. This time delay is important for understanding observations and communication in space missions.

• Speed of light, c = 3.00 × 10⁸ m/s (in vacuum)
• Speed is constant and does not depend on the source or observer

The time taken for light to travel a certain distance can be calculated using the relationship between speed, distance, and time:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

When calculating light travel time, the speed used is the speed of light, c. Therefore: \[ t = \frac{d}{c} \] where: t is time in seconds (s), d is distance in metres (m), and c is speed of light (3.00 × 10⁸ m/s).

• Time (s) = Distance (m) ÷ Speed (m/s)
• Use consistent SI units for accuracy

The average distance from the Sun to Earth is about 1.5 × 10¹¹ metres. To find the time light takes to travel this distance:

\[ t = \frac{1.5 \times 10^{11} \text{ m}}{3.00 \times 10^{8} \text{ m/s}} = 500 \text{ s} \]

Convert seconds to minutes: \[ 500 \text{ s} \div 60 = 8.33 \text{ minutes} \] which is approximately 8 minutes and 20 seconds. This means when we see the Sun, we are seeing it as it was 8 minutes and 20 seconds ago.

• Substitute values carefully with units
• Convert seconds to minutes for easier interpretation

1. Using inconsistent units: Distances must be in metres and speed in metres per second to get time in seconds.

2. Forgetting to convert seconds to minutes or hours when interpreting results.

3. Incorrect rearrangement of the formula, e.g., mixing up numerator and denominator in the time calculation.

• Always check units before calculating
• Write down the formula clearly before substituting
• Convert final answers to appropriate time units

Light travel times help us understand the scale of the Solar System and the delay in receiving information from space probes or observing celestial events.

For example, light takes about 4.3 hours to reach Neptune from the Sun, showing how vast the distances are.

These calculations assume light travels in a vacuum; in other media, the speed is slower but this is negligible for space distances.

• Light travel time increases with distance
• Useful for timing signals in space exploration
• Assumes vacuum conditions for speed of light

The Solar System consists of the Sun, eight planets, dwarf planets, moons, asteroids, comets, and other small bodies. Among these, the Sun contains approximately 99.86% of the total mass. This means that nearly all the mass in the Solar System is concentrated in the Sun, making it the dominant gravitational influence.

The planets and other bodies together make up only about 0.14% of the Solar System's mass. For example, Earth’s mass is about 6 × 10^24 kg, which is tiny compared to the Sun’s mass of approximately 2 × 10^30 kg.

• Sun’s mass ≈ 2 × 10^30 kg
• Sun contains about 99.86% of Solar System mass
• Planets and other bodies have much smaller combined mass

Gravity is a force of attraction between two masses. The strength of this force depends on the masses involved and the distance between them, as described by Newton’s law of universal gravitation:

F = G × (m1 × m2) / r², where: F is the gravitational force (newtons, N), G is the gravitational constant (6.67 × 10⁻¹¹ N m²/kg²), m1 and m2 are the masses (kilograms, kg), and r is the distance between the centres of the two masses (metres, m).

Because the Sun’s mass is so large, it exerts a strong gravitational pull on the planets, which keeps them moving in their orbits around the Sun. Without this force, the planets would move in straight lines and drift away into space.

• Newton’s law of universal gravitation: F = G(m1 m2)/r²
• Gravitational force depends on mass and distance
• Sun’s large mass creates strong gravitational pull

Planets orbit the Sun because the Sun’s gravity provides the centripetal force necessary to keep them moving in curved paths. The balance between the planet’s forward velocity and the Sun’s gravitational pull results in an elliptical orbit.

If the Sun’s gravity were weaker or absent, planets would not follow these orbits and would instead travel in straight lines due to inertia.

The orbital motion can be understood by equating the gravitational force to the centripetal force required for circular motion: F = m v² / r, where m is the planet’s mass, v is orbital speed, and r is orbital radius.

• Planets move forward but are pulled inward by Sun’s gravity
• Gravitational force acts as centripetal force
• Planets follow elliptical orbits around the Sun

Earth orbits the Sun at an average distance of about 1.5 × 10^11 m (1 astronomical unit). The Sun’s mass is about 2 × 10^30 kg.

Using Newton’s law of gravitation and centripetal force, the orbital speed v of Earth can be calculated:

Set gravitational force equal to centripetal force: G M m / r² = m v² / r, where M is Sun’s mass, m is Earth’s mass, r is orbital radius, v is orbital speed. Cancel m and rearrange for v: v = √(G M / r) = √((6.67 × 10⁻¹¹)(2 × 10^30) / (1.5 × 10^11)) ≈ 3.0 × 10^4 m/s (30 km/s). This matches the observed orbital speed of Earth around the Sun.

• Earth’s orbital radius ≈ 1.5 × 10^11 m
• Sun’s mass ≈ 2 × 10^30 kg
• Earth’s orbital speed ≈ 3.0 × 10^4 m/s

To draw a diagram showing the Sun’s mass and planetary orbit:

1. Draw a large circle near the centre labelled 'Sun'.

2. Draw a smaller circle representing a planet orbiting around the Sun in an elliptical or near-circular path, with the Sun at one focus of the ellipse (or centre if circular). Mark the orbital radius r from the Sun’s centre to the planet’s position on the orbit.

• Sun at centre or focus of orbit
• Planet shown on elliptical or circular orbit
• Label orbital radius r

Gravity is a force of attraction between two masses. The Sun, having a very large mass, exerts a strong gravitational pull on objects in the Solar System such as planets, comets, and asteroids.

This gravitational force acts towards the centre of the Sun and provides the necessary centripetal force to keep these objects moving in curved paths called orbits.

Without the Sun's gravitational pull, objects would move in straight lines at constant speeds according to Newton's first law of motion and would leave the Solar System.

• Gravitational force is always attractive and acts along the line joining two masses.
• The Sun's gravity pulls planets towards it, preventing them from flying off into space.
• This force acts as the centripetal force required for circular or elliptical orbits.

The gravitational force (F) between two masses (m₁ and m₂) separated by a distance (r) is given by Newton's law of universal gravitation:

\[ F = G \frac{m_1 m_2}{r^2} \] where: G is the gravitational constant \(6.67 \times 10^{-11} \, \mathrm{N \cdot m^2 / kg^2}\).

In the Solar System, m₁ can be the mass of the Sun and m₂ the mass of a planet or other object, with r being the distance between their centres.

• Force decreases with the square of the distance between objects.
• Force increases with the product of the two masses.
• This explains why planets closer to the Sun experience a stronger gravitational pull.

An object moving in a circular orbit of radius r at speed v requires a centripetal force directed towards the centre of the circle, given by:

\[ F_c = \frac{m v^2}{r} \] where m is the mass of the orbiting object.

In the Solar System, the Sun's gravitational force provides this centripetal force, so: \[ G \frac{M_{sun} m}{r^2} = \frac{m v^2}{r} \] where \(M_{sun}\) is the Sun's mass.

• The mass of the orbiting object cancels out, showing orbital speed depends on Sun's mass and orbital radius.
• Planets closer to the Sun orbit faster due to stronger gravitational force.
• This balance keeps planets in stable orbits.

Planets and other bodies orbit the Sun in elliptical paths, with the Sun at one focus of the ellipse.

The gravitational force varies slightly as the distance changes, causing the orbital speed to change: objects move faster when closer to the Sun and slower when farther away.

This variation is explained by conservation of energy and angular momentum in orbital mechanics.

• Orbits are not perfect circles but ellipses.
• Orbital speed is highest at perihelion (closest point) and lowest at aphelion (farthest point).
• The Sun's gravitational pull continuously changes the direction of velocity, keeping the object in orbit.

The Sun's gravitational attraction is fundamental to the structure and stability of the Solar System.

It ensures that planets, comets, and other bodies remain in orbit rather than drifting away.

Understanding this force allows prediction of orbital periods, speeds, and distances, which are essential in astronomy and space exploration.

• Gravitational force is the centripetal force for orbits.
• Orbital characteristics depend on mass and distance.
• This principle applies universally to all gravitational systems.

The paths followed by planets, minor planets (such as dwarf planets and asteroids), and comets around the Sun are not perfect circles but ellipses. An ellipse is a closed curve resembling a flattened circle, defined mathematically as the set of points where the sum of the distances from two fixed points (called foci) is constant.

This elliptical shape of orbits was first described by Johannes Kepler in his first law of planetary motion, which states: "The orbit of every planet is an ellipse with the Sun at one of the two foci."

• An ellipse has two foci; the sum of distances from any point on the ellipse to the foci is constant.
• Orbits of planets and comets are elliptical, not circular.
• Kepler's First Law defines the elliptical nature of planetary orbits.

In an elliptical orbit, the Sun is located at one focus of the ellipse, not at the geometric centre. This means the distance between the orbiting body and the Sun changes as the body moves along its orbit.

When the orbit is nearly circular, the two foci are very close together, so the Sun appears almost at the centre of the orbit. However, for more elongated ellipses, the Sun is noticeably off-centre.

This off-centre position of the Sun explains why planets move faster when they are closer to the Sun (at perihelion) and slower when they are farther away (at aphelion), as described by Kepler's second law.

• Sun is at one focus of the elliptical orbit, not the centre.
• Distance between planet and Sun varies during orbit.
• Nearly circular orbits have the Sun close to the centre.

Ellipse: A closed curve with two foci where the sum of distances from any point on the curve to the foci is constant.

Focus (plural: foci): Fixed points used to define an ellipse; the Sun occupies one focus in planetary orbits.

Perihelion: The point in the orbit closest to the Sun, where the planet moves fastest due to stronger gravitational attraction and conservation of angular momentum.

• Ellipse: flattened circle with two foci.
• Focus: fixed point defining ellipse shape; Sun is at one focus.
• Perihelion: closest approach to the Sun in orbit.

Because the Sun is not at the centre of the orbit, the orbital speed of a planet or comet changes during its orbit. It moves faster when nearer the Sun and slower when farther away.

This variation in speed is a consequence of the conservation of angular momentum and gravitational forces acting between the Sun and the orbiting body.

Understanding elliptical orbits is essential for predicting planetary positions and spacecraft trajectories.

• Orbital speed varies with distance from the Sun.
• Planets move faster at perihelion and slower at aphelion.
• Elliptical orbits explain observed planetary motion accurately.

To draw an elliptical orbit: 1. Draw two points on a horizontal line to represent the foci; label one as the Sun's position. 2. Using a string fixed at both foci, trace the ellipse by keeping the string taut with a pencil. 3. The orbiting body moves along this ellipse, with the Sun at one focus, not the centre.

This construction helps visualise why the Sun is off-centre and how the orbit shape affects orbital dynamics.

• Two foci define the ellipse; Sun at one focus.
• Use string method to draw ellipse accurately.
• Orbiting body moves along the ellipse, varying distance from Sun.

Orbital distance is the average distance from a planet to the Sun, usually measured in millions of kilometres (10^6 km) or astronomical units (AU), where 1 AU ≈ 1.5 × 10^8 km (the average Earth-Sun distance).

The orbital period (duration) is the time a planet takes to complete one full orbit around the Sun, measured in Earth days or years.

Kepler's Third Law relates orbital distance (r) and orbital period (T): T² ∝ r³, where T is in years and r in AU. This means planets farther from the Sun have longer orbital periods due to weaker solar gravitational attraction and slower orbital speeds.

• Orbital distance affects solar energy received, influencing temperature.
• Orbital period determines length of a planet's year.
• Kepler's Third Law: T² = k r³ (k is constant for all planets orbiting the Sun).

Density (ρ) is mass per unit volume, measured in kilograms per cubic metre (kg/m³). It indicates the planet's composition and internal structure.

Density is calculated by ρ = mass ÷ volume. For a spherical planet, volume V = (4/3)πr³, where r is the planet's radius.

Terrestrial (rocky) planets have higher densities (~3000–5500 kg/m³) due to metallic and rocky materials, while gas giants have lower densities (~700–1600 kg/m³) because they are mostly hydrogen and helium gas.

• Density reveals whether a planet is rocky or gaseous.
• Calculate density using mass and radius measurements.
• Higher density implies heavier elements and solid surfaces.

Surface temperature depends mainly on the planet's distance from the Sun and its atmosphere's ability to retain heat.

Planets closer to the Sun receive more solar radiation, leading to higher temperatures.

Atmospheric composition and greenhouse gases can raise surface temperature beyond what distance alone predicts (e.g., Venus).

• Temperature decreases with increasing orbital distance, generally.
• Atmospheric effects can cause deviations from expected temperatures.
• Surface temperature is measured in kelvin (K) or degrees Celsius (°C).

Gravitational field strength (g) at a planet's surface is the force per unit mass experienced by an object, measured in newtons per kilogram (N/kg).

It depends on the planet's mass (M) and radius (R) according to the formula: g = GM/R², where G = 6.67 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant.

A stronger gravitational field helps a planet retain an atmosphere and affects surface weight and pressure.

• Calculate g using planet mass and radius.
• Gravitational field strength decreases with increasing radius.
• Earth's g ≈ 9.8 N/kg is a reference value.

By comparing orbital distance and period, one can verify Kepler's Third Law and understand planetary motion.

Density and surface temperature data together indicate planet type and habitability potential.

Gravitational field strength data help predict atmospheric retention and surface conditions.

• Use data tables to compare planets systematically.
• Plot graphs of orbital period squared vs orbital distance cubed to check Kepler's law.
• Relate density and temperature to planetary classification (rocky or gaseous).

Gravitational field strength (g) at a point in space is defined as the gravitational force per unit mass experienced by a small test mass placed at that point. It is measured in newtons per kilogram (N/kg).

The Sun, having the largest mass in the Solar System, creates a gravitational field that extends throughout the system. The strength of this field decreases with increasing distance from the Sun.

Mathematically, the gravitational field strength due to the Sun at a distance r from its centre is given by: g = GM/r² where: G = universal gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²), M = mass of the Sun (approximately 2.0 × 10³⁰ kg), r = distance from the Sun's centre (in metres).

• Gravitational field strength is a vector quantity directed towards the Sun.
• It decreases with the square of the distance from the Sun.

Planets orbit the Sun due to the gravitational force acting as the centripetal force that keeps them moving in curved paths.

For a planet of mass m orbiting at radius r with orbital speed v, the gravitational force provides the centripetal force: F_gravity = F_centripetal => GMm/r² = mv²/r

Rearranging to find orbital speed: v = √(GM/r) This shows that orbital speed decreases as the square root of the orbital radius increases.

• Orbital speed is inversely proportional to the square root of the distance from the Sun.
• Planets closer to the Sun move faster in their orbits.

The Sun's gravitational pull weakens with distance, so planets farther away experience less force pulling them inward.

To maintain a stable orbit, the centripetal force required is less for planets at larger radii, so their orbital speeds are lower.

This explains why Mercury, the closest planet to the Sun, has the highest orbital speed, while Neptune, the farthest, moves the slowest.

• Weaker gravitational field at greater distances reduces centripetal force.
• Lower centripetal force means lower orbital speed is sufficient.

Given: G = 6.67 × 10⁻¹¹ N·m²/kg², M = 2.0 × 10³⁰ kg, r = 1.5 × 10¹¹ m (average Earth-Sun distance).

Calculate Earth's orbital speed v: v = √(GM/r) = √[(6.67 × 10⁻¹¹)(2.0 × 10³⁰) / (1.5 × 10¹¹)]

First calculate numerator: (6.67 × 10⁻¹¹)(2.0 × 10³⁰) = 1.334 × 10²⁰ Then divide by r: 1.334 × 10²⁰ / 1.5 × 10¹¹ = 8.89 × 10⁸ Finally, square root: v = √(8.89 × 10⁸) ≈ 2.98 × 10⁴ m/s = 29.8 km/s.

• Use correct units for G, M, and r.
• Substitute values carefully to avoid errors.

To draw a diagram illustrating the Sun's gravitational field and planetary orbits:

1. Draw a large circle labelled 'Sun' at the centre.

2. Draw concentric circles around the Sun representing planetary orbits at increasing radii (e.g., Mercury, Earth, Neptune). Label each orbit with the planet's name and approximate distance from the Sun in metres or astronomical units (AU). 1 AU ≈ 1.5 × 10¹¹ m is the Earth-Sun distance. 3. At points on each orbit, draw arrows pointing towards the Sun representing gravitational field vectors. Make the arrows shorter for orbits farther from the Sun to indicate weaker field strength. 4. Next to each orbit, write the approximate orbital speed, showing that it decreases with increasing orbital radius.

• Concentric circles represent orbits at different distances.
• Arrows towards Sun show gravitational field direction and relative strength.

Objects such as planets, comets, and asteroids orbit the Sun in elliptical paths, not perfect circles. An ellipse is an oval shape with two focal points; the Sun is located at one focus of the ellipse.

The distance between the orbiting object and the Sun changes continuously during the orbit. The closest point to the Sun is called perihelion, and the farthest point is called aphelion.

Because the distance varies, the gravitational force between the Sun and the object also changes, affecting the object's speed along its orbit.

• Ellipse has two foci; Sun is at one focus.
• Perihelion: closest point to Sun.
• Aphelion: farthest point from Sun.

Gravitational force between two masses is given by Newton’s law of universal gravitation:

\[ F = \frac{GMm}{r^2} \] where \(G\) is the gravitational constant, \(M\) is the mass of the Sun, \(m\) is the mass of the orbiting object, and \(r\) is the distance between their centres.

When the object is nearer the Sun (smaller \(r\)), the gravitational force is stronger, pulling the object more and causing it to accelerate, increasing its speed. When the object is farther away, the force is weaker, and the object moves more slowly.

• Gravitational force \(F = GMm/r^2\).
• Force stronger at perihelion (smaller \(r\)).
• Object moves faster when closer to Sun.

The total mechanical energy of the orbiting object is the sum of its kinetic energy (KE) and gravitational potential energy (GPE):

\[ E_{total} = KE + GPE \] where \( KE = \frac{1}{2}mv^2 \) and \( GPE = -\frac{GMm}{r} \).

The negative sign in GPE indicates that the object is bound to the Sun by gravity. As the object moves closer to the Sun, \(r\) decreases, so \(GPE\) becomes more negative (decreases). To conserve total energy, \(KE\) must increase, so the object’s speed \(v\) increases. Conversely, when the object moves away, \(r\) increases, \(GPE\) increases (less negative), and \(KE\) decreases, so speed decreases.

• Total energy \(E = KE + GPE\) is conserved.
• \(KE = \frac{1}{2}mv^2\), \(GPE = -GMm/r\).
• Closer to Sun: \(r\) decreases, \(GPE\) decreases, \(KE\) increases, speed increases.
• Farther from Sun: \(r\) increases, \(GPE\) increases, \(KE\) decreases, speed decreases.

Using conservation of energy: at two points in the orbit,

\[ KE_1 + GPE_1 = KE_2 + GPE_2 \] or, substituting expressions,

\[ \frac{1}{2}mv_1^2 - \frac{GMm}{r_1} = \frac{1}{2}mv_2^2 - \frac{GMm}{r_2} \] where \(v_1, r_1\) and \(v_2, r_2\) are speeds and distances at two points in the orbit (e.g., perihelion and aphelion). Rearranging to find speed at one point if the other is known:

• \[ v_2 = \sqrt{v_1^2 + 2GM\left(\frac{1}{r_2} - \frac{1}{r_1}\right)} \]
• Speed increases as \(r\) decreases due to energy conservation.
• Mass \(m\) cancels out, so speed depends on \(G, M, r\).

To draw a diagram illustrating speed variation: 1. Draw an ellipse with two foci; mark one focus as the Sun.

2. Label the closest point on the ellipse to the Sun as perihelion and the farthest as aphelion.

3. Draw velocity vectors at perihelion and aphelion; the vector at perihelion should be longer, indicating higher speed, and shorter at aphelion, indicating lower speed.

• Ellipse with Sun at one focus.
• Label perihelion (closest) and aphelion (farthest).
• Velocity vector longer at perihelion, shorter at aphelion.