The Earth moves around the Sun in an elliptic trajectory due to gravity. If another Sun like star is brought near the Earth, what will be the shape of the trajectory ?

1. Newton's Law of Universal Gravitation (basic)

Welcome to your first step into the cosmos! To understand how planets move and why stars form, we must start with the invisible thread that holds the universe together: Gravity. While thinkers had wondered about the falling of objects for centuries, the scientific revolution reached its peak when Isaac Newton formulated his theory of gravitation Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119. He proposed that gravity isn't just a force on Earth, but a universal force that acts between every single object in existence that has mass.

Newton’s Law of Universal Gravitation states that the force of attraction between two bodies depends on two factors: their mass and the distance between them. Specifically, the force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers (F = G m₁m₂ / r²). This means if you double the distance between two planets, the pull between them doesn’t just halve; it becomes four times weaker! This force is measured in newtons (N) Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.65.

Interestingly, gravity is not perfectly uniform everywhere. Because Earth is not a perfect sphere, the gravity you feel is greater near the poles and less at the equator, simply because the equator is further from the Earth’s center FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19. Scientists even use gravity anomalies—differences between expected and measured gravity—to map the uneven distribution of mass within the Earth’s crust. To give you a sense of scale, look at how gravity varies across our neighborhood:

Celestial Body	Surface Gravity	Comparison to Earth
Sun	274 m/s²	28.0x Earth
Earth	9.8 m/s²	1.0x Earth
Moon	1.62 m/s²	~0.16x Earth

Source: Physical Geography by PMF IAS, The Solar System, p.23

Sources: Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119; Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.65; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.), The Solar System, p.23

2. Kepler's Laws of Planetary Motion (basic)

Hello! Today we are diving into the fundamental rules that govern how everything moves in our solar system. For a long time, humanity believed planets moved in perfect circles. It was Johannes Kepler who realized that nature is slightly more complex—and much more interesting. He gave us three laws that describe the motion of planets around the Sun.

The First Law (The Law of Orbits) tells us that a planet does not travel in a circle, but in an ellipse. An ellipse is like a stretched circle with two central points called foci. The Sun sits at one of these two foci, not at the dead center Physical Geography by PMF IAS, The Solar System, p.21. This means that throughout the year, the distance between the Earth and the Sun changes. We call the closest point perihelion and the farthest point aphelion.

The Second Law (The Law of Equal Areas) explains how fast a planet moves. It states that a line connecting the planet and the Sun sweeps out equal areas in equal intervals of time Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.257. To keep the "area" the same when the planet is close to the Sun, it must move faster. When it is far away (at aphelion), it moves slower. This has a fascinating effect on our calendar: because Earth is farther from the Sun during the Northern Hemisphere's summer, it moves slower in its orbit, making our summer about 92 days long—slightly longer than our winter, which is about 89 days Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.256.

The Third Law (The Law of Periods) provides a mathematical link between a planet's distance from the Sun and its orbital period. It states that the square of the time it takes to orbit (T²) is proportional to the cube of its average distance from the Sun (a³). In simple terms: the farther a planet is from the Sun, the significantly longer it takes to complete one "year" Physical Geography by PMF IAS, The Solar System, p.21.

Kepler's Law	What it defines	Key Takeaway
1st Law	The Shape	Orbits are ellipses, not circles.
2nd Law	The Speed	Planets move faster when closer to the Sun.
3rd Law	The Duration	Distant planets have much longer years (T² ∝ a³).

Sources: Physical Geography by PMF IAS, The Solar System, p.21; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.256-257

3. The Physics of the Two-Body Problem (intermediate)

At its heart, the Two-Body Problem is the study of how two massive objects interact with one another through the force of gravity. In a pure two-body system, we assume these two objects are the only things in the universe, pulling on each other according to Newton's Law of Gravitation (F = Gm₁m₂/r²). This setup is the foundation of orbital mechanics because it is analytically solvable—meaning we can use math to predict exactly where both objects will be at any point in the future. In our neighborhood, the Earth-Sun relationship is treated as a two-body system where the Earth follows a stable, predictable path called an orbit Science-Class VII NCERT, Earth, Moon, and the Sun, p.176.

The motion in a two-body system is governed by Kepler’s Laws of Planetary Motion. These laws tell us that the orbit of a planet is not a perfect circle, but an ellipse, with the sun at one of the two foci Physical Geography by PMF IAS, The Solar System, p.21. Because the two bodies are technically pulling on each other, they actually revolve around a common center of mass called the barycenter. In the case of the Sun and Earth, the Sun is so massive that the barycenter lies deep within the Sun itself, making it look like the Sun is stationary while the Earth does all the moving. However, in binary star systems, where two stars have comparable masses, they can be seen clearly orbiting this empty point in space between them Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.11.

While the two-body problem gives us beautiful, closed-form solutions like ellipses or parabolas, real life is rarely that simple. As soon as you introduce a third body—such as a second sun or a nearby giant planet—the system becomes chaotic. This is known as the 'Three-Body Problem.' Unlike the stable two-body dance, three-body interactions are non-integrable and highly sensitive to initial conditions. We see hints of this complexity when astronomers noticed 'irregular gravitational effects' on known planets, which actually led to the mathematical discovery of Neptune and Pluto Certificate Physical and Human Geography, The Earth's Crust, p.3. This highlights that while the two-body model is our primary tool for understanding stability, the universe is often a more complex 'N-body' puzzle.

System Type	Trajectory Shape	Predictability
Two-Body	Conic sections (Ellipse, Circle, Parabola)	High (Deterministic)
Three-Body	Complex, non-repeating loops	Low (Chaotic)

Sources: Science-Class VII NCERT, Earth, Moon, and the Sun, p.176; Physical Geography by PMF IAS, The Solar System, p.21; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.11; Certificate Physical and Human Geography, The Earth's Crust, p.3

4. Gravitational Equilibrium: Lagrange Points (exam-level)

To understand Lagrange points, we must first look at the Three-Body Problem. In a simple 'two-body' system, like the Earth orbiting the Sun, the physics is predictable and follows a stable elliptical path Physical Geography by PMF IAS, Chapter 19, p.255. However, when we introduce a third, much smaller body (like a satellite), the gravitational interactions become incredibly complex. Lagrange points are the 'sweet spots' in space where a small object can stay in a fixed position relative to two larger orbiting bodies. At these five specific points, the gravitational pull of the two large masses (e.g., the Sun and Earth) precisely equals the centrifugal force felt by the smaller object. Imagine a cosmic tug-of-war where the ropes are perfectly balanced, allowing the satellite to 'hover' with minimal fuel consumption. This equilibrium is essential for modern space exploration. For instance, India’s Aditya L1 mission is designed to sit at the L1 point to maintain a continuous, unobstructed view of the Sun Science Class VIII NCERT, Keeping Time with the Skies, p.185.

Point	Location	Stability & Use
L1	Between Sun and Earth	Unstable; ideal for solar observation (Aditya L1).
L2	'Behind' the Earth	Unstable; perfect for space telescopes (James Webb) looking into deep space.
L3	Hidden behind the Sun	Unstable; rarely used as it is always hidden from Earth.
L4 & L5	60° ahead/behind in orbit	Stable; objects here (like Trojan asteroids) stay naturally trapped.

While L1, L2, and L3 are mathematically points of equilibrium, they are 'unstable'—much like balancing a marble on top of a mountain. A slight nudge requires the satellite to use small thrusters to stay in place. In contrast, L4 and L5 are like the bottom of a bowl; if an object drifts, gravity naturally pulls it back into the center.

Sources: Physical Geography by PMF IAS, Chapter 19: The Motions of The Earth and Their Effects, p.255; Science Class VIII NCERT, Keeping Time with the Skies, p.185

5. Multi-Star Systems and Orbital Perturbations (intermediate)

In our own solar system, we are used to a sense of cosmic order. According to Kepler’s First Law, planets move in stable, predictable ellipses with the Sun at one focus Physical Geography by PMF IAS, The Solar System, p.21. This is a classic example of a two-body system. Because the Sun is so much more massive than the Earth, the gravitational interaction is straightforward, allowing us to calculate the Earth's position thousands of years into the future with high precision. This predictability is what astronomers call a "closed-form solution."

However, the universe is rarely that simple. Many stars do not exist in isolation like our Sun; they are part of multi-star systems. For instance, our nearest neighbor, Alpha Centauri, is actually a triple-star system consisting of Alpha Centauri A, Alpha Centauri B, and Proxima Centauri Physical Geography by PMF IAS, The Solar System, p.37. In such environments, a planet is no longer pulled by a single dominant gravity source. Instead, it experiences orbital perturbations—complex gravitational "tugs" from multiple stars that constantly nudge the planet out of a simple elliptical path. If the stars are close enough, they can even strip material from one another, leading to phenomena like Novae Physical Geography by PMF IAS, The Universe, p.11.

When we introduce a third massive body into the equation, we enter the realm of the Three-Body Problem. Unlike the stable two-body system, the three-body problem is famously chaotic and non-integrable. This means there is no simple mathematical formula to describe the long-term motion of the bodies. The system becomes extremely sensitive to initial conditions—a tiny change in velocity or distance today could lead to the planet being flung into deep space or crashing into a star millions of years later. In such a system, the orbit ceases to be a clean circle or ellipse and becomes an unpredictable, wandering trajectory.

Feature	Two-Body System (e.g., Sun-Earth)	Three-Body System (e.g., Planet in Binary Star)
Predictability	Deterministic and stable	Chaotic and unpredictable
Orbital Path	Conic sections (Ellipse, Circle)	Complex, non-repeating trajectories
Math Nature	Integrable (solvable)	Non-integrable (requires simulation)

Sources: Physical Geography by PMF IAS, The Solar System, p.21; Physical Geography by PMF IAS, The Solar System, p.37; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.11

6. Complexity and Chaos: The Three-Body Problem (exam-level)

In our study of astrophysics, we often start with the elegant simplicity of the Two-Body Problem. When we look at the Earth orbiting the Sun, we see a system governed by Isaac Newton’s law of universal gravitation Themes in world history, Changing Cultural Traditions, p.119. In this scenario, the math is "closed-form" or analytical, meaning we have neat equations that predict a stable, repeating elliptical trajectory. This is why we can precisely calculate events like the perihelion (when Earth is closest to the Sun) and aphelion (when it is farthest) Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.255.

However, the universe is rarely that simple. When a third massive body is introduced—such as a second star in a binary system or the massive influence of Jupiter on smaller bodies—we enter the realm of the Three-Body Problem. Unlike the two-body system, the three-body problem is famously non-integrable. This means there is no general mathematical formula that can predict the exact positions of all three bodies forever. The gravitational tug-of-war becomes a chaotic system, where the motion of one body is constantly and unpredictably altered by the shifting positions of the other two.

The defining characteristic of this complexity is sensitivity to initial conditions. In a chaotic system, an infinitesimally small change in the starting position or velocity of a planet can lead to a completely different orbit millions of years later. While the orbits of major planets like Jupiter and Saturn appear stable, the movement of smaller objects, such as asteroids located between them Physical Geography by PMF IAS, The Solar System, p.33, can be significantly perturbed by these complex gravitational interactions.

Feature	Two-Body System (e.g., Earth-Sun)	Three-Body System
Predictability	Deterministic and stable over long periods.	Chaotic; unpredictable over long timeframes.
Math Solution	Analytical (closed-form) solutions exist.	Numerical (computational) simulations required.
Trajectory	Simple conic sections (Ellipses/Circles).	Irregular and highly complex paths.

In a three-body environment, the dynamics vary extremely over the trajectory. Standard linearized models used for simple orbits break down because the forces are constantly compounding in non-linear ways. If Earth were suddenly part of a three-body system, its path would cease to be a predictable ellipse and would instead become an undeterministic journey, potentially leading to it being flung out of the system or crashing into a star, depending on the specific masses and velocities involved.

Sources: Themes in world history, Changing Cultural Traditions, p.119; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.255; Physical Geography by PMF IAS, The Solar System, p.33

7. Solving the Original PYQ (exam-level)

You have just mastered the fundamentals of planetary motion and Kepler’s Laws, which explain why the Earth follows a stable, elliptical path around the Sun. This is a classic example of a two-body problem, where the gravitational interaction between two masses results in predictable, closed-form orbits. However, this question tests your ability to recognize when those fundamental rules are superseded by more complex dynamics. By introducing a second Sun-like star, the system evolves into the three-body problem, a famous challenge in physics where the gravitational pull from two massive objects on a smaller third body creates a chaotic system that lacks a simple geometric solution.

To arrive at the correct answer, you must realize that in a three-body environment, the Earth's trajectory is no longer governed by a single focus. Instead, it becomes extremely sensitive to initial conditions—meaning even a tiny change in position or velocity leads to vastly different outcomes over time. Because there is no general mathematical formula to predict these complex interactions, the trajectory is fundamentally undeterministic. While you might be tempted to choose Ellipse or Circle based on your knowledge of standard orbits, these shapes only exist in the equilibrium of a two-body system. Similarly, a Parabola represents a specific escape trajectory that still implies a level of predictability that a three-body system cannot guarantee.

UPSC often uses these types of questions to see if you can identify a paradigm shift in a problem. The common trap is to assume that adding more gravity simply results in a "different" predictable shape. In reality, as noted in Physical Geography by PMF IAS, the transition from a two-body to a three-body interaction moves the Earth from the realm of classical orbital mechanics into non-integrable dynamics, where the path becomes unpredictable and non-repeating.