Question

Decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain clearly; not all these have definitive answers, so your explanation is more important than your chosen answer. If the Sun suddenly became a \(1 M_{\text {Sun }}\) black hole, the orbits of the planets would not change at all.

Short answer

The statement makes sense; the gravitational force remains the same, so orbits do not change.

Step-by-step solution

Understanding the Problem

The problem asks us to determine if the claim that the planets' orbits would not change if the Sun became a black hole with the same mass makes sense. We have to evaluate the effects on planetary orbits.

Gravitational Influence

The gravitational force exerted by the Sun on the planets depends on its mass, not its composition or structure. A black hole with the same mass as the Sun exerts the same gravitational force on the planets.

Gravitational Force Equation

The force of gravity between two masses is given by Newton's law of gravitation: \[ F = G \frac{m_1 m_2}{r^2} \]where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses, and \(r\) is the distance between the centers of the two masses. Since \(m_1\), \(m_2\), and \(r\) do not change, the force remains the same.

Mass of the Sun

The problem specifies that the mass remains \(1 M_{\text{Sun}}\). Since the mass is the same, the gravitational influence at a given distance does not change, so the orbits, which depend on this gravitational force, remain the same.

Conclusion

Given the mass of the black hole is equivalent to the mass of the Sun, the gravitational force affecting the planets would not change, and therefore, the orbits of the planets would not change either.

Key concepts

Gravitational Force

Gravitational force is a fundamental interaction in nature that pulls two masses toward each other. It is responsible for keeping planets in orbit around stars, like Earth around the Sun. The magnitude of this force depends on two main factors: the masses involved and the distance between the centers of these masses.

If we consider two objects, such as a planet and a star, the gravitational force (\( F \) ) between them is calculated using the equation derived from Newton's law of gravitation: \[ F = G \frac{m_1 m_2}{r^2} \] Here, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between their centers. Gravitational force is key to understanding how celestial bodies move in space, maintaining their orbits due to the constant pull toward one another. As long as the mass and distance don't change, the gravitational force remains constant.

Newton's Law of Gravitation

Newton's law of gravitation is a cornerstone of classical physics, explaining how objects interact with each other through the force of gravity. Formulated by Sir Isaac Newton in the 17th century, this law states that every point mass attracts every other point mass in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them.

This can be mathematically expressed as: \[ F = G \frac{m_1 m_2}{r^2} \] Where \( F \) represents the gravitational force, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between their centers.

• Proportional to the masses: This means if the mass of either object increases, the gravitational force also increases.
• Inversely proportional to the distance squared: If the distance between objects increases, the gravitational force rapidly decreases.

Newton's law offers profound insights into planetary motions, including how the Sun's gravity governs the orbits of planets in our solar system. Understanding this allows us to predict planetary paths and explore the universe's mechanics.

Black Holes

Black holes are fascinating and mysterious objects in space, formed when massive stars collapse under their own gravity. They have an incredibly strong gravitational pull that not even light can escape, hence the term "black" hole. The border beyond which nothing can escape is known as the event horizon.

Despite their peculiar nature, black holes can have the same gravitational influence as other celestial bodies of equal mass. If, hypothetically, the Sun were to transform into a black hole with the same mass, its gravitational pull on the planets would remain unchanged. This is because gravitational force depends on mass, not composition.

Consequently, the planets in our solar system would continue orbiting the black hole just as they orbit the Sun today, provided the mass distribution relative to their orbits doesn't change. Black holes remind us that while gravity might seem straightforward in formulaic terms, it governs some of the most intriguing phenomena in the universe.