Problem 29 - College Physics

Question

In January \(2005,\) the Huygens probe landed on Saturn's moon Titan, the only satellite in the solar system having a thick atmosphere. Titan's diameter is \(5150 \mathrm{~km},\) and its mass is \(1.35 \times 10^{23} \mathrm{~kg} .\) The probe weighed \(3120 \mathrm{~N}\) on the earth. What did it weigh on the surface of Titan?

Step-by-Step Solution

Answer

The probe weighs approximately 1157.2 N on Titan.

Step 1: Understand the Concept of Weight

Weight is the force with which an object is pulled towards the center of a celestial body due to gravity. It is calculated as \( W = m \cdot g \), where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity on that body.

Step 2: Calculate Acceleration Due to Gravity on Titan

The formula for gravitational acceleration \( g \) on the surface of a celestial body is \( g = \frac{{G \cdot M}}{{R^2}} \), where \( G = 6.674 \times 10^{-11} \text{ N m}^2/ ext{kg}^2 \) is the gravitational constant, \( M \) is the mass of Titan, and \( R \) is the radius of Titan. Titan's radius \( R \) is half of its diameter, so \( R = \frac{5150 \text{ km}}{2} = 2575 \times 10^3 \text{ m} \). Substitute \( M = 1.35 \times 10^{23} \text{ kg} \) to find \( g \).

Step 3: Find the Mass of the Huygens Probe

First, find the mass of the probe on Earth using \( W = m \cdot g \), where \( W = 3120 \text{ N} \) and \( g = 9.81 \text{ m/s}^2 \) (Earth's gravity). Solving for \( m \) gives \( m = \frac{W}{g} = \frac{3120}{9.81} \approx 318 \text{ kg} \).

Step 4: Calculate the Weight of the Probe on Titan

Substitute the mass \( m \) of the probe and the gravitational acceleration \( g \) on Titan into the weight formula \( W = m \cdot g \). Use the \( g \) value calculated in step 2 to find \( W \) on Titan.

Key Concepts

Huygens Probe

The Huygens Probe was an important piece of the Cassini-Huygens mission to Saturn. Managed and launched by the European Space Agency (ESA), it was designed to study the atmosphere and surface of Saturn’s largest moon, Titan. This mission was significant because Titan is unique among moons in our solar system for having a dense atmosphere, primarily composed of nitrogen, similar to Earth's. When the probe successfully landed on Titan’s surface in January 2005, it provided invaluable data and images, offering insights into the atmospheric conditions and surface characteristics.

Understanding the conditions of Titan was crucial for the Huygens Probe mission, particularly because Titan’s thick atmosphere and surface chemistry have the potential to give us clues about the early Earth and the processes that lead to the development of life. By studying the data from Huygens, scientists hope to comprehend better the natural processes involving hydrocarbons, methane, and nitrogen cycles. This mission extends beyond merely sending a spacecraft; it drives deep into comparative planetology to unravel mysteries about the solar system's diversity.

Titan's Gravity

Titan's gravity differs from what we experience on Earth due to its smaller mass and size. Gravity on any celestial body is determined by its mass and radius, and Titan's figures make its gravity significantly less than Earth's. For instance, Titan's mass is about 1.35 times 10^{23} kg, and with a diameter of 5150 km, this yields specific gravitational attributes using the formula for gravitational acceleration:

• The gravitational acceleration on Titan, given by the formula \( g = \frac{G \cdot M}{R^2} \), where \( G \) is the gravitational constant, \( M \) is Titan's mass, and \( R \) is its radius (half of its diameter).
• Substituting Titan's values, one can calculate Titan's gravity, which is roughly about 1.35 m/s^2, much weaker compared to Earth's 9.81 m/s².

The implications of this weaker gravity are profound. Objects weigh less on Titan compared to Earth, which means astronauts or probes landing on Titan would feel lighter. This lower gravitational force also influences how the atmosphere is retained and contributes to its fascinating atmospheric phenomena.

Weight Calculation

Calculating the weight of an object, like the Huygens Probe, on a celestial body involves understanding the distinction between mass and weight. Weight is not just a static property like mass; it varies depending on the gravitational pull of the body on which the object resides.

The formula used for calculating weight is \( W = m \cdot g \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the gravitational acceleration on the celestial body:

• To find the mass of the probe using its Earth weight (3120 N), use Earth's gravity \( g = 9.81 \text{ m/s}^2 \), calculating \( m = \frac{3120}{9.81} = 318 \text{ kg} \).
• Once the mass is determined, compute the weight on Titan by using Titan's gravity: \( W_{\text{Titan}} = m \cdot g_{\text{Titan}} = 318 \text{ kg} \cdot 1.35 \text{ m/s}^2 \), yielding the new weight.

This calculation illustrates how gravity's strength changes weight even though mass remains constant, underpinning fundamental physics concepts that are crucial for astronomy and space exploration.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a key factor in calculating gravitational forces between two masses. Its value is approximately \( 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \), and it essentially bridges the understanding between gravitational forces and mass relationships across the universe.

To determine gravitational acceleration on celestial bodies such as Titan, \( G \) plays a central role in the equation \( g = \frac{G \cdot M}{R^2} \):

• This formula highlights how \( G \) is used alongside the body's mass \( M \) and radius \( R \) to compute the local gravitational force field.
• It showcases the universal applicability of gravitational calculations, allowing scientists to predict and understand gravitational interactions not only within our solar system but also beyond it.

Realizing \( G \)'s importance is crucial for all gravitational studies, reflecting Newton's law of universal gravitation and reinforcing how it underpins the celestial mechanics governing planet and satellite motion.