Problem 7 - Astronomy: At Play in the Cosmos

Question

By how many orders of magnitude does the gas giant planet Jupiter \(\left(L_{1} \approx 10^{8} \text { meters }\right)\) exceed the size of human beings \(\left(L_{1 \mathrm{iB}} \approx 10^{0} \text { meters }\right) ?\) a. 2 b. 4 c. 6 d. 8 e. 10

Step-by-Step Solution

Answer

Jupiter exceeds the size of humans by 8 orders of magnitude (option d).

Step 1: Understand the Concept of Orders of Magnitude

The order of magnitude is a way to express the scale or size of a quantity relative to another. It is typically calculated using logarithms base 10. For two quantities, the order of magnitude difference is the difference between the exponents when the quantities are expressed in powers of 10.

Step 2: Express the Sizes in Powers of 10

From the problem, Jupiter's size is approximately \(L_1 \approx 10^8\) meters and the size of a human being is approximately \(L_{1 \mathrm{iB}} \approx 10^0\) meters. These are already expressed in terms of powers of 10, with exponents 8 and 0 respectively.

Step 3: Calculate the Order of Magnitude Difference

The order of magnitude difference is calculated by subtracting the exponent of the smaller quantity from that of the larger quantity: \(8 - 0 = 8\).

Step 4: Interpret the Result

Since the difference in exponents is 8, Jupiter exceeds the size of a human being by 8 orders of magnitude. Therefore, the correct answer is 8, which corresponds to option d.

Key Concepts

Jupiter

Jupiter is the largest planet in our solar system. It is known as a "gas giant" because it is composed mostly of hydrogen and helium, similar to the sun. Unlike Earth, which has a solid surface, Jupiter doesn't have a solid surface that you can walk on. Instead, it has a thick atmosphere filled with swirling gases and immense storms. Think of the famous Great Red Spot, a massive storm larger than Earth that has been active for centuries. Jupiter's size is truly impressive:

• It has a diameter of about 139,820 kilometers (around 86,881 miles).
• It could fit more than 1,300 Earths inside it.
• It completes a rotation in just under 10 hours, though a complete orbit around the Sun takes nearly 12 Earth years.

This immense size and rapid rotation contribute to its powerful magnetic field and vast number of moons, at least 79, including the four largest—Io, Europa, Ganymede, and Callisto.

Gas Giants

Gas giants are a fascinating category of planets. In our solar system, Jupiter and Saturn are the most notable members. These planets are distinct from terrestrial planets like Earth:

• They lack a solid surface to stand on; instead, their atmospheres are composed of dense clouds and gases.
• They have thick, heavy atmospheres mostly made of hydrogen and helium, similar to the sun.
• Despite their size, they have a relatively small solid core, possibly made of rock and metal.

Gas giants tend to have a large number of moons and often possess ring systems. Saturn, famous for its beautiful rings, demonstrates this feature well. These belts are composed of ice and rock particles. This family of planets shows that not all worlds are like Earth, paving the way for understanding planetary formation and diversity both within and beyond our solar system.

Human Size Comparison

When we talk about size, humans are on the much smaller end of the scale compared to objects like planets. An average human being is about 1 to 2 meters tall. In terms of orders of magnitude—a mathematical concept used to compare sizes or quantities—our planet Earth is several steps above us, but Jupiter is another story entirely. To compare Jupiter with a human:

• Jupiter's diameter is approximately 10⁸ meters.
• The average human size is roughly 10⁰ meters.

This comparison shows that Jupiter exceeds the size of a human by about 8 orders of magnitude. This exponential difference illustrates not just Jupiter's vastness but also serves to highlight the vast scale of celestial bodies compared to our everyday human experiences. Understanding these size differences offers a perspective on our place in the universe. It encourages curiosity about the scale and scope of everything beyond our little corner of the cosmos.