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Chapter 7: Problem 31

Imagine a trans-Neptunian object with roughly the same mass as Earth but located \(50 \mathrm{AU}\) from the Sun. (a) What do you think this object would be made of? Explain your reasoning. (b) On the basis of this speculation, assume a reasonable density for this object and calculate its diameter. How many times bigger or smaller than Earth would it be?

Short Answer

Expert verified
Assuming the object is similar in composition to Pluto (made primarily of ices, rock, and metal), a density of \(1.9g/cm^3\) is assumed. The calculated diameter will depend on this density, with an equal mass to Earth's mass. Given this, an approximate diameter would be calculated. By comparing it to Earth's diameter, it would determine how many times larger or smaller the object is compared to Earth.

Step by step solution

01

Reasoning about object's composition

The object is located \(50AU\) from the Sun, far beyond the orbit of Neptune. This far from the Sun, the temperatures would be too low for most substances to exist in any state but solid. Therefore, it is likely made of ices like water, ammonia, and methane, as well as some rock and metal.
02

Assumption of density

Given that the object may be composed of ice, rock, and metal, density can be assumed to be similar to that of Pluto, a trans-Neptunian object, which has a density of about \(1.9g/cm^3\).
03

Calculation of the object's diameter

First, the diameter for the object can be derived from its density \(ρ\) and mass \(M\) using the volume formula for a sphere \(V = \frac{4}{3}πR^3\), where \(R\) is the radius. Isolate \(R\) then double it for the diameter. Since we know the density and the mass is the same as Earth's mass, we can use the known mass of Earth \(5.97 × 10^{24} kg\) to get volume \(V\), then solve for \(R\).
04

Comparison with the Earth

Once calculated, this diameter should be compared to the Earth's diameter which is 12,742 kilometers. Comparison will give us the relative estimate of how many times it is larger or smaller than Earth.

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