Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

In "Natural History", March, \(1996,\) Neil de Grass Tyson discusses the discovery of an astronomical object called a "brown dwarf". "We have suspected all along that brown dwarfs were out there. One reason for our confidence is the fundamental theorem of mathematics that allows you to declare that if you were once \(3^{\prime} 8^{\prime \prime}\) tall and are now \(5^{\prime} 8 "\) tall, then there was a moment when you were \(4^{\prime} 8^{\prime \prime}\) tall (or any other height in between). An extension of this notion to the physical universe allows us to suggest that if round things come in low-mass versions (such as planets) and high-mass versions (such as stars) then there ought to be orbs at all masses in between provided a similar physical mechanism made both. What fundamental theorem of mathematics is being referenced in the article about the astronomical objects called brown dwarfs? What implicit assumption is being made about the sizes of astronomical objects? (For future consideration: Is the number of 'orbs' countable?)

Step-by-Step Solution

Answer

The Intermediate Value Theorem is referenced, assuming continuous mass distribution. The number of such objects would depend on formation and detection mechanisms.

Step 1: Identify the Mathematical Theorem

The article references a theorem that describes a continuous transition from one height or mass to another. This theorem is the Intermediate Value Theorem. It states that for any continuous function, for any value between the function's extremes, there exists at least one input that will produce that value. This aligns with the description of going from 3'8" to 5'8" and passing through every height in between.

Step 2: Understand the Implicit Assumption

The article assumes a continuous range of masses for astronomical objects between planets (low mass) and stars (high mass). This assumption implies that just as a person grows through all intermediate heights continuously, there should be astronomical objects at every mass between planets and stars, given a suitable physical mechanism.

Step 3: Consider Future Implications

The article suggests an infinite or continuous set of possibilities for masses, as implied by the theorem. When considering whether the number of such 'orbs' or brown dwarfs is countable, this would depend on the mechanisms of formation and detection, which aren't directly addressed by the theorem itself.

Key Concepts

continuous function

A continuous function is a core concept in mathematics, especially when discussing the Intermediate Value Theorem. Essentially, a function is continuous if you can draw its graph without lifting your pencil from the paper. For a function to be continuous, every small change in the input results in a small change in the output.
This means that as you move along the input values, the output values change smoothly without any abrupt jumps or breaks.
Imagine a ball smoothly rolling down a hill; it does not teleport from one point to another but travels through every spot in between.

• Continuity ensures predictability in real-world situations.
• It's a crucial property when modeling anything that changes gradually over time or space, such as height, temperature, or mass.

In the context of astronomical objects, this idea suggests that if you have two celestial bodies of different masses, there is a possibility of having objects with every mass in between, provided they were formed similarly.

astronomical objects

Astronomical objects encompass a wide range of entities in the universe, including planets, stars, asteroids, moons, and more. These objects vary greatly in terms of size, composition, and mass.
A fascinating aspect of these objects is how they represent stages in the continuum of cosmic scale and activity. For instance, planets and stars are both exceptionally common but differ greatly in terms of mass and the processes that sustain them.
When considering the Intermediate Value Theorem in terms of astronomy, we observe the concept of a continuous spectrum of forms and sizes.

• Planets and stars can be thought of as the low and high ends of a mass spectrum.
• Brown dwarfs fit somewhere in the middle, bridging a gap in characteristics and formation scenarios between planets and stars.
• The hypothesis suggests every possible mass might exist in some form in the universe, given appropriate conditions and physical laws.

Understanding these objects and their continuous distribution helps astronomers to better model the components and evolution of galaxies.

mass of planets and stars

The mass of planets and stars varies vastly, influencing the characteristics and behavior of each celestial body. Stars typically possess much larger masses than planets, as they undergo nuclear fusion, the process that lights them up brightly in space.
Planets, having smaller masses, do not have this energetic process, resulting in different life cycles and roles in the universe. When discussing mass in terms of the Intermediate Value Theorem:

• The theorem would imply that if low mass bodies like planets exist, and high mass bodies like stars also exist, intermediate mass objects should theoretically exist.
• Brown dwarfs are a great example, as they hold more mass than planets but not enough to sustain nuclear fusion like stars.
• This intermediate class of masses highlights the variety and complexity of objects found in the universe.

Understanding these differences is crucial for astronomers as they classify the myriad of astronomical objects, study their formation and development, and predict their future.