Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

A bit of a difficult exercise. For any location, \(\lambda\) on Earth, let Annual Daytime at \(\lambda, A D(\lambda),\) be the sum of the lengths of time between sunrise and sunset at \(\lambda\) for all of the days of the year. Find a reasonable formula for \(A D(\lambda)\). You may guess or find data to suggest a reasonable formula, but we found proof of the validity of our formula a bit arduous. As often happens in mathematics, instead of solving the actual problem posed, we found it best to solve a 'nearby' problem that was more tractable. The \(365.24 \ldots\) days in a year is a distraction, the elliptical orbit of Earth is a downright hinderance, and the wobble of Earth on its axis can be overlooked. Specifically, we find it helpful to assume that there are precisely 366 days in the year (after all this was true about 7 or 8 million years ago), the Earth's orbit about the sun is a circle, the Earth's axis makes a constant angle with the plane of the orbit, and that the rays from the sun to Earth are parallel. We hope you enjoy the question.

Step-by-Step Solution

Answer

The formula for annual daytime at any latitude is approximately 4392 hours under the given assumptions.

Step 1: Understand the Assumptions

We assume a simplified model of the Earth where there are 366 days in a year, the Earth's orbit around the sun is circular, Earth's axis tilt is constant, and sunlight rays are parallel. This ignores variations like the elliptical orbit and axial precession, assuming regularity in day length variations.

Step 2: Define the Problem Mathematically

We need to determine the total daylight hours at a given latitude \( \lambda \) over the entire year, which we express as a function \( A D(\lambda) \). Daylight hours depend on the angle of Earth's tilt and the latitude, varying as the Earth orbits the Sun.

Step 3: Consider the Circular Orbit Assumption

In a circular orbit, Earth's position defines a day of the year, and due to the constant axial tilt, the daylight change is periodic with the maximum and minimum daylight occurring at solstices. The daylight duration can thus be modeled by a sinusoidal function of Earth's orbit position around the Sun.

Step 4: Formulate the Daylight Equation for a Single Day

For simplicity, consider the daylight at a specific day \(D\) in the year. The daylight duration at latitude \(\lambda\) is affected by the tilt angle \(\theta\) and can be formulated as \( T(D, \lambda) = 12 + 12 \cdot \sin( (2\pi D / 366) + f(\lambda) ) \), where \( f(\lambda) \) accounts for latitude effect.

Step 5: Calculate the Annual Daytime

The annual daytime \( A D(\lambda) \) is the sum of daily daylight hours across all 366 days, given by \( A D(\lambda) = \int_0^{366} T(D, \lambda) \, dD \). Substituting \( T \) gives \( A D(\lambda) = \int_0^{366} 12 + 12 \cdot \sin( (2\pi D/366) + f(\lambda) ) \, dD \).

Step 6: Simplify the Integral

The integral can be simplified using properties of periodic functions. Since the sine function is periodic and symmetrical, the integral of the sine function over a full period (one year) around a complete cycle is zero, simplifying the equation to \( A D(\lambda) = 366 \times 12 = 4392 \) hours.

Step 7: Write the Final Formula

Based on the circular orbit and symmetrical day length model, the reasonable formula for annual daytime at latitude \( \lambda \) is \( A D(\lambda) = 4392 \) hours, independent of latitude due to our assumptions.

Key Concepts

Circular Orbit Assumption

When we talk about the Circular Orbit Assumption in mathematical modeling of Earth's sunlight exposure, we mean simplifying Earth's path around the Sun as a perfect circle.
This helps sidestep the complexities introduced by Earth's true elliptical orbit, which can affect the duration of daylight. With a circular path, we anticipate uniform movement of Earth around the Sun, making calculations more predictable.
It's an essential tool for simplifying the problem, as it assumes that the pace of seasonal changes and variations in day length follow a consistent pattern throughout the year.

Daylight Duration Equation

The core of finding how much daylight a location receives involves working with a Daylight Duration Equation. This equation estimates hours between sunrise and sunset for any day of the year.
In the simplified Earth model, this can be described using trigonometric functions due to the periodic nature of daylight changes through the year.
The key equation for a given day is formulated as \( T(D, \lambda) = 12 + 12 \cdot \sin( (2\pi D / 366) + f(\lambda) ) \).
Here, the term \( \sin(... ) \) signifies the oscillating change in daylight, while \( f(\lambda) \) adjusts for the latitude-specific effect on daylight patterns.

Latitude Effect

Latitude plays a crucial role in how much daylight different places on Earth receive across the year. This is known as the Latitude Effect.
Simply put, locations farther from the equator experience bigger differences in day length between summer and winter, as the angle of sunlight changes more dramatically with the seasons.
In our equation, this is represented by the function \( f(\lambda) \), which adjusts the daylight duration for varying latitudes.
This reflects the fact that the same latitude at opposite times of the year will have significantly different daylight durations, resulting from the way Earth tilts relative to the Sun.

Periodic Functions

In our study of daylight duration, periodic functions play a vital role. These functions repeat their values in regular intervals, much like the changing lengths of day and night.
The sine function used in our model equation is the perfect example, representing the natural periodic fluctuations seen throughout the year.
This mirrors how, despite day-to-day changes, daylight and night follow a continuous cyclical pattern that repeats yearly.
Periodic functions make it possible to handle these cyclical changes mathematically, allowing predictions of day lengths across different days with accuracy.

Simplified Earth Model

The Simplified Earth Model allows us to make easier calculations by stripping away some of the more erratic natural behaviors of the planet.
In this model, we ignore certain things like the actual elliptical shape of the orbit or the wobble of Earth on its axis.
Instead, we assume a constant orbital distance and a non-changing tilt of the axis, simplifying equations and making them more straightforward.
This approximation makes it easier to generalize daylight calculations for educational purposes, even though it might not capture every nuance of real-world Earth parameters.