Understanding Latitude Differences: Geographic vs. Geocentric

This question asks about the point on Earth where the difference between geographic latitude and geocentric latitude is greatest. To solve this, we need to understand what these two types of latitude represent and how the Earth's shape affects them.

Defining Geographic and Geocentric Latitude

• Geocentric Latitude ($\phi_c$): Imagine a line drawn from the exact center of the Earth to a point on the surface. Geocentric latitude is the angle this line makes with the equatorial plane. It assumes the Earth is a perfect sphere.
• Geographic Latitude ($\phi_g$): This is also called geodetic latitude. Instead of a line from the center, it uses a line that is perpendicular (normal) to the Earth's surface at that point. This line is then measured relative to the equatorial plane.

Why is there a Difference? Earth's Shape

The Earth isn't a perfect sphere. It bulges slightly at the equator and is flattened at the poles, forming an oblate spheroid. Because of this bulge, the line perpendicular to the surface (the normal) at most points doesn't point directly to the Earth's center. This difference in direction is what causes the difference between geographic and geocentric latitudes.

Mathematical Relationship and Maximum Difference

The relationship between geographic latitude ($\phi_g$) and geocentric latitude ($\phi_c$) for an oblate spheroid is approximately:

$$ \tan(\phi_c) = (1 - e^2) \tan(\phi_g) $$

Here, '$e^2$' represents the eccentricity squared of the Earth's shape, a small value (around 0.0067). The difference is $\Delta\phi = \phi_g - \phi_c$.

Calculus can be used to find the latitude where $\Delta\phi$ is maximum. The maximum difference occurs when:

$$ \tan(\phi_g) = \frac{a}{b} $$

Where '$a$' is the Earth's equatorial radius and '$b$' is its polar radius. For the Earth, this condition gives a geographic latitude ($\phi_g$) of approximately $45.1^\circ$.

Analyzing the Options

Let's see how the difference behaves:

• At $0^\circ$ latitude (Equator): The surface is normal to the radius, so $\phi_g = \phi_c = 0^\circ$. The difference is $0^\circ$.
• At $90^\circ$ latitude (Poles): The surface is normal to the radius, so $\phi_g = \phi_c = 90^\circ$. The difference is $0^\circ$.
• At $45^\circ$ latitude: Here, the difference between $\phi_g$ and $\phi_c$ is significant. As calculated, the theoretical maximum difference occurs around $45.1^\circ$.
• At $23.5^\circ$ latitude: A difference exists, but it's less than the difference at $45^\circ$.

Conclusion

The difference between geographic and geocentric latitudes starts at zero at the equator, increases as you move towards the poles, reaches its maximum value around $45^\circ$ latitude, and then decreases back to zero at the poles. This behavior is a direct consequence of the Earth's shape as an oblate spheroid.