Question

The measured declination and inclination of the paleomagnetic field in Lower Cretaceous rocks at \(45.5^{\circ} \mathrm{N}\) and \(73^{\circ} \mathrm{W}\) are \(D=154^{\circ}\) and \(I=-58^{\circ}\). Determine the paleomagnetic pole position.

Short answer

The paleomagnetic pole position is approximately \(33.6^{\circ} \mathrm{S}\) and \(47^{\circ} \mathrm{E}\).

Step-by-step solution

Calculate the Paleolatitude

Use the inclination value to find the paleolatitude using the formula \( \tan(I) = 2\tan(\text{paleolatitude}) \). Solve for paleolatitude: \( \text{paleolatitude} = \tan^{-1}\left(\frac{\tan(-58^{\circ})}{2}\right) \). Compute this to get the paleolatitude as approximately \(-33.6^{\circ}\).

Determine the Paleolongitude

The declination, \( D = 154^{\circ} \), needs to be adjusted by adding or subtracting \(180^{\circ}\) to align with the current location longitude. For \( D = 154^{\circ} \), the corrected value becomes \(154^{\circ} + 180^{\circ} \approx 334^{\circ}\). Adjust within the \([0^{\circ}, 360^{\circ}]\) range if needed. Since the site longitude is \(73^{\circ} \mathrm{W}\), the paleolongitude is \(73^{\circ} + (334^{\circ} - 180^{\circ}) = 227^{\circ} \approx 47^{\circ} \mathrm{E} \).

Locate the Paleomagnetic Pole Position

Combine the paleolatitude and paleolongitude to express the paleomagnetic pole position. Paleolatitude indicates a southern latitude, so the paleomagnetic pole position calculated is approximately \(33.6^{\circ} \mathrm{S}\) and \(47^{\circ} \mathrm{E}\).

Key concepts

Paleolatitude calculation

Paleolatitude calculation is an essential process used to determine the ancient latitude of a location based on the Earth's past magnetic field. It employs the idea that the inclination of the magnetic field can inform us about the latitude. The inclination, denoted by \(I\), is the angle made by the magnetic field with the horizontal.
The formula used here is \( \tan(I) = 2 \tan(\text{paleolatitude}) \). This formula expresses the relationship between the Earth's magnetic field inclination and the latitude at which rocks formed. To find the paleolatitude, the solution involves rearranging the formula to solve for paleolatitude:
\[ \text{paleolatitude} = \tan^{-1}\left(\frac{\tan(I)}{2}\right) \].
For the problem given, with an inclination of \(-58^{\circ}\), we compute:
\[ \text{paleolatitude} = \tan^{-1}\left(\frac{\tan(-58^{\circ})}{2}\right) \],
resulting in a paleolatitude of approximately \(-33.6^{\circ}\). This negative value suggests the location was south of the equator at that time.

Paleolongitude determination

Determining the paleolongitude involves adjusting the measured declination to reflect the ancient geographic position. The declination, \(D\), tells us how much the magnetic north has shifted from the geographic north at a specific location and can be used to derive the paleolongitude.
First, adjust the declination by accounting for the earth's current position. This is done by adding or subtracting \(180^{\circ}\) to line up with the present-day reference. For a measured declination of \(154^{\circ}\), the corrected declination is:
\(154^{\circ} + 180^{\circ} = 334^{\circ}\).

Since this value should fit within a \([0^{\circ}, 360^{\circ}]\) range, the corrected declination remains \(334^{\circ}\).
Next, use the site’s current longitude, in this case, \(73^{\circ} \text{W}\). Since west is negative in the conventional setting, the paleolongitude calculation becomes:
\[ 73^{\circ} + (334^{\circ} - 180^{\circ}) = 227^{\circ} \].
Within the west-east spectrum, this results in \(47^{\circ} \text{E}\), indicating that the ancient longitude was shifted eastward.

Paleomagnetic pole position

Paleomagnetic pole position is a method to locate where the magnetic north or south pole was on Earth in the geologic past. This helps scientists understand continental drift and plate tectonics by providing insights into how landmasses have shifted over millions of years.
In the context of the exercise, combining the previous calculations gives us the paleomagnetic pole's position. The paleolatitude calculation pointed to \(-33.6^{\circ}\), confirming a southern location. Similarly, the determined paleolongitude \(47^{\circ} \text{E}\) identified the eastward position on the globe.

Thus, the paleomagnetic pole was located approximately at \(33.6^{\circ} \text{S}\) and \(47^{\circ} \text{E}\). This indicates that the measured location was once positioned more southward and eastward relative to today, reflecting ancient continental positions.