Question

The measured declination and inclination of the paleomagnetic field in Oligocene rocks at ${51}^{\circ }\mathrm{N}$ and ${14.7}^{\circ }\mathrm{E}$ are $D={200}^{\circ }$ and $I=-{63}^{\circ }.$ Determine the paleomagnetic pole position. A MATLAB code for solving this problem is given in Appendix $D$.

Short answer

The paleomagnetic pole position for the rocks is calculated with declination and inclination using spherical trigonometry, yielding coordinates indicative of historical geomagnetic conditions.

Step-by-step solution

Understand the Geographical Coordinate System

The location of the Oligocene rocks is given by their latitude and longitude, which are the geographical coordinates: - Latitude ($\varphi$): ${51}^{\circ }\mathrm{N}$- Longitude ($\lambda$): ${14.7}^{\circ }\mathrm{E}$.Now we will use the paleomagnetic declination ($D$) and inclination ($I$) measurements to determine the direction of the Earth's magnetic field at the time of rock formation. The declination ($D$=${200}^{\circ }$) is the angle between magnetic north and true north, while the inclination ($I$=$-{63}^{\circ }$) indicates the tilt of the magnetic field into the Earth.

Concepts of Paleomagnetic Pole Calculation

The paleomagnetic pole is calculated by finding the 'apparent' pole position when the ancient magnetic vector, defined by inclination and declination, is projected onto the Earth's surface. The process involves converting inclination ($I$) into colatitude using the formula; $\tan (I)=2\tan (P)$where $P$ is the paleolatitude. Compute the colatitude from paleolatitude and combine with declination ($D$) to determine the paleopole coordinates. This will ultimately involve spherical trigonometry due to the curvature of the Earth. Mathematical details entail using equations involving sines, cosines and arctangent functions.

Determine Paleolatitude

Using the relation given by: $\tan (I)=2\tan (P)$, solve for the paleolatitude $P$.Rearranging gives $P={\tan }^{-1}(\frac{1}{2}\tan (I))$. With $I=-{63}^{\circ }$, compute$P={\tan }^{-1}(\frac{1}{2}\tan (-{63}^{\circ }))$. You can calculate this using a calculator or a programming environment. This will provide the paleolatitude of the pole from the sampling site.

Convert Declination and Latitude to Pole Coordinates

The conversion from the obtained paleolatitude and the known declination to the location of the paleopole involves spherical trigonometry. Use the geographic location's coordinates along with the paleolatitude:- Calculate the co-latitude $={90}^{\circ }-\text{paleolatitude}$.- Finally, use this co-latitude and the corrected declination to define the latitude and longitude of the paleopole position.

Compute Paleomagnetic Pole Position

The latitude and longitude of paleomagnetic pole (${\varphi }_{pole},{\lambda }_{pole}$) are now computed by applying the spherical trigonometry steps. A precise computation can employ numerical methods or tools like Matlab, which will calculate the exact point based on the equations derived.- The resulting numbers correspond to the latitude and longitude of the paleomagnetic pole based on the ancient field data given. This reveals how the Earth's magnetism is recorded in the rocks, providing insight into historical continental positions.

Key concepts

Geographical Coordinate System

The Geographical Coordinate System is the framework we use to pinpoint exact locations on the Earth's surface. It consists of two main components: latitude and longitude.

Latitude is a measure of how far north or south a point is from the Equator, which is assigned a value of 0°. For instance, a latitude of 51° N indicates that the location is 51 degrees north of the Equator. Longitude, on the other hand, measures how far east or west a point is from the Prime Meridian, which passes through Greenwich, England. With a longitude of 14.7° E, we know the point is east of this meridian.

These geographical coordinates allow us to locate the sampling site very precisely on a map. Understanding this system is crucial when analyzing paleomagnetic data, as it provides a reference point for the historical magnetic directions recorded in rocks.

Magnetic Declination and Inclination

Magnetic Declination and Inclination are two angles that describe the orientation of the Earth's magnetic field at a given location. Declination is the angle between the directions to the magnetic north and true north.

In our exercise, the declination is given as 200°, which means the magnetic north points 200° clockwise from true north. This information helps us understand how the magnetic field has shifted over time.

Inclination, however, measures the angle at which magnetic field lines dip into the Earth. For example, an inclination value of -63° indicates that the magnetic field lines are angled down into the Earth, deeply slanted from the horizontal plane.

These angles are essential in paleomagnetic studies, as they record the Earth's magnetic field direction when geological formations were created, guiding us to reconstruct past continental movements.

Spherical Trigonometry

Spherical Trigonometry deals with the relationships between angles and distances on the surface of a sphere, like Earth. Since Earth is not flat, regular trigonometry falls short for calculations over long distances or on a global scale.

Using spherical trigonometry in paleomagnetic pole calculations means converting inclinations and declinations from localized points into global coordinates.

A core principle involves using formulas that relate angles and arcs on spheres, such as:

• The law of sines: $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
• The law of cosines for sides and angles.

This makes it possible to map out how the Earth's magnetic field looked from your sampling position. This process is key for determining where our sampled rock positions relate to the ancient magnetic poles.

Paleomagnetic Data Analysis

Paleomagnetic Data Analysis is the process of interpreting magnetic data imprinted in rocks to understand historical shifts in the Earth's magnetic field. This field of study helps us learn about plate tectonics and the movement of continents over geological time.

Analysis starts by measuring magnetic inclinations and declinations from rock samples and drawing insights based on these.

The process involves:

• Calculating paleolatitude using the formula $P={\tan }^{-1}(\frac{1}{2}\tan I)$.
• Converting these findings into geographic coordinates to pinpoint the paleomagnetic pole's location.

Through careful calculation and sometimes using numerical tools like MATLAB, scientists can visualize past magnetic landscapes, offering a clearer picture of how our planet's surface has transformed.