The auroral zone is defined by the last field line from the Earth that returns to the Earth. If the field of the earth extends no further than 10RE in the sunward direction, at what angle (in degrees) in the picture above does the last field line that returns to the earth on the sunward side leave the polar regions of the earth at 1RE, assuming that the fields are always described by r=R0sin2(θ)? In the drawing, this is equivalent to asking what is the angle with respect to the vertical of a line from the center of the earth to the point that the red curve intersects the light blue circle.
To determine the angle at which the last field line that returns to the Earth leaves the polar regions on the sunward side, we need to find the intersection point between the red curve and the light blue circle and measure the angle with respect to the vertical.
To find the intersection point, we'll solve the equation of the red curve and the equation of the light blue circle simultaneously.
The equation of the red curve can be represented as:
r = R0 * sin^2(θ)
The equation of the light blue circle can be represented as:
r = 1RE
To solve for the intersection point, we equate the two equations:
R0 * sin^2(θ) = 1RE
Simplifying the equation and solving for θ:
sin^2(θ) = 1RE / R0
θ = arcsin(sqrt(1RE / R0))
Once we have the value of θ, we can measure the angle with respect to the vertical using a protractor or by calculating the difference between θ and 90 degrees.
Please note that the values of R0 and RE should be provided in order to obtain a specific solution.