Date modified: Wed 2023-05-24 . 04:00 PM
Date created: Sat 2023-10-07 . 09:02 PM
# Celestial navigation - Part 4 Rotation, sunrise, longitude, and time. (TO BE UPDATED) ## The rotation. As the Earth rotates, a person standing on the equator goes around in a circle with radius that is the radius of the Earth. Standing at different latitudes, however, one travels in a circle of a different radius (less than or equal to the radius of the Earth) as the Earth rotates. ![[---images/---assets/---icons/question-icon.svg]] Find the radius of this circle if one is at a latitude of $34^\circ N$ (approximate latitude of Los Angeles). Assume that the radius of Earth is 3960 miles. Use the above diagrams to help you visualize. ![[---images/---assets/---icons/question-icon.svg]] This means as the Earth rotates, one has a **different linear speed** at different latitudes, despite the angular speed of Earth is always the same (what is this angular speed approximately?) Find the linear speed for someone at the equator versus someone at a latitude of $34^\circ N$. ## Finding distance at the same latitude with sunrises. Now that we know the linear speed on Earth depending on the latitude, we can use this to discover distance between two points with the same latitude ! ![[---images/---assets/---icons/exclaim-icon.svg]] ![[---images/---assets/---icons/question-icon.svg]] Phoenix, Arizona is roughly directly due east from Los Angeles, California, they both roughly have the same latitude of $34^\circ N$. **Using the internet, find out when the sun rises in Phoenix, AZ, and when the sun rises in Los Angeles, CA for today.** Now using the estimate of Earth's radius $R_\text{Earth}\approx 3690$ miles, estimate the distance between Los Angeles and Phoenix. ![[---images/---assets/---icons/question-icon.svg]] Let us do this symbolically. Let us say you are at a location $L$ with a latitude of $\alpha$ degree north. You walk eastward or westward for $1$ mile. What is the time difference $\Delta T$ in the sunrises at these two locations? Write $\Delta T$ as a function of $L$ and $\alpha$. ## Longitude and time. Now let us briefly speak of the imaginary **longitude** lines on Earth. Unlike the latitude, which has a natural reference point for zero degrees (the equator), there is no natural choice of where "zero degree longitude", or **prime meridian** is. This prime meridian is chosen arbitrarily, and various countries did not always agree on where it is. It is now set at Greenwich, England. In Hipparchus's day he set the prime meridian that goes through Alexandria. Finding what is one's longitude relative to a reference longitude (namely, finding out how far east-west one has travelled) posed a traditionally difficult problem. Especially among the maritime powers as the need for navigation at sea. Countries set out prizes to solve the problem of determining one's longitude (relative to another), a famous one is the Longitude act of 1714 by Great Britain. And most solution involved **the precise measurement of time!** From the above exercise this might not be too surprising -- If you know the local time of a reference location, and synchronously your own local time at a particular location, then you can work out the difference in longitude.