Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Celestial navigation - Part 3 Is the North Star always the north star? Currently the North Star that is located at the celestial north pole is Polaris, but is this always the case? In 127 BCE, the ancient Greek astronomer Hipparchus notices that stars were not always in the same position every year. It is said that he had about 700 years or so worth of data, and that he found that Earth **precesses** about $1^\circ$ every year. We will see what this means. To start, let us describe the geometry of earth's rotational axis first. ## Axial tilt of Earth's rotational axis. As the Earth orbits around the sun, we can imagine that there is a plane that contains this orbit. This plane is called the **elliptical plane** or the **orbital plane**. And as it turns out, the self-rotational axis of Earth is not perpendicular to this plane, but rather has an **axial tilt** at about $23.4^\circ$ from the **normal** (or the **orbital pole**) to the elliptical plane: ![[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 13.34.39.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 13.34.39.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Celestial_navigation_3 2023-05-22 13.34.39.excalidraw.dark.svg|dark exported image]]%% ![[---images/---assets/---icons/question-icon.svg]] **Summer solstice** is (approximately) the point in the year where that hemisphere experiences the **longest day**, while **winter solstice** is the point where the hemisphere experience the **shortest day**. In between the solstices, there are two **equinox** points, a spring and autumn one. During an equinox, the day and night are approximately the same duration (equi = equal, nox = night). In above diagram, which of the four labeled points $A,B,C,D$ represent summer solstice, winter solstice, spring equinox, and autumn equinox for **northern hemisphere**? What about for **southern hemisphere**? The axial tilt of Earth is responsible for season's on Earth! During the solstices, the axial tilt is tilting directly towards (or away from) the sun. ![[---images/---assets/---icons/question-icon.svg]] (Challenge) Los Angeles has a latitude of about $34^\circ$N. Approximate the length of day during summer solstice. What about on winter solstice? Hint: The following diagram may help you. Draw in a line that represents the latitude $34^\circ$N. Fill in all the angles that you know. Think about when does night time begin, and that at $34^\circ$N, the circle it travels has a different radius than that of the equator. How much of that small circle is spent in darkness and in day? ![[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 14.47.57.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 14.47.57.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Celestial_navigation_3 2023-05-22 14.47.57.excalidraw.dark.svg|dark exported image]]%% Look up on the internet to see how close your prediction is. (Search for summer solstice for Los Angeles, and see how long that day is.) ## Axial precession. The rotational axis of Earth actually doesn't always point in the same direction, but rather it slowly sweeps out a cone shape. This motion is called **precession**, it wobbles in addition to rotation: ![[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 15.15.12.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 15.15.12.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Celestial_navigation_3 2023-05-22 15.15.12.excalidraw.dark.svg|dark exported image]]%% So for now the celestial north pole is pointing at Polaris, but it will be gradually pointing at different stars (if there is one!) The brightest start approximately opposing Polaris on this precession path is Vega of Lyra. ![[---images/---assets/---icons/question-icon.svg]] Why does spinning around a tilted axis around sweeps out a cone? Can you visualize or manifest why this is true? ![[---images/---assets/---icons/question-icon.svg]] Using Hipparchus's estimate of about $1^\circ$ of precession every 70 years, approximately how long would it take for Earth's rotational axis to precess one full cycle? ![[---images/---assets/---icons/question-icon.svg]] What would this precession mean for the seasons on Earth for each hemisphere? ![[---images/---assets/---icons/question-icon.svg]] The distance from Earth to Polaris is about 323 light-years away. If we imagine Earth's precession cone where the base touches Polaris and the vertex is at Earth, what is the **volume** of this cone? (Hint: Again use the axial tilt of $23.4^\circ$) For curiosity, here is the imaginary circular path traced out by Earth's celestial north pole as the rotational axis precesses, going through various constellations and stars. It shows which stars will be our next "North Star". The central cross shows the position of the orbital pole. ![[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 15.51.48.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-22 15.51.48.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Celestial_navigation_3 2023-05-22 15.51.48.excalidraw.dark.svg|dark exported image]]%% ![[---images/---assets/---icons/question-icon.svg]] Approximately how many years from now will the North Star be approximately Vega of Lyra? Approximately how many years ago was the North Star Thuban of constellation Draco? ![[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-23 04.47.28.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 2/---files/Celestial_navigation_3 2023-05-23 04.47.28.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Celestial_navigation_3 2023-05-23 04.47.28.excalidraw.dark.svg|dark exported image]]%%