Physics Problem: Estimating Solar Power

The Sun (in our solar system) has a power output of 3.9 x 1026 Watts. The Earth orbits in a circular orbit with a radius of 1.5 x 1011 meters. The Earth’s axis of rotation is tilted 23 degrees relative to the plane of the orbit.
What power strikes a 1 m2 solar panel a point R?
I'm working through chapter summaries for introductory physics (algebra-based). I'm using the Openstax online (free) textbook College Physics. You can access this book here openstax.org/details/books/co...
Here is the playlist with all the videos and problems for this course.
• Online Physics 2 2024

[@fizixx]

That's funny that you're talking about the sun. My parents said they always liked to call me 'sun', because I was so bright. ☀ 😄 🥸

[@kthwkr]

A typical fossil fuel plant has 1,000 MegaWatt turbine generators. But that's power. Energy is power times time. So energy is MegaWattHours(MWHrs). A fossil fuel plant runs 24 hours per day, 365 days a year. So that turbine outputs 1,000x24x365MWHrs energy per year. To get the same energy output from a solar facility it will need to be about 100 sq miles. But a typical US state might have 50 1,000MW turbines. So 100sq miles x 50 = 5,000 sq miles. That's a lot of land to be covered up and have absolutely nothing growing on it.
To duplicate my calculations be sure to take into account that a solar facility has to have access roads between the rows of panels. And those roads are about the same width as the panels. So immediately half the energy is lost to the road instead of to solar panels. Also, integrate the curve of the incident energy during the day. It goes up and then down sort of like a bell curve shape. Integrate the area under that curve. That's the energy. Divide by the max watt rating of the panels. That gives you a time. Now you can construct a rectangular box that is max watt high and about 4 hours wide. Now we don't have to do any more integral calculus. Just assume 4 hours per day at max watt. Now examine the weather for the year. Do an integral for the year. Construct another rectangular box. And in my state it's about 250 clear days per year out of 365. End result is a 100 megaWatt solar facility produces about the same energy per year as a 100/9MegaWatt fossil fuel turbine.
5,000 sq miles is nearly 10% of the land area of my whole state. Most people will go crazy at the idea of covering up 10% of the land of the state and removing from that area all plant life.