Satellite Problem...?

brokenwatch

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Jul 18, 2019
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The problem:
A satellite in synchronous orbit with the earth (it always stays above a fixed point on the earth's surface)
has a radius of orbit 6.5 times the earth's radius (3950 miles).

The satellite relays a signal from one point on earth to another on earth (so, in a distance that's an arc).

The sender & receiver signal must be within a line of sight w/the satellite.

What's the maximum (arc) distance on the surface of the earth between the sender & receiver for this satellite? Round to the nearest mile.

My work:
The earth's radius is r=3950, and the distance of the orbit radius extended all the way to the center of the earth is R=3950+(6.5*3950)=29625.
The satellite's lines of sight over the earth are tangent to it, so touch at 90 degree angles with two radii from Earth's center to those points.
Two right triangles are formed by the lines of sight, the two radii, and the distance from the satellite to the earth's center, where R=29625 is the hypotenuse of both triangles.

So, to find the arc distance between the two points (where the lines of sight touch Earth) I used:
s = 3950(2cos^-1(3950/29625)), which is approx. 11,353 miles.

Now, the problem is, the answer is approximately 11,189 miles. How am about 164 miles off?
 

lev888

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If satellite's radius of orbit is 6.5 times the earth's radius, then why do we need to add 3950 to 6.5*3950? If it's a radius it already corresponds to the distance to the earth's center.
 

brokenwatch

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The diagram I'm looking at has a line that extends from the satellite above the earth to its center/core. It labels the portion of the line from the satellite to the point on the earth's surface as "r x 6.5." Of course, the length of the line in full would be r + (r x 6.5).

"r x 6.5" is how far above earth the satellite is. It calls this the satellite's "radius of orbit." Sorry about the confusion. It's not my words. I just ignored that to solve the problem. The only radius of a circle I see is that of the earth's radius, which is given as r=3950 mi.
 

lev888

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Whoever made the diagram doesn't understand the difference between orbit altitude and orbit radius. I plugged in 6.5*3950 instead of 29625 in your formula, got 11189.06 on wolframalpha.com.
 

brokenwatch

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Ah, thank you for breaking that down. A simple curly bracket along the length of the line to reference its entirety in the diagram would have made it less confusing, as I've no great depth of experience with astronomical topics.
Thanks again!
 

lev888

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Welcome.

I've no great depth of experience with astronomical topics.
Thanks again!
Neither do I. But radius has a meaning in math and you don't need a diagram to know that it's the distance from a point on a circle to its center.
 

wolf

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The distance from the center of the Earth to the satellite would be 25,675 miles and not 29,625.
The problem states that the satellite "has radius of orbit 6.5 times the earth's radius (3950 miles)", which to me means 6.5 * 3950 equals 25.675.


25675​
 

brokenwatch

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Neither do I. But radius has a meaning in math and you don't need a diagram to know that it's the distance from a point on a circle to its center.
What about the radius (one of two), such as the semimajor axis, of an ellipse? As in a satellite in orbit above Earth?
Or, regular polygons' radii (extending from the center to a vertex)?
 

brokenwatch

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The distance from the center of the Earth to the satellite would be 25,675 miles and not 29,625.
The problem states that the satellite "has radius of orbit 6.5 times the earth's radius (3950 miles)", which to me means 6.5 * 3950 equals 25.675.


25675​
Awesome, wolf! It seems we all agree on 25675 miles, then. 😊
 

wolf

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Okay, brokenwatch, I decided NOT to leave in the middle of all this. (See attached graphic)

sine (A) = 3,950 / 25,675
sine (A) = 2 / 13
sine (A) = 0.1538461538
arc sine (0.1538461538) = 8.8499 Degrees = Angle A

Therefore Angle B = 81.1501 Degrees

Central Angle = 162.3002
Central Angle / 360 = 0.4508338889

Circle Circumference = 2 * PI * 3,950 = 24,818.58 miles

So, the arc distance = 0.4508338889 * 24,818.58 miles = 11,189.06 miles

Which appears to be exactly the answer that lev888 said.
 

Attachments

brokenwatch

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Okay, brokenwatch, I decided NOT to leave in the middle of all this. (See attached graphic)

sine (A) = 3,950 / 25,675
sine (A) = 2 / 13
sine (A) = 0.1538461538
arc sine (0.1538461538) = 8.8499 Degrees = Angle A

Therefore Angle B = 81.1501 Degrees

Central Angle = 162.3002
Central Angle / 360 = 0.4508338889

Circle Circumference = 2 * PI * 3,950 = 24,818.58 miles

So, the arc distance = 0.4508338889 * 24,818.58 miles = 11,189.06 miles

Which appears to be exactly the answer that lev888 said.
Sweet, wolf! 😎 Cool diagram. Aye. We all agree on 25,675mi for the "orbit radius" and about 11,189 miles for the arc distance, then.
 

wolf

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brokenwatch, I'm glad you like the diagram. I have a website www.1728.org for math and science help and I make all the diagrams with Microsoft Paint - it's pretty good considering it comes free with Windows.
So, I guess we finally have the answer? Great!!!
 
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