Q. 53 Measurements indicate that Earth... [FREE SOLUTION]

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Chapter 9: Q. 53 (page 772)

Measurements indicate that Earth鈥檚 orbital eccentricity is $0.0167$ and its semimajor axis is $1.00000011$ astronomical units.
(a) Write a Cartesian equation for Earth鈥檚 orbit.
(b) Give a polar coordinate equation for Earth鈥檚 orbit, assuming that the sun is the focus of the elliptical orbit.

Short Answer

Expert verified

Part (a) The cartesian equation is $\frac{x^{2}}{(1.00000011)^{2}} + \frac{y^{2}}{(0.9998606552)^{2}} = 1$ .

Part (b) The polar equation is $r = \frac{1.00000011}{1 + 0.0167 \cos 胃}$ .

Step by step solution

Part (a) Step 1. Given information.

The eccentricity is $0.0167 .$

The semi-major axis isrole="math" localid="1649526544989" $1.00000011 astronomicalunits.$

Part (a) Step 2. Explanation.

Now,

$e = \frac{\sqrt{A^{2} - B^{2}}}{A} A = 1.00000011, e = 0.0167 0.0167 = \frac{\sqrt{(1.00000011)^{2} - B^{2}}}{1.00000011} 0.0167 (1.00000011) = \sqrt{(1.00000011)^{2} - B^{2}} 0.0167000018 = \sqrt{1.00000022 - B^{2}} (0.0167000018)^{2} = {(\sqrt{(1.00000022) - B^{2}})}^{2} 0.00027889006 = (1.00000022) - B^{2} B^{2} = 0.99972133 A^{2} = (1.00000011)^{2} = 1.00000022 T h e r e f o r e, \frac{x^{2}}{1.00000022} + \frac{y^{2}}{0.99972133} = 1 or \frac{x^{2}}{(1.00000011)^{2}} + \frac{y^{2}}{(0.9998606552)^{2}} = 1$

Part (b) Step 1. Explanation.

Now, the standard form,

$r = \frac{e u}{1 + e \cos 胃} T h e r e f o r e, r = \frac{1.00000011}{1 + 0.0167 \cos 胃}$

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Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Explanation.

Part (b) Step 1. Explanation.

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91影视

Measurements indicate that Earth鈥檚 orbital eccentricity is 0.0167and its semimajor axis is 1.00000011astronomical units. (a) Write a Cartesian equation for Earth鈥檚 orbit.(b) Give a polar coordinate equation for Earth鈥檚 orbit, assuming that the sun is the focus of the elliptical orbit.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Explanation.

Part (b) Step 1. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Logic and Functions

Probability and Statistics

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.