Mars's orbit is quite elliptical, so at different times when...

Name: Problem 1, Chapter 40: Pathways to Astronomy
Uploaded: 2020-07-23T20:01:53-08:00
Duration: 5 min 9 s
Channel: Eric Mockensturm
Description: Mars's orbit is quite elliptical, so at different times when it is in opposition, its distance from Earth can vary substantially. From the data given in this chapter, estimate the largest and smallest distance to Mars at opposition and when it is on the far side of the Sun. Assume that the Earth's orbit is circular at 1 AU.

Mars's orbit is quite elliptical, so at different times when it is in opposition, its distance from Earth can vary substantially. From the data given in this chapter, estimate the largest and smallest distance to Mars at opposition and when it is on the far side of the Sun. Assume that the Earth's orbit is circular at 1 AU.

Key Concepts

Astronomical Unit (AU)

An astronomical unit (AU) is a standard unit of measurement in astronomy defined as the average distance between the Earth and the Sun. It is used to express distances within the solar system, allowing for easier comparison and calculation of planetary distances relative to the Earth.

Perihelion and Aphelion

These terms refer to the closest and farthest points in a planet's elliptical orbit around the Sun, respectively. At perihelion, a planet is closest to the Sun, while at aphelion, it is furthest. These points determine the range of distances that a planet, such as Mars, will have relative to Earth during different alignments.

Heliocentric and Geocentric Distances

This concept involves understanding the measurements of distances from the Sun (heliocentric) versus from the Earth (geocentric). Calculating geocentric distances, especially when a planet is either in opposition or on the far side of the Sun, requires combining its orbital distance from the Sun with Earth’s orbital position.

Elliptical Orbit

An elliptical orbit is the path followed by a planet around the Sun that is not a perfect circle but an ellipse. This shape causes the distance between the planet and the Sun to vary throughout its orbit, influencing observable planetary distances from Earth at different times.

Opposition

Opposition occurs when a planet is directly opposite the Sun in the sky as seen from Earth. This alignment generally results in the planet being at its closest approach, especially when its position in its elliptical orbit coincides with its nearest point to the Sun.

Transcript

00:01 In this problem, we're thinking about how to get, um, produce a transfer orbit from the earth to mars.

00:10 So in this case, our, the sun is our central body, and it is here.

00:17 And then we have earth's orbit here, and mars is orbit here.

00:20 And we're assuming that they're, that they're circular and co -planar, which is not exactly the case.

00:28 And we know that the mean distances, which we'll use as a radius from the sun, are this much for earth and this much for mars.

00:41 So we need to, basically, we want to launch a spacecraft here that then intercepts mars at this point.

00:50 And we need to figure out how much energy we need to add because it's now, it's, this whatever body is here, is orbiting.

01:00 The sun in a circular orbit and we need to now make it orbit the sun in a polybine elliptic orbit so that it reaches out here and then we need to change the speed here so that it goes now into a circular orbit with mars or reduce the speed so that it matches that of mars.

01:21 So what we have as a kinetic energy at a and again it's just circumferential at both a and b so we don't have any radial components because these are the maximum or minimum of the or the maximum here and the minimum here of the elliptic orbit.

01:38 We have the potential energy at a, kinetic energy at b and the potential energy at b.

01:45 Again, we're not worried, we're not going to worry about the gravity of the planet in this case.

01:52 So we have our conservation of angle momentum.

01:58 So all the force on this body acts just through the, through the center of the sun at all points.

02:06 Obviously as it gets closer and closer to mars and farther and farther and and it's close to earth, that's not going to be a good assumption.

02:15 But again, we're just using a first approximation to try to figure out exact kind of orders of magnitude here...

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