Distances to Venus The elongation αof a planet is the angle formed by the planet, earth, and sun

Distances to Venus The elongation αof a planet is the angle...

Key Concepts

Planetary Geometry

Planetary geometry involves the application of geometric and trigonometric methods to understand the spatial relationships and motions of planets in the solar system. This includes analyzing angular separations, such as elongation angles, and using laws like the Law of Cosines to determine distances between celestial bodies based on their positions relative to one another.

Law of Cosines

The Law of Cosines is a fundamental relation in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. In the context of astronomy, this law is often applied to triangular configurations formed by the Sun, Earth, and a planet in order to calculate unknown distances, such as the distance from Earth to the planet when certain angles or distances are known.

Astronomical Unit (AU)

The astronomical unit (AU) is a standard unit of measurement used in astronomy, defined as the average distance between the Earth and the Sun. It is approximately 149.6 million kilometers. The AU provides a convenient scale for measuring and comparing distances within the solar system, making it useful for problems involving planetary distances and orbital mechanics.

Elongation Angle

The elongation angle is the angle between the line connecting the observer (Earth) to a planet and the line connecting the observer to the Sun. It is a crucial concept in observational astronomy, as it helps determine the relative positions of a planet with respect to the Sun and Earth. This angle is particularly significant when studying the apparent paths of the inner planets in the sky.

Transcript

00:01 So basically our picture kind of looks like this, where we have, let's say that's where venus is, but venus might also be here.

00:16 So there are two tracks where it could be, because we're going to have that short side, long side angle situation.

00:23 Sun is here, earth is here.

00:25 And we know the distance from e to s is one au, astronomical unit.

00:32 And we know that the distance from venus to the sun is 0 .723aus.

00:40 And again, we're going to have two settings in which that would take place.

00:46 And we'll see what these distances are from earth.

00:52 So we want to find this distance, d1, from earth to venus, or from earth to venus when it's in our further position.

01:01 And we know that this angle is 39 .4 degrees for this particular problem.

01:10 And so we can see that we don't have any right triangle situation.

01:13 So we're going to have to use the law of signs.

01:16 And what can i find? i can find i know this side and i don't know the angle opposite.

01:25 And i know this side and i know the angle opposite.

01:29 So i can use law of signs because i have a side angle pair.

01:32 That's the way i like to say it.

01:34 We have a side and the angle opposite, and we have a known side, but we don't know the angle opposite.

01:40 So we're happy to find these two angles first.

01:42 So let's say we have the sign of angle v is to its side opposite, and its side opposite is actually this one, a u, and then we have the sign of the 39 degree, 0 .4 degree angle, and the side opposite that is that 0 .7 .2.

02:03 Aus.

02:04 So the sign of 34 .4 degrees over .723 aus.

02:13 And so when we do this quotient and we end up, we get that the sign of angle v ends up coming out to be, i believe it's 0 .879 and hit our inverse sign.

02:26 And our inverse sign is going to give us one of the answers and then we have to be smart to find the other.

02:32 So when we take the inverse sign of this value, we're going to get an angle for a measure of angle v, and we're going to get 61 .39 degrees.

02:43 Make sure your calculator was in degree mode.

02:45 Now, which of these angles am i, did i just find? i found this one.

02:49 I'll mark it with a star.

02:54 But there's another one, and we want this one as well.

03:00 And that's that angle b that's red.

03:02 And it is the supplement.

03:04 If i take that 180 degrees and subtract away that 61 .39 degrees, that's this little red angle right here.

03:13 And that little red angle comes out to be 118 .61 degrees.

03:21 So we have our two triangle settings.

03:23 We're going to have, i'll just label this one to be a red one...