Planets' Distances from the Sun
These methods (used by Copernicus) again assume that all planets have circular orbits. This is not true. But it still gives us a close answer to the average distance of each planet from the sun as the planet’s orbits are ALMOST CIRCLES.
Finding Distances for Inferior Planets (Venus and Mercury)
Tangents:
A tangent to point A on a circle is a line that just touches the circle at point A and at no other points. It can’t cross through the circle. The blue line below is a tangent to the circle at point A.
Finding Distances for Inferior Planets (Venus and Mercury)
Tangents:
A tangent to point A on a circle is a line that just touches the circle at point A and at no other points. It can’t cross through the circle. The blue line below is a tangent to the circle at point A.
A tangent line at point A will always meet a radial line at point A (a line going from the centre of the circle to point A) at right angles.
To prove this, consider this: if we tilt the blue line (pivoting on the point A) one way or the other to change the angle with the radius to something other than 90 degrees, then the line will cross through the circle, intersecting at 2 points, rather than just touching it once, and will no longer be a tangent.
OPTIONAL: Watch this video proving the 90 degree rule.
To prove this, consider this: if we tilt the blue line (pivoting on the point A) one way or the other to change the angle with the radius to something other than 90 degrees, then the line will cross through the circle, intersecting at 2 points, rather than just touching it once, and will no longer be a tangent.
OPTIONAL: Watch this video proving the 90 degree rule.
RULE: TANGENT AND RADIUS LINES MUST MEET AT 90 DEGREES.
Maximum Elongation:
Inferior planets like Venus and Mercury will have maximum elongation (appear as far from the sun in the sky as possible) when a line joining Earth to Venus is tangent to the circle.
This picture shows Venus at various points relative to Earth on its orbit. Notice that the widest angle occurs when the Earth/Venus line is tangent and NOT cutting through the circle (Venus’s orbit).
When Venus is at maximum elongation we have this scenario:
- Line a is tangent to the inner dotted circle (Venus’s orbit) touching the point where Venus is.
- The line b is the radius line of Venus’s orbit, touching the point where Venus is.
- Thus, the lines a and b must meet at right angles (90 degrees).
- The line c is the Sun-Earth line and so represents the distance between the Sun and Earth. We call this distance 1 AU (1 astronomical Unit).
- θ is the angle between the Sun and Venus in the sky as seen from Earth at this point in time. This is just the maximum elongation.
- The line b is the distance between the sun and Venus. This is what we are looking for. Let’s call it x.
Now our triangle can be redrawn with 1AU, x, and θ labelled.
Using trig ratios, we could say sin θ = x/1. We can solve for x. Our answer will be a fraction which is the distance in the unit AU.
Note our answer will tell us what fraction of the Earth/Sun distance Venus will be from the sun.
If it was 0.5, it would mean Venus orbits at half the distance compared to Earth.
If it was 0.25, it would mean Venus orbits at ¼ the distance compared to Earth.
So, AMAZINGLY, you only need to do one measurement to calculate how far Venus is away from the sun. Find the angle at maximum elongation. How do you do that? Watch Venus every day and record its elongation. Over time, you will find its maximum elongation. On Starry Night, we can just speed things up and watch the Elongation on the Info Bar (Left Hand Side). When the elongation stops getting bigger and starts getting smaller again, we know we have found the maximum elongation. Control time step by step to find the maximum elongation going backwards and forwards in time as need be. Let some Year 9 Trigonometry do the rest.
Note our answer will tell us what fraction of the Earth/Sun distance Venus will be from the sun.
If it was 0.5, it would mean Venus orbits at half the distance compared to Earth.
If it was 0.25, it would mean Venus orbits at ¼ the distance compared to Earth.
So, AMAZINGLY, you only need to do one measurement to calculate how far Venus is away from the sun. Find the angle at maximum elongation. How do you do that? Watch Venus every day and record its elongation. Over time, you will find its maximum elongation. On Starry Night, we can just speed things up and watch the Elongation on the Info Bar (Left Hand Side). When the elongation stops getting bigger and starts getting smaller again, we know we have found the maximum elongation. Control time step by step to find the maximum elongation going backwards and forwards in time as need be. Let some Year 9 Trigonometry do the rest.
Finding Distances for Superior Planets (Mars, Jupiter, Saturn etc.)
Superior Planets are a little harder because they don’t have maximum elongations. There is still a way.
Remember in what we just did, that, Venus (the closer planet) appeared at maximum elongation on Earth (the planet further away) AT THE SAME TIME that the sun and Earth appeared at 90 degrees to each other on Venus.
Remember in what we just did, that, Venus (the closer planet) appeared at maximum elongation on Earth (the planet further away) AT THE SAME TIME that the sun and Earth appeared at 90 degrees to each other on Venus.
When trying to work with Earth and Mars, Earth now takes the role of the closer planet and Mars takes the roll of the planet further away. So we know Earth will appear at maximum elongation for a Mars observer WHEN Mars and the Sun are at right angles as seen from Earth. In other words, when Mars’ elongation is 90 degrees. We can then use the same triangle idea.
Only problem will be this: we won’t know the angle theta (which will now be the elongation of the Earth as seen from Mars). However, we CAN work out the unmarked angle (bottom right corner where the sun is). Notice that this angle is the angular separation of the two planets as seen from the sun. How do we work this angle out?
Well, we know the rate at which the Earth sweeps out an angle as it orbits the sun. And we know the rate at which Mars sweeps out an angle as it orbits the sun.
Mars’ period is 687 days and so Mars sweeps out 0.524 degrees per day.
Earth’s period is 365.24 days and so sweeps out 0.986 degrees per day.
Earth is sweeping out an extra .986 - 0.524 = 0.462 degrees of orbit each day.
So if we time how long it takes between Earth and Mars being lined up (with Earth in the middle, sun and Mars are separated by 180 degrees as seen from Earth) until our triangle forms (so when sun and Mars are at 90 degrees as seen from Earth), we can simply multiply the number of days this took by 0.462.
Alternatively (and getting the same answer), we could work out how much angle Earth swept out during this time (between Mars and sun being at 180 degrees until Mars and sun being at 90 degrees), how much angle Mars swept out during this time, and find how much extra Earth swept out. This amount of course is the separation between Earth and Mars as seen from Mars.
The following picture demonstrates this (next page).
Only problem will be this: we won’t know the angle theta (which will now be the elongation of the Earth as seen from Mars). However, we CAN work out the unmarked angle (bottom right corner where the sun is). Notice that this angle is the angular separation of the two planets as seen from the sun. How do we work this angle out?
Well, we know the rate at which the Earth sweeps out an angle as it orbits the sun. And we know the rate at which Mars sweeps out an angle as it orbits the sun.
Mars’ period is 687 days and so Mars sweeps out 0.524 degrees per day.
Earth’s period is 365.24 days and so sweeps out 0.986 degrees per day.
Earth is sweeping out an extra .986 - 0.524 = 0.462 degrees of orbit each day.
So if we time how long it takes between Earth and Mars being lined up (with Earth in the middle, sun and Mars are separated by 180 degrees as seen from Earth) until our triangle forms (so when sun and Mars are at 90 degrees as seen from Earth), we can simply multiply the number of days this took by 0.462.
Alternatively (and getting the same answer), we could work out how much angle Earth swept out during this time (between Mars and sun being at 180 degrees until Mars and sun being at 90 degrees), how much angle Mars swept out during this time, and find how much extra Earth swept out. This amount of course is the separation between Earth and Mars as seen from Mars.
The following picture demonstrates this (next page).
From this, we can use trigonometry to find x, the distance between the sun and Mars.
ACTIVITY
Using Starry Night, find the distances between the sun and the planets Mercury, Venus, Mars, Jupiter, Saturn. Compare your answers with the average distances given from a reliable source. Calculate your percentage error.
Using Starry Night, find the distances between the sun and the planets Mercury, Venus, Mars, Jupiter, Saturn. Compare your answers with the average distances given from a reliable source. Calculate your percentage error.