Heliocentric polar orbit crossing the Earth's orbit twice

In summary, the problem is that in order for a space probe to have a highly eccentric orbit and still "meet" the Earth twice every time it travels along the perihelion, it would need to have a nearly circular orbit similar to that of Earth. This is due to the fact that the Earth's orbit is slightly eccentric and the perihelion of the probe's orbit must be at a distance of 1 AU in order to cross the Earth's orbit. Additionally, a highly eccentric orbit out of the plane of the planets with the same semi-major axis as Earth cannot cross the Earth's orbit twice. Therefore, it is not possible for the probe to have the desired orbit as described in the conversation.
  • #1
xpell
140
16
Sorry, the title's length didn't allow me to explain this better and I need it for a story that I'm writing, if you're so kind to help me. I've been trying it hard to solve it myself but I've been unable to.

The problem looks simple but it isn't (for me): Please assume we have a space probe of negligible mass in a highly eccentric, heliocentric polar orbit (so it doesn't experience heavy perturbation caused by other solar system bodies; let's say it "goes up and down" the Sun.) I need this probe to "meet" (experience a close encounter with) the Earth twice every time it travels along the perihelion (i.e. 6 months of travel between each extreme of the Sun-focus latum rectum), like in this drawing:
heliopolar6months.jpg


Could anyone please help me to solve the following:?
  • Dimensions of the ellipse (major and minor axis and/or eccentricity);
  • Total orbital period;
  • Maximum velocity at the perihelion; and, most important:
  • Velocity (relative to the Earth) while "crossing" the Earth's orbit "upwards" and, 6 months later, "downwards."
Thank you in advance for any help! (And please move if this is not the appropriate forum...)
 
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  • #2
In an eccentric orbit, the probe would have both a shorter path and a higher speed than Earth. To make that work, the probe needs a nearly circular orbit similar to the one of Earth.

The close encounters with Earth will make the orbit unstable.
 
  • #3
Thank you, Mfb!

mfb said:
The close encounters with Earth will make the orbit unstable.

Let's assume that the probe has some maneouvering/stationkeeping capability to compensate this, please.

mfb said:
In an eccentric orbit, the probe would have both a shorter path and a higher speed than Earth. To make that work, the probe needs a nearly circular orbit similar to the one of Earth.

Heck, I didn't think so, but you're most probably right... is there any formula or something that I could use to calculate that, please?

And no highly eccentric orbit that would cover the "perihelion leg" in such a time because of a very high velocity (even if the perihelion is quite far away!)?
 
  • #4
Note: the Earth has a slightly eccentric orbit (0.017), I'll ignore this here, it does not change the conclusions.
xpell said:
And no highly eccentric orbit that would cover the "perihelion leg" in such a time because of a very high velocity (even if the perihelion is quite far away!)?
The perihelion has a maximal distance of 1 AU, otherwise your orbit doesn't cross the one of Earth. A circular orbit with a radius of 1 AU matches your required time, everything else does not.

There are formulas to calculate things like the time spent between two points of the orbit, and various websites show to calculate that. It's not necessary in this case.

Unrelated: to make it worse, to match the orbital period, the semi-major axis has to be the same. An eccentric orbit out of the plane of the planets with the same semi-major axis as Earth cannot cross its orbit twice.

xpell said:
Let's assume that the probe has some maneouvering/stationkeeping capability to compensate this, please.
Hmm...
 
  • #5
mfb said:
Note: the Earth has a slightly eccentric orbit (0.017), I'll ignore this here, it does not change the conclusions.
The perihelion has a maximal distance of 1 AU, otherwise your orbit doesn't cross the one of Earth. A circular orbit with a radius of 1 AU matches your required time, everything else does not.

There are formulas to calculate things like the time spent between two points of the orbit, and various websites show to calculate that. It's not necessary in this case.

Of course you're right, Mfb. Silly me didn't notice on the spot that my only possible orbit is another "Earth's orbit" more or less inclined relative to the ecliptic, does it?

mfb said:
Unrelated: to make it worse, to match the orbital period, the semi-major axis has to be the same. An eccentric orbit out of the plane of the planets with the same semi-major axis as Earth cannot cross its orbit twice.

Hmm...

Just out of curiosity, would you be so kind to elaborate this a bit more please? I'm interested!
 
  • #6
xpell said:
my only possible orbit is another "Earth's orbit" more or less inclined relative to the ecliptic, does it?
Right.
xpell said:
Just out of curiosity, would you be so kind to elaborate this a bit more please? I'm interested!
Draw an ellipse and a circle with the same semi-major axis and focal point on a plane. They will intersect each other twice, but the intersection points are not on opposite sides of the central body. There is no way to rotate the ellipse out of the plane while keeping the two intersection points. You can rotate around the (sun - one intersection) axis, but then the second intersection goes away.
 
  • #7
mfb said:
Draw an ellipse and a circle with the same semi-major axis and focal point on a plane. They will intersect each other twice, but the intersection points are not on opposite sides of the central body. There is no way to rotate the ellipse out of the plane while keeping the two intersection points. You can rotate around the (sun - one intersection) axis, but then the second intersection goes away.
Done and understood. Thank you very much, Mfb. :)
 
  • #8
xpell said:
The problem looks simple but it isn't (for me): Please assume we have a space probe of negligible mass in a highly eccentric, heliocentric polar orbit (so it doesn't experience heavy perturbation caused by other solar system bodies; let's say it "goes up and down" the Sun.) I need this probe to "meet" (experience a close encounter with) the Earth twice every time it travels along the perihelion (i.e. 6 months of travel between each extreme of the Sun-focus latum rectum), like in this drawing:

Can't happen. If it's orbital period is half the Earth's orbital period, then the orbit has to be smaller and perigee will be closer to the Sun than the Earth.

You could set up an orbit that has half the Earth's orbital period and is close to the Earth at apogee. But every other orbit, the Earth won't be there - the Earth will be on the other side of the Sun.

As mentioned, the only way to get close to the Earth twice a year is to have a circular orbit the same size as the Earth.
 
  • #9
BobG said:
Can't happen. If it's orbital period is half the Earth's orbital period, then the orbit has to be smaller and perigee will be closer to the Sun than the Earth.

You could set up an orbit that has half the Earth's orbital period and is close to the Earth at apogee. But every other orbit, the Earth won't be there - the Earth will be on the other side of the Sun.

As mentioned, the only way to get close to the Earth twice a year is to have a circular orbit the same size as the Earth.
Thank you too very much, BobG. :)
 

1. What is a heliocentric polar orbit?

A heliocentric polar orbit is a type of orbit around the sun in which a satellite or spacecraft passes over the north and south poles of the sun. This type of orbit is often used for scientific missions as it allows for a full view of the sun's polar regions.

2. How does a heliocentric polar orbit cross the Earth's orbit twice?

A heliocentric polar orbit can cross the Earth's orbit twice because the Earth's orbit is tilted at an angle of 23.5 degrees relative to the sun's equator. This means that as the satellite or spacecraft orbits the sun, it will pass over the Earth's orbit twice, once when it is above the Earth's northern hemisphere and once when it is above the Earth's southern hemisphere.

3. What are the advantages of a heliocentric polar orbit crossing the Earth's orbit twice?

One advantage of this type of orbit is that it allows for a more comprehensive study of the sun's polar regions. Additionally, by crossing the Earth's orbit twice, the satellite or spacecraft can gather data from different perspectives, which can provide a more complete understanding of the sun's behavior.

4. How is a heliocentric polar orbit crossing the Earth's orbit twice achieved?

To achieve this type of orbit, a spacecraft must first be launched into a low Earth orbit. Then, using its own propulsion system or a gravity assist from another planet, it can gradually increase its orbit until it reaches a heliocentric polar orbit that crosses the Earth's orbit twice.

5. What types of missions use a heliocentric polar orbit crossing the Earth's orbit twice?

Scientific missions that study the sun, such as the Solar and Heliospheric Observatory (SOHO) and the Solar Dynamics Observatory (SDO), use this type of orbit. Other missions, such as the Deep Space Climate Observatory (DSCOVR), use a similar orbit to study the Earth's climate and space weather.

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