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View Full Version : What is the Average Distance Between Earth and Sun?

mkline55
2013-May-14, 07:41 PM
That sounds simple enough. But the question is ambiguous, and the answers are worse. Universe Today (http://www.universetoday.com/14437/how-far-is-earth-from-the-sun/) says 150 million km. Really? Exactly 150 million? What an amazing coincidence! Good enough for public consumption anyway. About.com (http://geography.about.com/od/learnabouttheearth/a/earthfacts.htm) says 149,669,180 km. They took a rounded number of miles, 93,020,000 and converted that to kilometers to make it look like it's accurate to the nearest 1.0 kilometer. Wiki (http://wiki.answers.com/Q/What_is_the_distance_between_Earth_and_the_Sun) says 149,597,870.7 km. Wiki presents it like an absolute fact. The problem with the Wiki number is that it is nothing more than an average of the aphelion (152,097,701 km) and perihelion (147,098,074 km). Since the orbit is an elipse, Earth actually spends more time beyond this average than within this average. So, is it really the average distance? At any given point in time, is Earth more likely to be beyond the average than within it?

None of the sources states whether the measurements are from the surface or center of either Earth or sun. Do we simply assume center of earth to surface of sun, since that is the most logical meaning for distance to an object from a smaller orbiting object? that extra 0.1 kilometer might make a difference.

Is there room for ambiguity? With earth and sun, it's not too bad due to the low eccentricity of the orbit. It's not too bad for 5th grade astronomy. Anyone care to launch a \$200 million vehicle relying on that number? No problem, Bob. You have enough fuel for 93 million miles. You've got a 50/50 chance.

How about Halley's comet? Use the same method as the Wiki source to determine the average distance between Halley's comet and the sun. 35 AU or so vs. 0.6 AU. Is the average distance between it and the sun about 18 AU? Is the distance more than 18 AU half the time, and less than 18 AU the rest of the time, or is it more like 70 years outside that range for every 5 years inside? Is that any way to calculate an average distance?

Amber Robot
2013-May-14, 08:19 PM
At any given point in time, is Earth more likely to be beyond the average than within it?

Average is not the same as median.

antoniseb
2013-May-14, 08:30 PM
I'd go with 150,000,000 km, unless someone wanted a more exact value for a real reason, in which case I'd supply the specifics of the orbit, and let the person calculate what they really needed.
If you're looking for real details don't forget to take into account the influences of the Moon, and the other planets. If you're getting down to precision in meters, these are definitely something you have to take into consideration. ... likewise the distance of the true Sun from the average Sun.

Glom
2013-May-14, 11:14 PM
Average orbital radius is usually taken to mean the semi-major axis which is the mean of the aphelion and perihelion.

I would say it is exactly 1.0 AU ;)

Centaur
2013-May-15, 12:18 AM
The astronomical unit (AU) was recently standardized by the IAU to be precisely 149,587,870.7 km. It is that length (rounded to 0.1 km) for which the Gaussian gravitational constant (k) takes the value 0.01720209895 when the units of measurement are the metric astronomical units of length, mass and time. The mean semi-major axis (a) of the Earth’s orbit actually has a length of 1.000001018 AU or 149,588,023.0 km. The distance averaged over time between the centers of the Earth and Sun measured with J2013.5 mean eccentricity (e) of 0.01670295 is actually a (1 + eČ / 2) which is 1.000140512 AU or 149,608,889.7 km.

If that is mind boggling, just go with antoniseb’s 150 million km. ;)

grapes
2013-May-15, 03:36 AM
How about Halley's comet? Use the same method as the Wiki source to determine the average distance between Halley's comet and the sun. 35 AU or so vs. 0.6 AU. Is the average distance between it and the sun about 18 AU? Is the distance more than 18 AU half the time, and less than 18 AU the rest of the time, or is it more like 70 years outside that range for every 5 years inside? Is that any way to calculate an average distance?
Like Glom, I'd say yes. Aphelion plus perihelion equals major axis, so half of that is the semi-major axis, and Kepler's laws depend upon the semi-major axis. Two objects with the same semi-major axis will have the same period.

Centaur
2013-May-15, 04:01 AM
Taking Comet Halleys J1994.13 eccentricity (e) of 0.9671429085 and semi-major (a) axis of 17.83414429, then its distance from the Sun averaged over time would be a (1 + eČ / 2) or 26.17486509 AU.

mkline55
2013-May-15, 12:58 PM
The astronomical unit (AU) was recently standardized by the IAU to be precisely 149,587,870.7 km. It is that length (rounded to 0.1 km) for which the Gaussian gravitational constant (k) takes the value 0.01720209895 when the units of measurement are the metric astronomical units of length, mass and time. The mean semi-major axis (a) of the Earth’s orbit actually has a length of 1.000001018 AU or 149,588,023.0 km. The distance averaged over time between the centers of the Earth and Sun measured with J2013.5 mean eccentricity (e) of 0.01670295 is actually a (1 + eČ / 2) which is 1.000140512 AU or 149,608,889.7 km.

If that is mind boggling, just go with antoniseb’s 150 million km. ;)

Thanks Centaur. It's not mind boggling at all.

The point of the post is that "average distance between earth and sun" is ambiguous. It could mean a lot of different things. None of the sources I found made any distinction of what constituted their definitions of the concept. I'm still not certain if any of them mean the distance from the center of Earth to the surface of the sun, or from the surface of Earth to the surface of the sun, etc.

If I used the same logic as is commonly used for this statistic, I could say that the average age of all living humans is exactly 1/2 the age of the oldest living human. In addition, no country on Earth has an average age higher than the average of the entire population of the planet. Am I calculating the average wrong? It sure sounds wrong. But, not if the logic for "average distance" is used. Choosing the high and the low, then calculating the midpoint is not the same as calculating the average.

Centaur
2013-May-15, 02:31 PM
Thanks Centaur. It's not mind boggling at all.

You're welcome. At least you understand the reality and should be able to live with the common usage of the terms. I should note that the rather cryptic definition for AU is based on a theoretical solution in which only the Sun and Earth exist, and the Earth is a point of negligible mass moving in a perfectly circular orbit. In the context of your concerns, the measured distance is almost always between the centers of the bodies. The exception is when an object is not far from the surface of the Earth, and a given distance could be ambiguous without full disclosure of its meaning.

Hornblower
2013-May-15, 03:10 PM
As an exercise in mathematics or statistics we could define "average" in a variety of ways, including time-weighted ones, depending on what is most useful for the task at hand. For calculating orbits, the semimajor axis length, which is the straight mean between the maximum and minimum separation, is a good choice.

Shaula
2013-May-15, 03:39 PM
If I used the same logic as is commonly used for this statistic, I could say that the average age of all living humans is exactly 1/2 the age of the oldest living human.
You could but you would be wrong. You are assuming a symmetric distribution implicitly when you do this. There is no justification for this assumption (at the very least because there is a hard lower bound on the distribution that skews it but more generally because the distribution of ages is highly non-symmetric thanks to the fact that is is essentially the combination of several distributions in a non-trivial way) so the assumptions you use to generate your mean are wrong and your conclusion is wrong.

grapes
2013-May-15, 04:11 PM
You could but you would be wrong. You are assuming a symmetric distribution implicitly when you do this. There is no justification for this assumption (at the very least because there is a hard lower bound on the distribution that skews it but more generally because the distribution of ages is highly non-symmetric thanks to the fact that is is essentially the combination of several distributions in a non-trivial way) so the assumptions you use to generate your mean are wrong and your conclusion is wrong.
I think that was their point! :)

There is some justification for using the semi-major axis as the average distance, as we've said, and the formula for the time-averaged distance has been presented. "Average" can be a lot of different things, just ask anybody (mean, median, mode, etc.). I guess the only thing we have to clear up is whether the distance is surface-to-surface or center-to-center. Almost invariably, in astrononomical terms, it is center-to-center, because the effect of most bodies can be well approximated by a point mass at the center of the body.

Unless you want to argue what "center" means. :)

mkline55
2013-May-15, 06:30 PM
That is exactly the point. Ambiguity. I could argue that center to center is a poor assumption. How long is a string reaching from Earth to sun? That is the implied distance question. "How far is . . .?" At both ends, the string reaches the target at the surface. Hence, it's easy to assume the intended meaning is exactly that. Additionally, you would not wrap the string halfway around Earth just to measure from the far side. How far must a lander go to reach Mars? Are you planning to launch from the center of Earth and hit the center of Mars, or would surface to surface be more meaningful?

Centaur
2013-May-15, 06:59 PM
I could argue that center to center is a poor assumption.

A motivation for Newton to invent integral calculus was to prove that for the purpose of determining the motions of celestial bodies affected by the force of gravity, all of a spherical body’s mass can be assumed to be located at its center. This further assumes that the matter at a set radius from the center within any particular spherical shell is of uniform density. This is a good approximation for most planets and stars, and results in quite precise predictions of future positions. It allows for relatively simple calculations compared to having to consider every atom within a celestial body. Hence the continued standard of center to center measurements. As long as one understands this, one can make adjustments for other considerations.

Shaula
2013-May-15, 09:16 PM
That is exactly the point. Ambiguity. I could argue that center to center is a poor assumption. How long is a string reaching from Earth to sun? That is the implied distance question. "How far is . . .?" At both ends, the string reaches the target at the surface. Hence, it's easy to assume the intended meaning is exactly that. Additionally, you would not wrap the string halfway around Earth just to measure from the far side. How far must a lander go to reach Mars? Are you planning to launch from the center of Earth and hit the center of Mars, or would surface to surface be more meaningful?
So are you saying that because some popular websites give simplified versions of the answer or don't explicitly articulate the full range of assumptions/conveniences behind each number ... What? Because one thing we can be fairly sure of. Anyone launching a probe is not going to use the numbers in Wikipedia and hope.

mkline55
2013-May-16, 03:44 PM
So are you saying that because some popular websites give simplified versions of the answer or don't explicitly articulate the full range of assumptions/conveniences behind each number ... What? Because one thing we can be fairly sure of. Anyone launching a probe is not going to use the numbers in Wikipedia and hope.

I am saying that the failure of popular sources to give reliable and consistent information undermines the integrity of all sources. Three different sources gave three different answers. The science textbook I first encountered this in had the same problem. 93 million miles was the answer. It was given as if there were no considerations at all, and 93 million was the exact number. Most students just say it's far away, and they go on to study liberal arts. I wondered if the length of a mile was based on orbit of the Earth or what else caused it to be an exact multiple of 1 million miles. See the difference?

Shaula
2013-May-16, 04:42 PM
I am saying that the failure of popular sources to give reliable and consistent information undermines the integrity of all sources. Three different sources gave three different answers. The science textbook I first encountered this in had the same problem. 93 million miles was the answer. It was given as if there were no considerations at all, and 93 million was the exact number. Most students just say it's far away, and they go on to study liberal arts. I wondered if the length of a mile was based on orbit of the Earth or what else caused it to be an exact multiple of 1 million miles. See the difference?
I think you are somewhat over-blowing it. The failure of popular sources to go into the level of detail required to launch a space probe undermines all sources? No it doesn't. They are popular sources. Approximate. Just like medical popularisations tend not to go into epigenetics, evolution stuff doesn't go into lateral gene transfer and HSPs etc etc. They are tailored to an audience, there is always more detail to be had if you really want it. But giving a book length answer to "how far away is the Sun" would put off a lot of people who might otherwise go "Wow, I'd like to know more".

grapes
2013-May-16, 06:43 PM
I am saying that the failure of popular sources to give reliable and consistent information undermines the integrity of all sources. Three different sources gave three different answers. The science textbook I first encountered this in had the same problem. 93 million miles was the answer. It was given as if there were no considerations at all, and 93 million was the exact number. Most students just say it's far away, and they go on to study liberal arts. I wondered if the length of a mile was based on orbit of the Earth or what else caused it to be an exact multiple of 1 million miles. See the difference?
If it is given as 93,000,000 miles, there are two significant digits--it's been rounded to 93 million. Of course, it's 91.5-94.5 million over the course of a year, but that's the semi-major axis. If it were an exact measurement, it would be 93,000,000.0, no?

mkline55
2013-May-16, 07:05 PM
If it is given as 93,000,000 miles, there are two significant digits--it's been rounded to 93 million. Of course, it's 91.5-94.5 million over the course of a year, but that's the semi-major axis. If it were an exact measurement, it would be 93,000,000.0, no?

No. Unless you are defining a mile in fractions of the average distance between the center of the earth and the center of the sun over a 1-year period, yes?

Centaur
2013-May-16, 10:04 PM
No. Unless you are defining a mile in fractions of the average distance between the center of the earth and the center of the sun over a 1-year period, yes?

Come down to Earth. Average distances between astronomical objects are for people who simply want a rough idea. Anyone sending a spacecraft to Mars couldn't care less about the averages. It's the value at the time of rendezvous that would be important, not the average. Good writers understand their audiences and speak to their needs. In some contexts the readers require precision, but in most they do not. Those who do require precision have sources available to them. Perhaps you missed the math class in which significant digits and the rounding of numbers were discussed.

Ara Pacis
2013-May-17, 05:59 AM
I forget, at what grade level are the differences and uses for the mean average, median average, and modal average taught? I remember that I only learned about sig-figs specifically in 11th grade chemistry, although rounding numbers was much earlier.

I guess the best answer is by the time that you need to know the answer, you'll almost certainly already have received access to it.

Glom
2013-May-17, 07:50 AM
I really don't get what all the fuss is about. If someone asked me the distance between the Sun and Earth for curiosity, I'd say one hundred and fifty million kilometres.

What's the problem with that number in this context?

NEOWatcher
2013-May-17, 12:02 PM
The science textbook I first encountered this in had the same problem. 93 million miles was the answer.
It's hard to say if that's wrong or right because without knowing the target age or context there's no way to judge.

It was given as if there were no considerations at all
For that textbook, how big would it have been if they started to discuss considerations on all its topics?

and 93 million was the exact number.
It may be an exact number, but without saying it was exact, average, or about reference to a measurement, then I see no problem.

Grey
2013-May-17, 02:47 PM
Perhaps you missed the math class in which significant digits and the rounding of numbers were discussed.I've taught university-level introductory physics labs, and I'm sad to report that very few of the students I worked with had a decent understanding of dealing with significant figures and establishing error ranges on measurements.