Negative constant of proportionality

[Arkalius]

So I have a question for which I can't really find a good answer.

Mathematically speaking, we say a and b are inversely proportional if the relationship a * b = k for some constant k holds. Here, k is the constant of proportionality. Colloquially, when we say "a is inversely proportional to b", we expect that as a increases, b will decrease by some relative amount.

Now, consider what happens when k is a negative number. In this situation, exactly one of a or b must be negative as well. We also note that as the absolute value of one increases, the absolute value of the other must decrease. However, the actual value of both will vary in the same direction. If a goes up (ie if you add a positive value to it), so too does b (it gains positive value). Conversely, if a goes down (subtract from it) the same is true for b. However, to say in this situation that a is inversely proportional to b, while technically correct, suggests the inaccurate scenario that when a goes up, b goes down and vice versa.

Colloquially speaking we might want to say they are directly proportional but that mathematical relationship doesn't hold.

So how do we describe this relationship in a way that is both mathematically accurate, and which conveys the reality of how it works?

For context, I was looking for a way to explain the relationship between specific orbital energy and semi-major axis of the orbit. That relationship is expressed by:

E = - mu / (2 * a)

Where mu is the standard gravitational parameter of the central body. In this case, we can change the equation to E * a = - mu / 2, and since mu is always positive, we have a negative constant of proportionality. Thus either E or a is negative. E is negative when it is a closed orbit, and a is negative in an unbounded one (hyperbolic).

In this situation, if you increase the value of a (ie, add a positive value to it), the value of E also increases (as if you added a positive value to it). Saying they are inversely proportional seems to convey the opposite.

So, anyone have any insights in how to best communicate the relationship between E and a?

Side Note: for those wondering how you can have negative energy, specific orbital energy is considered the sum of specific kinetic energy and specific gravitational potential energy, which is always negative. This value represents the negative of the kinetic energy per kilogram required to just escape from the body in question (ie be slowed to a velocity of 0 by gravity at infinite distance). Thus, specific orbital energy represents the amount of kinetic energy per kilogram the object has in excess of what is needed to escape the body. So, in a closed orbit, this excess is actually a deficit and thus the value is negative.

Edited July 25, 2015 by Arkalius
Adding parenthesis

[YNM]

That's true indeed. It's the reason why a parabolic orbit will have a = infinite no matter where your closest distance is, and hyperbolic orbits will have negative value of a, that goes larger (absolute value), coupled to the closest distance. Semi major axis is just an expression (it's impossible to have negative length or distance, remember), and they're coupled to the fact that potential energy means "loss" of energy, or state the required energy to liberate the object. Just some adjustments and semantics, nothing else.

[Hcube]

For context, I was looking for a way to explain the relationship between specific orbital energy and semi-major axis of the orbit. That relationship is expressed by:

E = - mu / 2 * a

Where mu is the standard gravitational parameter of the central body. In this case, we can change the equation to E * a = - mu / 2

This bugs me. Did you maybe forget to add some () ?

Because you can change E=-mu/(2a) can be changed to Ea=-mu/2 , but E=-mu/2*a can not be changed to Ea=-mu/2

im sorry if this should be obvious to me but i don't know this formula so it sounds weird

very very interesting post though !

[K^2]

Hcube, yes, parenthesis are missing. It should be -mu/(2a)

Arkalius, still inversely proportional. Inverse proportion only talks about relationships between magnitudes. Double the one, and it halves the other. Fact that signs are opposite doesn't play into that part of it at all. As you've pointed out, it just means that proportionality constant is negative. Nothing wrong with inverse proportion being a negative one.