Translation:Determinatio attractionis quam in punctum quodvis positionis datae exerceret planeta si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita
Determination of the Attraction Which a Planet Would Exert on An Arbitrary Given Point, If Its Mass Were Distributed Throughout Its Entire Orbit in Proportion to the Time It Takes to Describe Each Individual Part (1818) by Carl Friedrich Gauss, translated from Latin by Wikisource
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4454449Determination of the Attraction Which a Planet Would Exert on An Arbitrary Given Point, If Its Mass Were Distributed Throughout Its Entire Orbit in Proportion to the Time It Takes to Describe Each Individual Part1818Carl Friedrich Gauss
1.
Secular variations, which the elements of a planetary orbit undergo due to the perturbation of another planet, are independent of the position of the latter in its orbit, and would be the same whether the perturbing planet moves in an elliptical orbit according to Kepler's laws or its mass is conceived to be evenly distributed throughout the orbit, so that equal masses are assigned to parts of the orbit described in equal time intervals, provided that the periods of revolution of the perturbed planet and the perturbing one are not commensurable. Although this elegant theorem has not yet been explicitly stated by anyone, it can at least be demonstrated easily from the principles of physical astronomy. Thus the following problem arises, which is worthy of attention both in itself and because of several techniques required for its solution: to determine exactly the attraction of the planetary orbit, or if one prefers, the elliptical ring, whose thickness at an arbitrary given position is infinitely small and varies according to the law just explained.
2.
Let denote the eccentricity of the orbit, and let denote the eccentric anomaly of an arbitrary point on the orbit. Then each element of the eccentric anomaly corresponds to an element of the mean anomaly and the element of mass of that portion of the orbit to which those elements correspond, will be to the whole mass (which we take to be unity) as is to where denotes the semicircumference of a circle of radius Therefore, assuming the distance from the attracted point to the point on the orbit to be the attraction produced by the element of the orbit will be
We will denote the major semi-axis by the minor semi-axis by and we will adopt that line as the abscissa, and the center of the ellipse as the origin. Thus we will have the abscissa of the point of the orbit will be and the ordinate will be Finally, we will denote the distance from the attracted point to the plane of the orbit by and the remaining coordinates parallel to the major and minor axes by and With these preparations, the attraction of an element of the orbit will be decomposed into two components parallel to the major and minor axes and a third perpendicular to the plane of the orbit, so that
where
Integrating these differentials from to yields partial attractions in directions opposite to the coordinate directions, and thus we obtain the total composite attraction, which can be referred to any other direction by a known method.
3.
The main task now is to replace with another variable, in order to simplify the radical quantity. To this end, we set
where the nine coefficients , etc., are not entirely arbitrary, but must satisfy certain conditions, which must be examined before we can proceed. First we observe that the substitution remains the same if all coefficients are multiplied by the same factor, so that without loss of generality, it is possible to assign a determined value to one of them, e.g. However, for the sake of elegance, all nine will remain indefinite for the time being. Furthermore, we note that values for which , or are proportional to , respectively must be excluded: otherwise, would no longer be undetermined. Therefore, cannot all vanish simultaneously.
It is clear that the coefficients , etc., must be related in such a way that the following expression becomes indefinite
It follows that the above function must have the form
Hence we obtain six conditional equations
(I)
From these equations result several others which are worth developing. For the sake of brevity, let us set
(II)
Then from various combinations of the equations (I), the following nine can be easily derived:
(III)
From the first three of these equations, we deduce
This is equivalent to
which, with the help of equations 2, 3, 4 in (I), can be transformed into
(IV)
It is equally easy to derive the following from equations (I):
(V)
We now provide a derivation of the first equation, based on whose example derivations of the others can be found. Specifically, equations 4, 2, 3 in (I) yield
which, when expanded, immediately yields the first equation in (V).
From the equations (V), we conclude that the value is not admissible in our discussion; otherwise, all nine quantities , etc., would necessarily vanish, i.e. the coefficients and and would all become proportional. In fact, the quantity cannot vanish, due to equation (IV); hence must be a positive quantity, since all of the coefficients , etc., must be real. Combining the first three equations in (III) with the first three in (V), the following new equations emerge, which clearly depend on the non-vanishing value of :
(VI)
Combining the other equations would produce the same result. Finally, we add the following three equations,
(VII)
which are easily derived from the equations (III). For example, the second, fifth, and eighth yield
Clearly, these equations also require the value to be excluded.[1]
Since, as mentioned above, we can multiply all coefficients , etc., by the same factor, the value of multiplied by the square of this factor will always result. Henceforth, we will always assume
,
from which it follows that either or It is evident, therefore, that the nine coefficients etc., among which there are six conditional equations, must be reducible to three independent quantities. This is most conveniently accomplished using three angles, as follows,
where the upper signs refer to the case and the lower ones to the case Nevertheless, the analytical treatment can be completed for the most part more elegantly without using these angles. Of course, it would not be difficult to assign a geometric interpretation to both these angles and the other auxiliary quantities occurring in this discussion; however, we leave this interpretation, which is not necessary for our purpose, to be explained by the knowledgeable reader.
4.
If the above values are substituted for and in the expression for the distance , then this expression obtains the form
where the coefficients , etc. must be determined in such a way that the six conditional equations are satisfied,
[1]
and so must be the others which follow from them,
,
and thus the problem will be generally determined. Therefore, if we denote the denominator of by and make the substitution
in the following function of the three quantities and
then we obtain
It would clearly be equivalent to say that substituting
in the following function () of the three variables and
,
results in the following function of the variables and
But from these formulas, with the help of the equations [1], it easily follows that
so it is clear that the function must be identical to
This leads us to six equations
[2]
From these twelve equations [1] and [2], we must determine the values of our unknowns , etc.
5.
By combining equations [1] and [2], the following equations are easily derived:
From this we further obtain:
[3]
[4]
We can also express the final equation as follows:
[5]
Similarly, by combining equations [1] and [2], we deduce:
and hence
[6]
[7]
[8]
and in a completely similar manner:
,
[9]
,
[10]
[11]
It is therefore clear that are the roots of the equation
[12]
which, when properly expanded, takes the form
[13]
6.
Regarding the nature of this cubic equation, the following observations are noteworthy.
I. From the last term of the equation, it follows that the equation has at least one real root, which can either be positive, or in the case equal to zero. Let us denote this real non-negative root by
II. Writing equation 12 in the form
subtracting the equation
and dividing by we obtain the following new equation, which is satisfied by the two remaining roots:
When arranged and solved in the usual way, this equation yields
[14]
Since the quantity under the radical sign is positive, or at least non-negative, this indicates that the two remaining roots are always real.
III. Subtracting the following equations from each other,
and dividing by we obtain an equation satisfied by the two remaining roots, of the form:
From this we see that if is positive, the equation cannot be satisfied by any positive value of Therefore, we conclude that our cubic equation cannot have more than one positive root.
IV. Thus, whenever is not among the roots of our equation, there will always be one positive root and two negative ones. However, when and hence is one of the roots, the equation satisfied by the remaining roots will be
from which these roots are found to be
Here, three cases need to be distinguished again.
Firstly, if the last term is positive (i.e., if the attracting point lies within the curve of the attracting ellipse), both roots, being real, will have the same sign. Since they cannot both be positive, they must be negative. Furthermore, this can also be concluded independently of what has already been shown, since the middle coefficient, which can be expressed as
is clearly positive in this case.
Secondly, if the last term is negative, indicating that the attracting point lies outside the curve, then one root will be positive, and the other negative.
Thirdly, if the last term vanishes, indicating that the attractor lies precisely on the circumference of the ellipse, then the second root will also be while the third will be
i.e. it will be negative. But this case, in which the attraction would become infinitely large, is physically impossible, and we exclude it from our discussion, at least for the moment.
7.
Using equations 1, 3, 4, 6, 7, 9, 10 to determine the coefficients we find that
[15]
Combining these equations with 5, 8, 11, we also obtain:
[16]
Since must be real, these expressions show that none of the quantities can be negative.
Therefore, in the case where is not must be set equal to the positive root of equation 13. Consequently, must be equal to one of the negative roots, and must be equal to the other.[2] However, which root to adopt for and which for is entirely arbitrary.
Whenever and the attractor lies outside the ellipse, the positive root of equation 13 must be set equal to and the negative root must be set equal either to with or to with The coefficient or can then be found using the formula
However, in the case already excluded, where the attractor is assumed to lie on the circumference of the ellipse, the coefficients and , or and , would become infinite, indicating that our transformation is not applicable in this case at all.
8.
Although formulas 15 and 16 could suffice to determine the coefficients more elegant formulas can be found. To this end, we will multiply equation [5] by which yields, after a slight reduction:
But from the nature of the cubic equation, we have:
the sum of roots:
the product of roots:
Hence, the preceding equation transforms into the following,
which can also be expressed as
Hence the value of the coefficient from the first formula in [15] transforms into the following:
[17]
A completely similar analysis yields:
[18]
[19]
Once the coefficients have been found, the remaining can be derived from formulas 3, 4, 6, 7, 9, 10.
9.
It is easy to understand that the signs of the radical expressions by which are determined can be taken arbitrarily. However, it is worth considering how the sign of the quantity is related to these signs. To this end, let us consider the third equation in III art. 3.
which through formulas 6, 7, 9, 10 transforms into
Now, considering equation 13, we easily deduce that
Hence, the preceding equation becomes
Combined with equation 17, this yields
Hence, it is clear that is taken to be the absolutely greatest negative root of the cubic equation, and at the same time all coefficients are taken positively, then will have the same sign as and the same will happen if these four conditions, or any two of them, are negated, but the opposite will happen if one or three of the conditions are negated. It is also worth noting the following relations, which are easily derived from those above:
10.
Our formulas can become indeterminate in certain cases, which need to be considered separately. First let us discuss the case where the negative roots and of the cubic equation become equal, and through formulas 18, 19, the coefficients seem to acquire infinite values, which are actually indeterminate.
Setting in formula 14, it is evident that to make the two values of i.e., and , be equal, it must be that:
Therefore, since is inherently either positive or it is easy to see that we must have
and
or
Substituting these values into equation 14 gives
Further substituting into the cubic equation 13 yields
Whenever this conditional equation holds simultaneously with the equation the case we are discussing arises. And since
formula 17 provides
and then formulas 3, 4 yield
The values of the coefficients remain indeterminate in this case, according to formulas 18, 19, and so do the values of the coefficients Nevertheless, all five of the remaining coefficients can be expressed in terms of one of them, e.g. by formula 6,
and so
;
but a more concise approach is the following. From
it follows that
and therefore, we can write
Then from the formulas
we find that
Here the value of the angle is arbitrary, and can be arbitrarily set to either or
11.
If and are not equal, then according to formulas 17, 18, 19, the values of coefficients cannot be indeterminate. On the other hand, if one of the quantities vanishes, then the value of the corresponding coefficient appears to be indeterminate according to formulas 6, 7, 9, 10. However, a closer look reveals otherwise.
Let us suppose that e.g. Then according to equation 18, and also, according to equation 7, (since ). Consequently, we must have However, if at the same time we have then the formula preceding the sixth in art. 5 yields which, when combined with yields:
These expressions cannot be indeterminate unless then we would fall into the case already considered in the previous article.
12.
Having determined the twelve quantities in full, we proceed to the determination of the differential Let us set
[20]
so that
[21]
[22]
From this, we deduce
and therefore
or
[23]
It is worth noting that the quantity is always positive when the coefficient is positive, and always negative when is negative. For when
it will always be the case that
regardless of the sign, is less than Thus we conclude that, whenever is a positive quantity, the variables and always increase together; however, when is negative, either variable must decrease when the other increases.
13.
The relationship between the variables and can be further clarified by the following reasoning. Setting so that we derive from equations 20, 21, 22
Likewise, from equations 21, 22 it follows that
If we set
,
then these equations take the following form:
From this, through division, due to we get:
Thus, not only have we arrived at the same conclusion as in the end of the previous article, but it has also become evident that if the value of increases by 360 degrees, the value of must either increase or decrease, depending on whether is a positive or negative quantity. Furthermore, by setting it is clear that
14.
Combining equations 20, 21, 22 with the equations from art. 5, we obtain:
For the sake of brevity, let us therefore set
so that
But we have
where the upper or lower sign holds depending on whether is positive or negative (since is always considered positive), i.e. depending on whether the coefficient is positive or negative. Hence,
where the ambiguous sign depends on the sign of the quantity .
To obtain the values of we need to integrate these differentials from the value of corresponding to up to the value corresponding to or even (which obviously produces the same results) from the value of corresponding to an arbitrary value of up to the value corresponding to this value of increased by it is permissible to integrate from to when is a positive quantity, or from to when is negative. Consequently, we have
where the integrations extended from to independently of the sign of
15.
It is easy to see that the integrals
extended from to yield the same values as those which the same integrals would obtain if extended from to , but with opposite signs; therefore, these integrals, when extended from to clearly become Hence, we conclude that
where the integrations extend from to Therefore, if the values of the integrals, with the same extension,
are denoted by and respectively, we will have
and thus our problem is completely solved.
16.
Regarding the quantities it is clear that each of them is
whenever but in all other cases, they can only be expressed in terms of transcendental functions. It is well known how they may be expressed in terms of series. We hope it will be pleasing to the reader if we take this opportunity to explain how these and other transcendental functions may be determined through a very efficient special algorithm, which we have frequently used many years ago and about which we intend to elaborate more extensively elsewhere.
Let be two positive quantities, and let us set
so that respectively represent the arithmetic and geometric mean of and We will always assume the geometric mean to be positive. Similarly, set
and so on, so that the series etc., and etc., converge very rapidly towards a common limit, which we will denote by We will simply call this the 'arithmetic-geometric mean' of and We will now demonstrate that is the value of the integral
extended from to
Proof. Let us assume that the variable is expressed in terms of another variable so that
It is easily understood that, as increases from to increases (although in unequal intervals) from to Having correctly performed this substitution, it is found that
and therefore the values of the integrals
are equal to each other if both variables are extended from to We can then continue in the same way, and it is clear the the values of the integrals will eventually be the same as that of
from to which is clearly Q. E. D.
17.
From the equation expressing the relationship between and
it is easily deduced that
and hence, with the aid of the same equation,
Multiplying this equation by
yields
Multiplying this equation by substituting and integrating from and to we have:
Moreover, since it is permissible to transform the definite integral on the right-hand side in the same manner, the integral
can be expressed by an infinitely convergent series as follows:
The numerical calculation is most conveniently carried out using logarithms, if we set
etc.
from which it follows that
etc.
and
18.
Using the method explained here, it is also possible to compute indefinite integrals (starting from a variable value ) with maximal efficiency. Specifically, if is assumed to be determined from just as was determined by and similarly is determined by , and so on, then for any given value of then the values of the terms of the series , etc., converge rapidly to the limit and we will have
But it suffices to mention these matters in passing here, as they are not necessary for our purpose.
19.
Now, if we already assume the values of quantities can easily be reduced to transcendentals . For if are the values of integrals
extended from to it is immediately evident that we have
[24]
Moreover, we have
Integrating this equation from to we get
[25]
Finally, combining equations 24 and 25, we conclude that
↑Perhaps it will not be superfluous to note that we have chosen the preceding analysis carefully and preferred it to another derivation of equations (III)-(VII), which, although it may have appeared somewhat more elegant, upon careful examination, was found to be susceptible to certain doubts that could not be resolved without difficulty.
↑Strictly speaking, it only follows from the preceding analysis that both and must satisfy equation 13, raising the question of whether it is permissible to assign both and to the same negative root, with the third root being completely ignored. However, it is easy to see that, if the second and third roots of the equation are unequal, then from it follows that and thus which contradicts the fourth equation in [1]. See what is said below regarding the case of two equal roots of equation 13.