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This question was underspecified, so I'm going to try again.

Osculating orbital elements are Keplerian elements of a hypothetical orbit around a specified center which would be tangent to a specified body's location and direction of motion.

Below I've used Horizons to generate osculating elements for Mercury around the solar system barycenter and around the Sun. Since gravity from all planets contribute to Mercury's motion both directly and by their effect on the movement of Sun and each other, neither is the correct orbit of Mercury.

Questions:

  1. Is it possible to determine the mass of the body that would be at each center to produce these orbital elements?
  2. Is it possible without using the period included in the output?

Output for Mercury from JPL's Horizons 

For 2458849.500000000 = A.D. 2020-Jan-01 00:00:00.0000 TDB 

Keplerian osculating element         barycenter (@0)           Sun (@sun)

Eccentricity                EC:  2.200014479776101E-01    2.056502791763153E-01
Inclination wrt XY-plane    IN:  7.072029216276680E+00    7.003793158072573E+00
Long. of Ascend. Nd, OMEGA  OM:  4.754470515340162E+01    4.830652204584498E+01
Argument of Perifocus, w    W:   2.767991909564741E+01    2.918253289128074E+01
Mean anomaly, M             MA:  1.899912221742112E+02    1.872506756791482E+02
True anomaly, nu            TA:  1.865562987479777E+02    1.848845489756656E+02
Semi-major axis, a (AU)     A:   3.772942492110727E-01    3.870976616195002E-01
Sidereal orbit period (day) PR:  8.448193239598780E+01    8.796891036601312E+01
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  • $\begingroup$ I don't think I fully understand your question. The mass that would result in an orbit with the elements given is just the mass of the Solar system located at the SSB for the first set and the mass of the Sun located at the current position of the Sun for the second set. Think of osculating elements like this: If at this moment, all perturbations disappear and you're only left with a central gravitational attraction, what would the orbit look like. The mass is therefore entirely decided by what central body you use for these osculating elements. $\endgroup$
    – Alexander Vandenberghe
    Jan 16, 2020 at 9:38
  • $\begingroup$ @AlexanderVandenberghe I am not sure that that is true. As I understand it, the osculating orbital elements provided by Horizons only provide a location $\mathbf{x}$ and a direction $\mathbf{v}/|v|$ (since the elements are tangent to the real orbit). Are you abel to demonstrate what you've said about masses either numerically or analytically, or is it more of an educated guess. $\endgroup$
    – uhoh
    Jan 16, 2020 at 14:19
  • $\begingroup$ It is the definition of an osculating orbit: "The orbit a satellite would have around the central body without any perturbations" $\endgroup$
    – Alexander Vandenberghe
    Jan 16, 2020 at 14:23
  • $\begingroup$ @AlexanderVandenberghe if this is the answer, can you post it as an answer along with a reference for that definition? My understanding is (or at least was) that it only reproduced the position and direction of motion. Thanks! $\endgroup$
    – uhoh
    Jan 16, 2020 at 14:24
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    $\begingroup$ @AlexanderVandenberghe Horizons gives me osculating elements for Voyager 1 around Earth and ISS around the Moon, and I'm having a hard time understanding what those orbits are (e.g. eccentricities of 5.1E+07 and 5.7E+03 and negative semi-major axes). Oh! Are those simply the hyperbolic orbits assuming that the central bodies are the actual Earth and Moon with their present locations and masses? $\endgroup$
    – uhoh
    Jan 16, 2020 at 14:41

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The answer to this question can be directly extracted from the definition of an osculating orbit.

the osculating orbit of an object in space at a given moment in time is the gravitational Kepler orbit (i.e. an elliptic or other conic one) that it would have around its central body if perturbations were absent.

The previous question that was linked here, How to find instantaneous positions of a central body of specified mass that results in an osculating orbit? , already has an answer explaining that keplerian elements are meaningless unless you know what central body they are referring to.

To calculate the osculating elements, the first thing you therefore need to do is selecting the central body. In the two different sets of element in your question the central bodies selected are thus the SSB and the Sun. Then using the position and mass of these central bodies, you use the position and velocity of the body of interest relative to the central body to solve the IVP for a kepler orbit.

If you check the two sets for the mass of the central body using:

$$M = \frac{4*a^3 *\pi^2}{G*T^2}$$

you get respectively 1.9963e30 kg and 1.9984e30 kg. This corresponds pretty much to Sun and Solar system mass.

As you apparently discovered yourself, you can calculate osculating elements with respect to any arbitrary body, but of course it makes zero sense to express the state of Voyager 1 as a Kepler orbit around Earth. In a mathematical sense any arbitrary state can be expressed as if it were a Kepler orbit around any arbitrary central body but we all know that Voyager is not actually orbiting the Earth in some strange hyperbolic orbit.

When it comes to determining the mass of the central body without the provided period, that is not possible. You need either of both to fully define the system.

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