### How was the mass of Venus determined?

• The mass of Venus seems rather complicated to determine to me:

• Venus doesn't have any satellites, so you can't just apply Kepler's third law (like you would with Jupiter or Saturn for instance) to determine its mass.

• The gravitational tug of Venus on the Sun is very small compared to that of other Jupiter or Saturn, so it seems like it would be difficult to extract what part of the Sun's proper motion is caused by Venus.

• There are few asteroids with orbits close to that of Venus, so not many objects who might have their trajectory modified by Venus.

Knowing Venus' radius, and assuming it has the same density as the Earth, you can get a pretty close estimate of its mass (85% of Earth's mass with this assumption, when the actual value is 82%). But that's a pretty strong assumption (the density of Earth and Venus only happen to be close by chance) and a rather unsatisfactory "guesstimate".

Nowdays, there are a few probes that have flown by Venus, so by looking at their trajectory, you can infer what Venus' gravitational field looks like. But those fly-bys are pretty recent. Did we know about Venus' mass before those fly-bys?

How was the mass of Venus measured for the first time?

Mercury's mean density is actually very close to Earth's, for what it's worth - 5.4 g/cm$^3$ as opposed to 5.5 g/cm$^3$.

@HDE226868 Wolfram|Alpha says 5.52 g/cm³ versus 5.24 g/cm³.

@gen-ℤreadytoperish though *handy*, WA isn't necessarily a good source for this kind of thing. Why doesn't Wolfram Alpha show low gravitational acceleration for the Hudson Bay?

@gen-ℤreadytoperish check the planets, *Mercury* has a density very similar to *Earth's*, *Venus* doesn't (well, 5.24 is not that far off, but the paragraph in the question makes it sound like *Mercury* is far denser than either Earth or Venus)

• How was the mass of Venus measured for the first time?

In the mid 19th century, Urbain Le Verrier's predicted of the existence of a then unknown planet beyond the orbit of Uranus. He even predicted this planet's orbit. The discovery of Neptune based on his predictions was perhaps his greatest accomplishment.

Le Verrier then went on to investigate Mercury. He used observations of Mercury, Venus, the Sun (as a stand-in for the Earth) and Mars and calculated that Mercury should precess by 532 arc seconds per century based on Newtonian mechanics. Along the way, he had to (and did) estimate the mass of Venus. There was a problem here; the observed precession of Mercury's orbit is 575 arc seconds per century, 43 arc seconds per century greater than his calculated value. This led Le Verrier to conjecture that there was a planet even closer to the Sun than Mercury.

Despite the failure to discover the non-existent planet tentatively named Vulcan, Le Verrier's estimate for the mass of Venus was fairly close to the correct figure, within a couple of percent. Once the cause of this 43 arc second per century discrepancy was discovered by Einstein, the mass of Venus was determined with even greater accuracy. Of course, once probes were sent into orbit about Venus, its mass was determined with greater accuracy yet.

Voted up your answer, however, was Le Verrier's remarkable work on Mercury the first attempt to determine the mass of Venus using observations of the orbit of the planets ? That is, are we reasonably sure that there was no earlier attempt ?

This doesn't make sense to me. Le Verrier would first have to estimate Venus's mass by assuming some density, presumably one equal to that of the earth. Then if the precession of Mercury came out about right, but a little off, couldn't he just adjust Venus's mass to reproduce the precession? Why would he invent a new planet instead?

@BenCrowell He would have known the paths of the planets and been able to estimate the masses required to explain perturbations. But that mass would need to be consistent over the full data fit, so any inconsistency (outside the margin of error) was put down (in the absence of GR) to a new planet. You'd have to vary the masses over time to explain it otherwise, I think.

@BenCrowell - The method of least square estimation was developed in the early 19th century. This is the technique Le Verrier used to discover "a planet with the point of his pen." This is the technique that the Jet Propulsion Laboratory uses to this day to compute its planetary ephemerides. Le Verrier made the mass of Venus one of the unknowns to be solved for in a least squares sense.

@StephenG As far as I can tell, Le Verrier was the first. While there is a priority dispute between Le Verrier and John Couch Adams regarding the mathematical prediction of Neptune's existence, I could not find anyone who claimed priority on the discovery of the 43 arc second per century discrepancy between the predicted and observed precession of Mercury. Getting there required estimating the mass of Venus as Venus is the largest perturber of Mercury's orbit.

Thanks for the response.

This answer describes *that* Le Verrier (and later, others) estimated the mass of Venus, but not *how* any of them did it.

Agreed with @Timbo - this answer only tangentially answers the question. How did Le Verrier actually determine the mass of Venus? It's implied that it was determined in the same way as Neptune was predicted - by "tweaking" the mass and/or adding mystery planets of required mass until the precession of neighbouring planets was well described using Newtonian mechanics, but this does not explain how Le Verrier did such computations without a computer (ie: this is an n-body problem) nor does it really address the methodology or measurement strategy.

In Le Verrier's paper linked above, he starts with "values of the masses which in the *Annales* have been used as starting-points" (Eq. (1)), which sounds like a reference to prior literature; he then looks for small corrections $\nu_1, \nu_2$, etc. to these values. This "starting point" corresponds to an estimate of $M_V = (2489/2817)M_\oplus$, which is about 8% larger than to the currently accepted value but is in the right ballpark. Where did this prior estimate come from?

More accurately, the number $2.489 \times 10^{-6}$ used in Eq. (1) appears to have been the best estimate of $M_V/M_\odot$ available to Le Verrier at the time, and *that* was only off by about 2%.

@DavidHammen: Whether he uses least squares is not the point. The point is that if he just has one number to fit, which is the anomalous secular trend of venus's perihelion, and he has one parameter that he can use to fit that, then there is nothing to prevent him from fitting that one parameter. It seems likely that StephenG has the right idea -- perhaps there was some more detailed time variation rather than just a single number for the secular rate of change. Downvoted.

This is a good answer but it does not actually describe how Venus's mass was estimated. I see discussion of this in the comments and I think that your answer would be improved by including this in your posted answer above.