Interiors of Terrestrial Planets in Metric-Affine Gravity
Abstract
:1. Introduction
2. Simple Model of Small Rocky Planets in Palatini Gravity
2.1. Planets’ Structure Equations
2.2. Internal Structure of Palatini Planets
3. Numerical Solutions
4. Conclusions
- Density profiles, as already noticed in our previous works, can significantly differ in modified gravity with respect to the Newtonian model. We observe not only lower values of central density and on the core–mantle boundary, but also the cores of the given exoplanets are bigger; that is, the cores are less dense in the case of Palatini gravity. Therefore, the observed transiting planets can have different structure for the same masses and radii than the one predicted in the usual way, and can affect the planet’s polar moment of inertia. The fact that internal structure of planets is affected by modifications of gravitational interaction is to be expected, since Equations (3) and (4), allowing one to compute the density profiles, change. This entails the fact that modifications of gravity introduce additional degeneracy when trying to determine planets’ internal composition by looking at the mass–radius relationship [23]. The values of internal pressure and core radius, giving the same total mass and radius, depend on the parameter . Therefore, this fact alone does not allow us to constrain alternative gravity models. What actually could help in distinguishing between different models would be collecting seismic data from Solar System planets, and investigating their density profiles. For example, Earth’s mass and radius are well known, as well as its internal composition, so, after having developed a more realistic model taking into account modifications of gravity, it will be possible to place a stringent constraint on values of .
- A similar situation happens when we plot the pressure curves obtained in this work: its central values decrease in modified gravity; however, when we approach the planet’s surface, the mantles do not differ much. This result derives from the fact that the additional term in Equation (15) for the pressure in the mantle is small, and smaller than the extra term appearing in the analogous equation for the core (Equation (18)).
- We also compared the numerical solutions for the pressure obtained from Equations (3) and (4) to the ones resulting from the analytical approach (which are approximated solutions). As one can see, the pressure drops roughly, similar to in the core, and then changes in a linear way in the mantle in the case of both numerical and analytical solution (although it is less pronounced for larger values of the parameter ). One notices that in the case of Newtonian gravity (), analytical (approximated) solution tends to provide smaller values than the numerical one. However, in the case of modified gravity, the effect is the reverse—approximated analytical solution provides larger values than the numerical one. This can be explained in the following way: the analytical approximation does not take into account the effect of modification of gravity in the CMF Formula (6), so it stays constant for various values of the parameter (as it depends of the mass and radius of the planet only, and these values do not change). On the other hand, the numerical method suggests that the size of the core and its mass grow in modified gravity, and hence the CMF must change. This combined effect of change in and CMF/CRF results in a bigger drop in internal pressure.
- Moreover, as already mentioned in the previous point, our numerical analysis revealed that the equation for the semiempirical CMF used in that work also depends on modified gravity. This is not a surprise, remembering the fact that for finding that relation, one uses the PREM model, which is based on Newtonian gravity.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
1 | The exoplanets of Mercury’s type, having cores with masses ∼0.7 of the total mass [14], are excluded from such an analysis. |
2 | |
3 | However, we will be equipped with the Mars ones, too, thanks to the Seismic Experiment for Interior Structure from NASA’s MARS InSight Mission’s seismometer [67]. |
4 | |
5 | But the results are similar for the other ones, too, with the more significant differences for larger planet’s masses with respect to the Newtonian solutions. |
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Kozak, A.; Wojnar, A. Interiors of Terrestrial Planets in Metric-Affine Gravity. Universe 2022, 8, 3. https://doi.org/10.3390/universe8010003
Kozak A, Wojnar A. Interiors of Terrestrial Planets in Metric-Affine Gravity. Universe. 2022; 8(1):3. https://doi.org/10.3390/universe8010003
Chicago/Turabian StyleKozak, Aleksander, and Aneta Wojnar. 2022. "Interiors of Terrestrial Planets in Metric-Affine Gravity" Universe 8, no. 1: 3. https://doi.org/10.3390/universe8010003