| Summary: | When the BepiColombo spacecraft arrives at Mercury in late 2025, it will be able to measure the orbit of the planet with unprecedented accuracy, allowing for more accurate measurements of the perihelion advance of the planet, as predicted by the Theory of General Relativity (GR). A similar effect is produced by the gravitational oblateness of the Sun through the zonal coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mrow><mn>2</mn><mo>⊙</mo></mrow></msub></semantics></math></inline-formula>. The gravitational field of the Sun has been hard to determine despite centuries of observations, causing great uncertainties in experiments on GR. Recent publications in heliophysics suggest that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mrow><mn>2</mn><mo>⊙</mo></mrow></msub></semantics></math></inline-formula> is not a constant, but a dynamic value that varies with solar magnetic activity. The aim of this paper is to analyse what the effect is of suggested higher-order effects of the solar gravitational field on experiments of the perihelion advance of Mercury as predicted by GR. The orbit of Mercury and observations of the MESSENGER and BepiColombo spacecraft are simulated, and parameters corresponding to gravitational theory, as well as the oblateness <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mrow><mn>2</mn><mo>⊙</mo></mrow></msub></semantics></math></inline-formula> including a time-variable component are estimated using a least-squares approach. The result of the estimation is that the amplitude of a periodic component can be found with an uncertainty of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3.7</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></semantics></math></inline-formula>, equal to 0.017% the value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mrow><mn>2</mn><mo>⊙</mo></mrow></msub></semantics></math></inline-formula>. From analysis of published experiments that used MESSENGER tracking data, it can already be deduced that the amplitude of the periodic variation cannot be higher than 5% of the value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mrow><mn>2</mn><mo>⊙</mo></mrow></msub></semantics></math></inline-formula>. It is also found that if a periodic component exists with an amplitude greater than 0.04% the value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mrow><mn>2</mn><mo>⊙</mo></mrow></msub></semantics></math></inline-formula> and it is not considered, it can lead to errors in the experiments of GR using BepiColombo data to the point that results falsely confirm or contradict the Theory of General Relativity.
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