Is There Still Something Left That Gravity Probe B Can Measure?
We perform a full analytical and numerical treatment, to the first post-Newtonian (1pN) order, of the general relativistic long-term spin precession of an orbiting gyroscope due to the mass quadrupole moment <inline-formula> <math display="inline"> <semantics> <msub>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-06-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/6/6/85 |
| _version_ | 1827714493719773184 |
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| author | Lorenzo Iorio |
| author_facet | Lorenzo Iorio |
| author_sort | Lorenzo Iorio |
| collection | DOAJ |
| description | We perform a full analytical and numerical treatment, to the first post-Newtonian (1pN) order, of the general relativistic long-term spin precession of an orbiting gyroscope due to the mass quadrupole moment <inline-formula> <math display="inline"> <semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> of its primary without any restriction on either the gyro’s orbital configuration and the orientation in space of the symmetry axis <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi mathvariant="bold-italic">k</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> of the central body. We apply our results to the past spaceborne Gravity Probe B (GP-B) mission by finding a secular rate of its spin’s declination <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> which may be as large as ≲30–40<inline-formula> <math display="inline"> <semantics> <mrow> <mspace width="0.166667em"></mspace> <mi>milliarcseconds</mi> <mspace width="0.166667em"></mspace> <mi>per</mi> <mspace width="0.166667em"></mspace> <mi>year</mi> <mspace width="0.166667em"></mspace> <mfenced separators="" open="(" close=")"> <mi>mas</mi> <mspace width="0.166667em"></mspace> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfenced> </mrow> </semantics> </math> </inline-formula>, depending on the initial orbital phase <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>. Both our analytical calculation and our simultaneous integration of the equations for the parallel transport of the spin 4-vector <i><b>S</b></i> and of the geodesic equations of motion of the gyroscope confirm such a finding. For GP-B, the reported mean error in measuring the spin’s declination rate amounts to <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>σ</mi> <mrow> <mover accent="true"> <mi>δ</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mi>GP</mi> <mo>−</mo> <mi mathvariant="normal">B</mi> </mrow> </msubsup> <mo>=</mo> <mn>18.3</mn> <mspace width="0.166667em"></mspace> <mi>mas</mi> <mspace width="0.166667em"></mspace> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> </inline-formula>. We also calculate the general analytical expressions of the gravitomagnetic spin precession induced by the primary’s angular momentum <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold-italic">J</mi> </semantics> </math> </inline-formula>. In view of their generality, our results can be extended also to other astronomical and astrophysical scenarios of interest like, e.g., stars orbiting galactic supermassive black holes, exoplanets close to their parent stars, tight binaries hosting compact stellar corpses. |
| first_indexed | 2024-03-10T18:59:26Z |
| format | Article |
| id | doaj.art-a05df1a182b24917a64daafe6d755c0d |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-10T18:59:26Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-a05df1a182b24917a64daafe6d755c0d2023-11-20T04:29:53ZengMDPI AGUniverse2218-19972020-06-01668510.3390/universe6060085Is There Still Something Left That Gravity Probe B Can Measure?Lorenzo Iorio0Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.)-Istruzione, Viale Unità di Italia 68, 70125 Bari, ItalyWe perform a full analytical and numerical treatment, to the first post-Newtonian (1pN) order, of the general relativistic long-term spin precession of an orbiting gyroscope due to the mass quadrupole moment <inline-formula> <math display="inline"> <semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> of its primary without any restriction on either the gyro’s orbital configuration and the orientation in space of the symmetry axis <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi mathvariant="bold-italic">k</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> of the central body. We apply our results to the past spaceborne Gravity Probe B (GP-B) mission by finding a secular rate of its spin’s declination <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> which may be as large as ≲30–40<inline-formula> <math display="inline"> <semantics> <mrow> <mspace width="0.166667em"></mspace> <mi>milliarcseconds</mi> <mspace width="0.166667em"></mspace> <mi>per</mi> <mspace width="0.166667em"></mspace> <mi>year</mi> <mspace width="0.166667em"></mspace> <mfenced separators="" open="(" close=")"> <mi>mas</mi> <mspace width="0.166667em"></mspace> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfenced> </mrow> </semantics> </math> </inline-formula>, depending on the initial orbital phase <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>. Both our analytical calculation and our simultaneous integration of the equations for the parallel transport of the spin 4-vector <i><b>S</b></i> and of the geodesic equations of motion of the gyroscope confirm such a finding. For GP-B, the reported mean error in measuring the spin’s declination rate amounts to <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>σ</mi> <mrow> <mover accent="true"> <mi>δ</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mi>GP</mi> <mo>−</mo> <mi mathvariant="normal">B</mi> </mrow> </msubsup> <mo>=</mo> <mn>18.3</mn> <mspace width="0.166667em"></mspace> <mi>mas</mi> <mspace width="0.166667em"></mspace> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> </inline-formula>. We also calculate the general analytical expressions of the gravitomagnetic spin precession induced by the primary’s angular momentum <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold-italic">J</mi> </semantics> </math> </inline-formula>. In view of their generality, our results can be extended also to other astronomical and astrophysical scenarios of interest like, e.g., stars orbiting galactic supermassive black holes, exoplanets close to their parent stars, tight binaries hosting compact stellar corpses.https://www.mdpi.com/2218-1997/6/6/85general relativity and gravitationexperimental studies of gravityexperimental tests of gravitational theoriessatellite orbitsharmonics of the gravity potential field |
| spellingShingle | Lorenzo Iorio Is There Still Something Left That Gravity Probe B Can Measure? Universe general relativity and gravitation experimental studies of gravity experimental tests of gravitational theories satellite orbits harmonics of the gravity potential field |
| title | Is There Still Something Left That Gravity Probe B Can Measure? |
| title_full | Is There Still Something Left That Gravity Probe B Can Measure? |
| title_fullStr | Is There Still Something Left That Gravity Probe B Can Measure? |
| title_full_unstemmed | Is There Still Something Left That Gravity Probe B Can Measure? |
| title_short | Is There Still Something Left That Gravity Probe B Can Measure? |
| title_sort | is there still something left that gravity probe b can measure |
| topic | general relativity and gravitation experimental studies of gravity experimental tests of gravitational theories satellite orbits harmonics of the gravity potential field |
| url | https://www.mdpi.com/2218-1997/6/6/85 |
| work_keys_str_mv | AT lorenzoiorio istherestillsomethingleftthatgravityprobebcanmeasure |