{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:AA34ILY5P5FJS5LAMJZOT6MFGA","short_pith_number":"pith:AA34ILY5","schema_version":"1.0","canonical_sha256":"0037c42f1d7f4a9975606272e9f985301de9604203f44773012b75632e53c132","source":{"kind":"arxiv","id":"1005.1752","version":2},"attestation_state":"computed","paper":{"title":"A Natural Symmetrization for the Plummer Potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["astro-ph.CO"],"primary_cat":"astro-ph.IM","authors_text":"Junichiro Makino, Takayuki R.Saitoh","submitted_at":"2010-05-11T08:33:57Z","abstract_excerpt":"We propose a symmetrized form of the softened gravitational potential which is a natural extension of the Plummer potential. The gravitational potential at the position of particle i (x_i,y_i,z_i), induced by particle j at (x_j,y_j,z_j), is given by: phi_ij = -G m_j/|r_ij^2+e_i^2+e_j^2|^1/2, where G is the gravitational constant, m_j is the mass of particle j, r_ij = |(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2|^1/2 and e_i and e_j are the gravitational softening lengths of particles i and j, respectively. This form is formally an extension of the Newtonian potential to five dimensions. The derivative"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1005.1752","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.IM","submitted_at":"2010-05-11T08:33:57Z","cross_cats_sorted":["astro-ph.CO"],"title_canon_sha256":"8eb483df96e129e6f3f1bacba73f3a8c5575bd8007976584b9ff1d88035897b8","abstract_canon_sha256":"cb9c1f2f86cf54a3c4c48b5cebb8fab069eea1281fbee10cab1235d9ce9324cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:07:34.445773Z","signature_b64":"W5vJQYgfqrgFzHkU9QmLax/brH3qUmhM6I5PYtKBx+MWFhdRYE6kh5ASfO9ACCIRR9GdT1qMvkn1CtN3zG1cAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0037c42f1d7f4a9975606272e9f985301de9604203f44773012b75632e53c132","last_reissued_at":"2026-05-18T02:07:34.445069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:07:34.445069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Natural Symmetrization for the Plummer Potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["astro-ph.CO"],"primary_cat":"astro-ph.IM","authors_text":"Junichiro Makino, Takayuki R.Saitoh","submitted_at":"2010-05-11T08:33:57Z","abstract_excerpt":"We propose a symmetrized form of the softened gravitational potential which is a natural extension of the Plummer potential. The gravitational potential at the position of particle i (x_i,y_i,z_i), induced by particle j at (x_j,y_j,z_j), is given by: phi_ij = -G m_j/|r_ij^2+e_i^2+e_j^2|^1/2, where G is the gravitational constant, m_j is the mass of particle j, r_ij = |(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2|^1/2 and e_i and e_j are the gravitational softening lengths of particles i and j, respectively. This form is formally an extension of the Newtonian potential to five dimensions. The derivative"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1752","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1005.1752","created_at":"2026-05-18T02:07:34.445174+00:00"},{"alias_kind":"arxiv_version","alias_value":"1005.1752v2","created_at":"2026-05-18T02:07:34.445174+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.1752","created_at":"2026-05-18T02:07:34.445174+00:00"},{"alias_kind":"pith_short_12","alias_value":"AA34ILY5P5FJ","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"AA34ILY5P5FJS5LA","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"AA34ILY5","created_at":"2026-05-18T12:26:05.355336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA","json":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA.json","graph_json":"https://pith.science/api/pith-number/AA34ILY5P5FJS5LAMJZOT6MFGA/graph.json","events_json":"https://pith.science/api/pith-number/AA34ILY5P5FJS5LAMJZOT6MFGA/events.json","paper":"https://pith.science/paper/AA34ILY5"},"agent_actions":{"view_html":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA","download_json":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA.json","view_paper":"https://pith.science/paper/AA34ILY5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1005.1752&json=true","fetch_graph":"https://pith.science/api/pith-number/AA34ILY5P5FJS5LAMJZOT6MFGA/graph.json","fetch_events":"https://pith.science/api/pith-number/AA34ILY5P5FJS5LAMJZOT6MFGA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA/action/storage_attestation","attest_author":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA/action/author_attestation","sign_citation":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA/action/citation_signature","submit_replication":"https://pith.science/pith/AA34ILY5P5FJS5LAMJZOT6MFGA/action/replication_record"}},"created_at":"2026-05-18T02:07:34.445174+00:00","updated_at":"2026-05-18T02:07:34.445174+00:00"}