{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:AWCD45E4AUAR5ZPJQYWYN27R7S","short_pith_number":"pith:AWCD45E4","canonical_record":{"source":{"id":"1608.04574","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.HE","submitted_at":"2016-08-16T12:40:49Z","cross_cats_sorted":[],"title_canon_sha256":"fac726e91852a6a75a98191ccf132664c76f1a5b8e793074d963379caa74458a","abstract_canon_sha256":"f91825b1f4b5c243792e97b090329f3f0110ac06920ae052ddfeb59ed06ee6e8"},"schema_version":"1.0"},"canonical_sha256":"05843e749c05011ee5e9862d86ebf1fcb9da2c63a6e8ed09b99a6a801ff54c2c","source":{"kind":"arxiv","id":"1608.04574","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.04574","created_at":"2026-05-18T01:00:45Z"},{"alias_kind":"arxiv_version","alias_value":"1608.04574v3","created_at":"2026-05-18T01:00:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04574","created_at":"2026-05-18T01:00:45Z"},{"alias_kind":"pith_short_12","alias_value":"AWCD45E4AUAR","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"AWCD45E4AUAR5ZPJ","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"AWCD45E4","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:AWCD45E4AUAR5ZPJQYWYN27R7S","target":"record","payload":{"canonical_record":{"source":{"id":"1608.04574","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.HE","submitted_at":"2016-08-16T12:40:49Z","cross_cats_sorted":[],"title_canon_sha256":"fac726e91852a6a75a98191ccf132664c76f1a5b8e793074d963379caa74458a","abstract_canon_sha256":"f91825b1f4b5c243792e97b090329f3f0110ac06920ae052ddfeb59ed06ee6e8"},"schema_version":"1.0"},"canonical_sha256":"05843e749c05011ee5e9862d86ebf1fcb9da2c63a6e8ed09b99a6a801ff54c2c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:45.067595Z","signature_b64":"Fcx0eTjog3RD6RHgrAZuNha7uP6vjmG6ssMxsBDdTU13HUqHrNSxAyym+YZcBLyB5OEoRg6Y2actBl9z4k6WBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05843e749c05011ee5e9862d86ebf1fcb9da2c63a6e8ed09b99a6a801ff54c2c","last_reissued_at":"2026-05-18T01:00:45.067251Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:45.067251Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.04574","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:00:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W0SjGO+xfXN2ornrSGcM6yux+F+dD/lvgNoyhmxcLe9a0pX1hIs9vYhhoeKFUUw+Xz1ss4n4J02hI5sE9cYxBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:40:11.621532Z"},"content_sha256":"f2b09d0ab5f53187e60022c054b22535e4f1568076cc190b9dd272a5b4e87ff7","schema_version":"1.0","event_id":"sha256:f2b09d0ab5f53187e60022c054b22535e4f1568076cc190b9dd272a5b4e87ff7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:AWCD45E4AUAR5ZPJQYWYN27R7S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximate analytical calculations of photon geodesics in the Schwarzschild metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"astro-ph.HE","authors_text":"Luigi Stella, Maurizio Falanga, Vittorio De Falco","submitted_at":"2016-08-16T12:40:49Z","abstract_excerpt":"We develop a method for deriving approximate analytical formulae to integrate photon geodesics in a Schwarzschild spacetime. Based on this, we derive the approximate equations for light bending and propagation delay that have been introduced empirically. We then derive for the first time an approximate analytical equation for the solid angle. We discuss the accuracy and range of applicability of the new equations and present a few simple applications of them to known astrophysical problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04574","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:00:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mKVeRVry9oFQ5TXTgeo4O7MoweLLXKzwNUeqjR9PZq/B2YgogZXCSQGKMv0S1ToVPrK55Dm/VBG4BD0MIYHVBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:40:11.621881Z"},"content_sha256":"e93d847af4c7e405afa0496f0eaa76729ed01e68973777834b71e5b47bdea5c1","schema_version":"1.0","event_id":"sha256:e93d847af4c7e405afa0496f0eaa76729ed01e68973777834b71e5b47bdea5c1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AWCD45E4AUAR5ZPJQYWYN27R7S/bundle.json","state_url":"https://pith.science/pith/AWCD45E4AUAR5ZPJQYWYN27R7S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AWCD45E4AUAR5ZPJQYWYN27R7S/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T03:40:11Z","links":{"resolver":"https://pith.science/pith/AWCD45E4AUAR5ZPJQYWYN27R7S","bundle":"https://pith.science/pith/AWCD45E4AUAR5ZPJQYWYN27R7S/bundle.json","state":"https://pith.science/pith/AWCD45E4AUAR5ZPJQYWYN27R7S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AWCD45E4AUAR5ZPJQYWYN27R7S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:AWCD45E4AUAR5ZPJQYWYN27R7S","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f91825b1f4b5c243792e97b090329f3f0110ac06920ae052ddfeb59ed06ee6e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.HE","submitted_at":"2016-08-16T12:40:49Z","title_canon_sha256":"fac726e91852a6a75a98191ccf132664c76f1a5b8e793074d963379caa74458a"},"schema_version":"1.0","source":{"id":"1608.04574","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.04574","created_at":"2026-05-18T01:00:45Z"},{"alias_kind":"arxiv_version","alias_value":"1608.04574v3","created_at":"2026-05-18T01:00:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04574","created_at":"2026-05-18T01:00:45Z"},{"alias_kind":"pith_short_12","alias_value":"AWCD45E4AUAR","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"AWCD45E4AUAR5ZPJ","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"AWCD45E4","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:e93d847af4c7e405afa0496f0eaa76729ed01e68973777834b71e5b47bdea5c1","target":"graph","created_at":"2026-05-18T01:00:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We develop a method for deriving approximate analytical formulae to integrate photon geodesics in a Schwarzschild spacetime. Based on this, we derive the approximate equations for light bending and propagation delay that have been introduced empirically. We then derive for the first time an approximate analytical equation for the solid angle. We discuss the accuracy and range of applicability of the new equations and present a few simple applications of them to known astrophysical problems.","authors_text":"Luigi Stella, Maurizio Falanga, Vittorio De Falco","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.HE","submitted_at":"2016-08-16T12:40:49Z","title":"Approximate analytical calculations of photon geodesics in the Schwarzschild metric"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04574","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f2b09d0ab5f53187e60022c054b22535e4f1568076cc190b9dd272a5b4e87ff7","target":"record","created_at":"2026-05-18T01:00:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f91825b1f4b5c243792e97b090329f3f0110ac06920ae052ddfeb59ed06ee6e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.HE","submitted_at":"2016-08-16T12:40:49Z","title_canon_sha256":"fac726e91852a6a75a98191ccf132664c76f1a5b8e793074d963379caa74458a"},"schema_version":"1.0","source":{"id":"1608.04574","kind":"arxiv","version":3}},"canonical_sha256":"05843e749c05011ee5e9862d86ebf1fcb9da2c63a6e8ed09b99a6a801ff54c2c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"05843e749c05011ee5e9862d86ebf1fcb9da2c63a6e8ed09b99a6a801ff54c2c","first_computed_at":"2026-05-18T01:00:45.067251Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:00:45.067251Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fcx0eTjog3RD6RHgrAZuNha7uP6vjmG6ssMxsBDdTU13HUqHrNSxAyym+YZcBLyB5OEoRg6Y2actBl9z4k6WBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:00:45.067595Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.04574","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2b09d0ab5f53187e60022c054b22535e4f1568076cc190b9dd272a5b4e87ff7","sha256:e93d847af4c7e405afa0496f0eaa76729ed01e68973777834b71e5b47bdea5c1"],"state_sha256":"38dcb883732c8dfd2c78770ce18041011ca0ed598833a35826a9a33e1c24e060"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GODSNylqDwNK2R24rB3MV1w3ohxAUgdF8EY07v5dXpwcUQTn/MtwesMCu/+FveT/IQE/6nw/WKb8CN95p9CTDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T03:40:11.624086Z","bundle_sha256":"fc66e80c736441381bfe05ee6536e60acabe9108025f0aeb2321784728bec97c"}}